Fourier and Fourier-Stieltjes algebras on locally compact groups

Mathematical Surveys

and Monographs

Volume 231

Fourier and Fourier-Stieltjes Algebras on Locally Compact Groups

Eberhard Kaniuth Anthony To-Ming Lau

Mathematical Surveys

and Monographs

Volume 231


Eberhard Kaniuth Anthony To-Ming Lau

EDITORIAL COMMITTEE

Robert GuralnickMichael A. Singer, Chair

Benjamin SudakovConstantin Teleman

Michael I. Weinstein

2010 Mathematics Subject Classification. Primary 43-02,43A10, 43A20, 43A30, 43A25, 46-02, 22-02.

For additional information and updates on this book, visitwww.ams.org/bookpages/surv-231

Library of Congress Cataloging-in-Publication Data

Names: Kaniuth, Eberhard, author. | Lau, Anthony To-Ming, author.Title: Fourier and Fourier-Stieltjes algebras on locally compact groups / Eberhard Kaniuth, An-

thony To-Ming Lau.Description: Providence, Rhode Island: American Mathematical Society, [2018] | Series: Mathe-

matical surveys and monographs; volume 231 | Includes bibliographical references and index.Identifiers: LCCN 2017052436 | ISBN 9780821853658 (alk. paper)Subjects: LCSH: Topological groups. | Group algebras. | Fourier analysis. | Stieltjes transform.

| Locally compact groups. | AMS: Abstract harmonic analysis – Research exposition (mono-graphs, survey articles). msc | Abstract harmonic analysis – Abstract harmonic analysis –Measure algebras on groups, semigroups, etc. msc | Abstract harmonic analysis – Abstractharmonic analysis – L1-algebras on groups, semigroups, etc. msc | Abstract harmonic analysis– Abstract harmonic analysis – Fourier and Fourier-Stieltjes transforms on nonabelian groupsand on semigroups, etc. msc | Abstract harmonic analysis – Abstract harmonic analysis –Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups. msc |Functional analysis – Research exposition (monographs, survey articles). msc | Topologicalgroups, Lie groups – Research exposition (monographs, survey articles). msc

Classification: LCC QA387 .K354 2018 | DDC 515/.2433–dc23LC record available at https://lccn.loc.gov/2017052436

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We dedicate this book to our wives, Ulla and Alice, for their lifetime of supportand exceptional patience during the preparation of the manuscript.

Contents

Preface ix

Acknowledgments xi

Chapter 1. Preliminaries 11.1. Banach algebras and Gelfand theory of commutative Banach

algebras 11.2. Locally compact groups and examples 61.3. Haar measure and group algebra 121.4. Unitary representations and positive definite functions 181.5. Abelian locally compact groups 241.6. Representations and positive definite functionals 281.7. Weak containment of representations 301.8. Amenable locally compact groups 33

Chapter 2. Basic Theory of Fourier and Fourier-Stieltjes Algebras 372.1. The Fourier-Stieltjes algebra BpGq 382.2. Functorial properties of BpGq 462.3. The Fourier algebra ApGq, its spectrum and its dual space 502.4. Functorial properties and a description of ApGq 572.5. The support of operators in V NpGq 602.6. The restriction map from ApGq onto ApHq 662.7. Existence of bounded approximate identities 722.8. The subspaces AπpGq of BpGq 782.9. Some examples 832.10. Notes and references 86

Chapter 3. Miscellaneous Further Topics 913.1. Host’s idempotent theorem 913.2. Isometric isomorphisms between Fourier-Stieltjes algebras 963.3. Homomorphisms between Fourier and Fourier-Stieltjes algebras 1013.4. Invariant subalgebras of V NpGq and subgroups of G 1073.5. Invariant subalgebras of ApGq and BpGq 1133.6. Comparison of ApG1q pbApG2q and ApG1 ˆ G2q 1173.7. The w˚-topology and other topologies on BpGq 1213.8. Notes and references 127

Chapter 4. Amenability Properties of ApGq and BpGq 1294.1. ApGq as a completely contractive Banach algebra 1294.2. Operator amenability of ApGq 132

vii

viii CONTENTS

4.3. Operator weak amenability of ApGq 1384.4. The flip map and the antidiagonal 1404.5. Amenability and weak amenability of ApGq and of L1pGq 1444.6. Notes and references 152

Chapter 5. Multiplier Algebras of Fourier Algebras 1535.1. Multipliers of ApGq 1535.2. MpApGqq “ BpGq implies amenability of G: The discrete case 1605.3. MpApGqq “ BpGq implies amenability of G: The nondiscrete case 1675.4. Completely bounded multipliers 1795.5. Uniformly bounded representations and multipliers 1865.6. Multiplier bounded approximate identities in ApGq 1915.7. Examples: Free groups and SLp2,Rq 1955.8. Notes and references 202

Chapter 6. Spectral Synthesis and Ideal Theory 2056.1. Sets of synthesis and Ditkin sets 2066.2. Malliavin’s theorem for ApGq 2106.3. Injection theorems for spectral sets and Ditkin sets 2116.4. A projection theorem for local spectral sets 2146.5. Bounded approximate identities I: Ideals 2206.6. Bounded approximate identities II 2286.7. Notes and references 234

Chapter 7. Extension and Separation Properties of Positive DefiniteFunctions 237

7.1. The extension property: Basic facts 2387.2. Extending from normal subgroups 2427.3. Connected groups and SIN-groups 2467.4. Nilpotent groups and 2-step solvable examples 2507.5. The separation property: Basic facts and examples 2577.6. The separation property: Nilpotent Groups 2647.7. The separation property: Almost connected groups 2687.8. Notes and references 273

Appendix A 277A.1. The closed coset ring 277A.2. Amenability and weak amenability of Banach algebras 280A.3. Operator spaces 282A.4. Operator amenability 284A.5. Operator weak amenability 287

Bibliography 291

Index 303

Preface

Let G be a locally compact group. Let CbpGq be the C˚-algebra of boundedcontinuous complex-valued functions on G with the supremum norm, and let C0pGq

be the closed ˚-subalgebra of CbpGq that consists of functions vanishing at infinity.

IfG is abelian, let pG be the dual group ofG, and let ApGq be all pf (Fourier transform

of f), f P L1p pGq (the group algebra of the dual group pG); and let BpGq be all pμ (the

Fourier-Stieltjes transform of μ), μ P Mp pGq (the measure algebra of pG). Then ApGq

is a subalgebra of C0pGq, and BpGq is a subalgebra of CbpGq. Furthermore, ApGq

(respectively, BpGq) with norm from L1p pGq (respectively, Mp pGq) is a commutativeBanach algebra called the Fourier (respectively, Fourier-Stieltjes) algebra of G.

In Chapter 2, we shall introduce and study some basic properties of Fourier andFourier-Stieltjes algebras, ApGq and BpGq, associated to a locally compact groupG based on the fundamental paper of Eymard [73]. BpGq will be identified asthe Banach space dual of the group C˚-algebra C˚pGq and a fair number of basicfunctorial properties will be presented. Similarly, for the Fourier algebra ApGq, theelements will be shown to be precisely the convolution products of L2-functionson G.

In Chapter 3, we shall study some further topics of ApGq and BpGq. Generaliz-ing the classical description of idempotents in the measure algebra of a locally com-pact abelian group, Host [129] has identified the integer-valued functions in BpGq.Host’s idempotent theorem, which has numerous applications, will be shown in thischapter. A natural question is whether either of the Banach algebras ApGq andBpGq determines G as a topological group. This question has been affirmativelyanswered by Walter [280]. If G1 and G2 are locally compact groups and BpG1q

and BpG2q (respectively, ApG1q and ApG2q) are isometrically isomorphic, then G1

and G2 are topologically isomorphic or anti-isomorphic.Amenable Banach algebras were introduced by B. E. Johnson. He showed the

fundamental result that a locally compact group is amenable if and only if the groupalgebra L1pGq is amenable. We present a proof of the “only if” part of Johnson’sresult in Chapter 4. In particular, if G is abelian, then ApGq, being isometrically

isomorphic to the L1-algebra of the dual group pG, is amenable. However, when Gis nonabelian, then ApGq need not be weakly amenable, even when G is compact.

In Chapter 4, we will also consider the completely bounded cohomology theoryof the Fourier algebra ApGq and of the Fourier-Stieltjes algebra BpGq. We willshow that ApGq, equipped with the operator space structure inherited from beingembedded into V NpGq˚, is a completely contractive Banach algebra. Using this, weestablish in this chapter the fundamental result, due to Ruan [245], that a locallycompact group G is amenable precisely when ApGq is operator amenable.

ix

x PREFACE

An important object associated to any (nonunital) commutative Banach alge-bra A is the multiplier algebra MpAq of A; that is, the algebra of all bounded linearmaps T : A Ñ A satisfying the equation T pabq “ aT pbq for all a, b P A. WhenA is faithful, then the map a Ñ Ta, where Tapbq “ ab for b P A, is a continuousembedding of A into MpAq.

Let G be a locally compact group. Then MpApGqq consists of all boundedcontinuous functions u on G such that uApGq Ď ApGq, and since ApGq is an idealin BpGq, BpGq embeds continuously into MpApGqq. If G is abelian, then as shownby Wendel [288], MpL1pGqq “ MpGq, and hence MpApGqq “ BpGq. It is notdifficult to see that this holds true, more generally, when G is amenable. One of theprofound achievements in abstract harmonic analysis has been that the converseholds; that is, MpApGqq “ BpGq forces G to be amenable. This was shown byNebbia [219] for discrete groups G and by Losert [201] for nondiscrete G. We willpresent these results in Chapter 5.

In Chapter 6, we study spectral synthesis and ideal theory for ApGq. A famoustheorem of Malliavin [207] states that spectral synthesis fails for ApGq whenever Gis any nondiscrete abelian locally compact group. Using this and a deep theoremof Zel1manov [293] ensuring the existence of infinite abelian subgroups of infinitecompact groups, we prove that for an arbitrary locally compact group G, undera mild additional hypothesis, spectral synthesis holds for ApGq if and only if G isdiscrete.

One of the most interesting problems in the ideal theory of a commutativeBanach algebra is to identify the closed ideals with bounded approximate identities.For Fourier algebras this problem is also treated in Chapter 6.

The Hahn-Banach extension theorem asserts that if E is a normed linear spaceand F is a closed linear subspace of E, then each continuous linear functional onF extends to a continuous linear functional on E. From this it follows that givenx P EzF , there exists a continuous linear functional φ on E such that φ “ 0 on Fand φpxq ‰ 0 (the Hahn-Banach separation theorem). In Chapter 7, we addressthe analogous properties for positive definite functions on locally compact groups.

Let G be an arbitrary locally compact group, and let H be a closed subgroup ofG. We show in Chapter 2 that the restriction map u Ñ u|H from ApGq into ApHq issurjective. The corresponding problem for Fourier-Stieltjes algebras is much moredelicate. We say that G has the extension property if for every closed subgroup H,each ϕ P P pHq admits an extension φ P P pGq (equivalently, BpHq “ BpGq|H). Thelargest class of locally compact groups sharing this extension property is formedby the groups with small conjugation invariant neighbourhoods of the identity, theSIN-groups. The converse implication is true for connected Lie groups and forcompactly generated nilpotent groups. More precisely, a connected Lie group hasthe extension property only if it is a direct product of a vector group and a compactgroup. On the other hand, there exists a compactly generated 2-step solvable groupwhich has the extension property, but fails to be a SIN-group.

Acknowledgments

The first author was very pleased to be a Pacific Institute of MathematicalSciences Distinguished Visiting Professor at the University of Alberta at Edmontonbetween 2007 and 2008 when the first thoughts of this work started. Subsequently,the first author’s visit to the University of Alberta was funded by NSERC grant ofAnthony To-Ming Lau, and the second author’s visit to the University of Paderbornwas supported by the funding of Eberhard Kaniuth from the university. The authorsare very grateful to Dr. Liangjin Yao for generously helping in the final preparationof this manuscript. We are very grateful to our friends Garth H. Dales, BrianForrest, Zhiguo Hu, Mehdi Monfared, Ali Ulger, and Matthew Wiersma for verycareful reading of the manuscript with many valuable suggestions. Without theirkind help, this book may not have been completed on time. We are also verygrateful to the American Mathematical Society for accepting our book. We wouldlike to thank Ina Mette for kindly inviting us to submit our book at a CanadianMathematical Society meeting in winter 2009 held at University of Windsor, andfor her patience. We would also like to thank Marcia Almeida and Becky Rivard ofthe American Mathematical Society for their very kind help in the final preparationof the manuscript for publication.

We would like to thank for referees of the first and second reviews of the bookand their many valuable comments.

Eberhard Kaniuth passed away recently in April 2017—an enormous loss tothe mathematical community. I enjoyed very much our over 20 years of researchcollaboration and his warm friendship. I would also like to thank Eberhard for hismany years of hard work in preparation of the book.

xi

CHAPTER 1

Preliminaries

This introductory chapter provides the notational conventions used throughoutthe book and presents all those basic results which are fundamental for developingthe theory of Fourier and Fourier-Stieltjes algebras and the various special topicstreated in this book. We refrain from giving proofs because these can be found inmany existing monographs. Without claiming completeness, we list a number ofsources for the following subjects.

‚ Gelfand theory of commutative Banach algebras: [151], [223]‚ Haar measure and group algebras: [78], [125]‚ Harmonic analysis on locally compact groups: [78], [125], [198], [241], [247],‚ amenability of locally compact groups: [107], [231], [236]‚ representation theory and positive definite functions: [60], [95]‚ weak containment of representations: [60], [74], [158].

1.1. Banach algebras and Gelfand theory ofcommutative Banach algebras

Let A be an algebra A over the complex number field. A is called unital if ithas an identity. An involution on A is a map a Ñ a˚ from A to A which satisfies

pa ` bq˚“ a˚

` b˚, pαaq˚

“ αa˚, pabq˚“ b˚a˚ and pa˚

q˚

“ a

for all a, b P A and α P C. A is then called a ˚-algebra. An element a P A is calledself-adjoint if a˚ “ a, and every a P A can be uniquely decomposed as a “ a1 ` ia2,where a1 and a2 are self-adjoint:

a1 “1

2pa ` a˚

q and a2 “1

2ipa ´ a˚

q.

If A is a ˚-algebra and I is a ˚-ideal in A, that is, a P I implies a˚ P I, then thequotient algebra A{I is a ˚-algebra with the involution pa ` Iq˚ “ a˚ ` I.

The algebra A is called a normed (Banach) algebra if it is endowed with asubmultiplicative norm with respect to which A is a normed (Banach) space. Ais called a normed ˚-algebra (Banach ˚-algebra) if it is a normed (Banach) algebrawith an involution such that }a˚} “ }a} for all a P A. Finally, a Banach ˚-algebrais called a C˚-algebra if the norm satisfies }a˚a} “ }a}2 for all a P A.

Example 1.1.1. (1) For an arbitrary set X, we denote by BpXq the set ofall bounded complex-valued functions on X. With pointwise operations, complexconjugation as involution and the supremum norm

}f}8 “ supt|fpxq| : x P Xu,

BpXq is a unital commutative C˚-algebra.

1

2 1. PRELIMINARIES

(2) For a topological space X, we denote by CpXq the space of all continuousfunctions on X and let CbpXq “ CpXq X BpXq. Moreover, C0pXq (resp. CcpXq)denotes the space all continuous function on X which vanish at infinity (resp. havecompact support). Then CbpXq, equipped with the supremum norm, is a C˚-algebra, C0pXq is a closed subalgebra of CbpXq and CcpXq is a dense subalgebraof C0pXq.

(3) Let H be a Hilbert space and let BpHq denote the set of all bounded linearoperators on H. With T Ñ T˚, where T˚ is the adjoint of T , and the operatornorm, BpHq is a unital C˚-algebra.

Theorem 1.1.2. [Gelfand-Naimark theorem] For every C˚-algebra A, thereexists a Hilbert space H such that A is isometrically ˚-isomorphic to some C˚-subalgebra of BpHq.

Theorem 1.1.3. [Gelfand-Mazur theorem] Let A be a unital Banach algebrain which every nonzero element is invertible. Then A is isomorphic to C.

An approximate identity for a normed algebra A is a net peαqα in A such that}eαa ´ a} Ñ 0 and }aeα ´ a} Ñ 0 for every a P A. The net peαqα is said to bebounded by c ą 0 if }eα} ď c for every α. Left and right approximate identities aredefined similarly.

Let A be a Banach algebra and X a left A-module. Then X is called a leftBanach A-module, if X is a Banach space and the module operation A ˆ X Ñ

X, pa, xq Ñ a ¨ x is a bounded bilinear mapping. The following theorem, which isdue to Cohen when X “ A and to Hewitt in the general case, is usually referred toas the Cohen-Hewitt factorization theorem.

Theorem 1.1.4. Let A be a Banach algebra having a left approximate identitybounded by d ą 0, and let X be a left Banach A-module. Then, given an element xof the closed linear span of A ¨ X and ε ą 0, there exist a P A and y P X with thefollowing properties: x “ a ¨ y, }a} ď d, y P A ¨ x and }x ´ y} ď ε.

Let I be an ideal of a normed algebra A. Then the closure I is an ideal, andif I is closed then the algebra A{I, equipped with the quotient norm }a ` I} “

inft}a ` b} : b P Iu is a normed algebra. If A is complete, then so is A{I.

Proposition 1.1.5. (See [248].) Let I and J be closed ideals of a normedalgebra A.

(i) If A has a bounded approximate identity, then the same is true for A{I.(ii) If both I and A{I have bounded approximate identities, then A has a

bounded approximate identity.(iii) If I and J have bounded approximate identities, then the ideals I XJ and

I ` J have bounded approximate identities. Furthermore, if one of thetwo ideals is closed, then I ` J is closed.

The following proposition is an application of Urysohn’s lemma.

Proposition 1.1.6. Let X be a locally compact Hausdorff space and, for eachsubset E of X let

IpEq “ tf P C0pXq : fpxq “ 0 for all x P Eu.

Then the map E Ñ IpEq is a bijection between the collections of all closed subsetsE of X and all closed ideals of C0pXq. Furthermore, C0pXq{IpEq is unital if andonly if E is compact, and IpEq is maximal if and only if E is a singleton.

1.1. BANACH ALGEBRAS AND GELFAND THEORY 3

Let A and B be Banach algebras, and let A b B denote their algebraic tensorproduct. Then the projective tensor norm γ on A b B is defined by

γpxq “ inf

#

nÿ

j“1

}aj}}bj} : x “

nÿ

j“1

aj b bj

+

,

where the infimum is taken over all such representations of x P AbB. The comple-tion of AbB with respect to γ is the projective tensor product of A and B, and isdenoted A bγB. Every x P A bγB can be represented in the form x “

ř8

j“1 aj bbj ,

whereř8

j“1 }aj}}bj} ă 8, and the norm of x is given by

γpxq “ inf

#

8ÿ

j“1

}aj}}bj} : x “

8ÿ

j“1

aj b bj

+

,

where again the infimum is taken over all such representations of x.

Proposition 1.1.7. (See [61].) Let A and B be Banach algebras. Then themultiplication on AbB extends uniquely to A bγB, turning A bγB into a Banachalgebra. If A and B are ˚-algebras, then A bγ B is a ˚-algebra, and if A and Bare commutative, then so is A bγ B. Moreover A bγ B has a bounded approximateidentity if and only if A and B both have bounded approximate identities.

Let A be a commutative Banach algebra and let σpAq denote the set of allalgebra homomorphisms from A onto C. Every ϕ P σpAq is bounded, actually}ϕ} ď 1 and }ϕ} “ 1 if A is unital. The space σpAq is equipped with the relativew˚-topology of A˚ and is called the spectrum or Gelfand space of A. Thus thistopology on σpAq is the weakest topology for which all the functions ϕ Ñ ϕpaq,a P A, are continuous. The Gelfand space of the projective tensor product of twocommutative Banach algebras turns out to be canonically homeomorphic to theproduct of the two Gelfand spaces.

The algebra A is said to be regular if the algebra of Gelfand transforms is aregular algebra of functions on σpAq in the sense that, given a closed subset E ofσpAq and some ϕ P σpAqzE, there exists a P A such that pa “ 0 on E and papϕq ‰ 0.Moreover, A is called Tauberian if the set of all a P A such that pa has compactsupport, is dense in A.

Proposition 1.1.8. Let A and B be commutative Banach algebras.

(i) Given ϕ P σpAq and ψ P σpBq, there exists a unique element ofσpA bγ Bq, denoted ϕ bγ ψ, such that

pϕ bγ ψqpa b bq “ ϕpaqψpbq

for all a P A and b P B.(ii) When σpAq ˆ σpBq carries the product topology, then the mapping

σpAq ˆ σpBq Ñ σpA bγ Bq, pϕ, ψq Ñ ϕ bγ ψ

is a surjective homeomorphism.(iii) A bγ B is regular if and only if A and B are regular.

For a P A, the continuous function pa : σpAq Ñ C, ϕ Ñ papϕq “ ϕpaq, is calledthe Gelfand transform of a, and the map Γ : A Ñ CpσpAqq, sending a to pa, isusually referred to as the Gelfand homomorphism or Gelfand representation of A.Let }pa}8 denote the sup norm of pa, a P A.

4 1. PRELIMINARIES

Let A be a normed algebra. For a P A, the number

rpaq “ inf t}an}1{n : n P Nu

is the spectral radius of a. Then the spectral radius formularpaq “ limnÑ8 }an}1{n holds. Moreover, if A is a commutative Banach algebra,then

rpaq “ sup t|γpaq| : γ P σpAqu.

Proposition 1.1.9. Let A be a commutative Banach algebra.

(i) σpAq is a locally compact Hausdorff space.(ii) If A is unital, then σpAq is compact.(iii) If σpAq is noncompact, then the w˚-closure of σpAq in A˚ is equal to

σpAq Y t0u.

Proposition 1.1.10. The Gelfand homomorphism has the following properties.

(i) For every a P A, }pa}8 “ limnÑ8 }an}1{n ď }a}.(ii) Γ maps A into C0pσpAqq.(iii) σpAq strongly separates the points of ΓpAq if A is semisimple.(iv) Γ is isometric if and only if }a2} “ }a}2 for all a P A.(v) If A is unital, then an element a P A is invertible if and only if pa never

vanishes on σpAq.

Lemma 1.1.11. Let A be a commutative Banach algebra and I a proper idealin A such that A{I has an identity.

(i) The closure I of I is a proper ideal.(ii) I is contained in a maximal proper ideal, that is, maximal proper ideal I

of A for which A{I is unital.(iii) If I is a maximal proper ideal, then it is closed.

Proposition 1.1.12. For a commutative Banach algebra A, the mapping

ϕ Ñ kerϕ “ ta P A : ϕpaq “ 0u

is a bijection between σpAq and the set of all maximal modular ideals of A.

For a closed ideal I of A, the Gelfand spaces of I and of A{I embed naturallyinto σpAq. More precisely, the following holds.

For any subset M of A, the hull hpMq of M is defined by

hpMq “ tϕ P σpAq : ϕpMq “ t0uu.

Associated with a subset E of σpAq is the closed ideal

kpEq “ ta P A : papϕq “ 0 for all ϕ P Eu

of A. The assignment E Ñ E “ hpkpEqq defines a closure operation on the collec-tion of all subsets E of σpAq. Therefore, there exists a unique topology on σpAq

such that E is the closure of E. This topology is called the hull-kernel topology onσpAq. In general, it is weaker than the Gelfand topology.

Proposition 1.1.13. Let I be a closed ideal of A, q : A Ñ A{I the quotienthomomorphism and hpIq “ tϕ P σpAq : ϕpIq “ t0uu.

(i) The map ϕ Ñ ϕ ˝ q is a homeomorphism from σpA{Iq onto the closedsubset hpIq of σpAq.

1.1. BANACH ALGEBRAS AND GELFAND THEORY 5

(ii) The map ϕ Ñ ϕ|I is a homeomorphism from the open subset σpAqzhpIq

of σpAq onto σpIq.

The commutative Banach algebra A is called semisimple if, for every a P A,pa “ 0 implies a “ 0.

Proposition 1.1.14. Let φ be a homomorphism from a commutative Banachalgebra into a semisimple Banach algebra. Then φ is continuous.

The preceding proposition implies that on a semisimple commutative Banachalgebra any two Banach algebra norms are equivalent.

Proposition 1.1.15. Let A be a commutative Banach algebra. Then A isregular if and only if the hull-kernel topology and the Gelfand topology on σpAq

coincide.

Proposition 1.1.16. Let A be regular commutative Banach algebra and sup-pose that I is an ideal of A and C is a compact subset of σpAq such that C XhpIq “

H. Then there exists an element a P I such that pa “ 1 on C and pa vanishes onsome neighbourhood of hpIq.

Let A be a commutative Banach ˚-algebra. Then A is called symmetric (or

self-adjoint) if xa˚ “ pa for all a P A. In other words, A is symmetric if the Gelfandhomomorphism is a ˚-homomorphism, when C0pσpAqq is endowed with complexconjugation as involution.

Proposition 1.1.17. Suppose that A is a commutative Banach ˚-algebra.

(i) If A is symmetric, then ΓpAq is dense in C0pσpAqq.(ii) If A is a commutative C˚-algebra, then Γ is an isometric isomorphism

onto C0pσpAqq.

Lemma 1.1.18. Let A and B be commutative Banach algebras and φ : A Ñ Ban algebra homomorphism such that γ ˝ φ P σpAq for every γ P σpBq. Then theadjoint map

φ˚ : σpBq Ñ σpAq, γ Ñ γ ˝ φ

is continuous and a homeomorphism provided it is a bijection.

Theorem 1.1.19. Let A be a regular commutative Banach algebra, C a closedsubset of σpAq and a P A such that |papγq| ě δ ą 0 for all γ P C. Then there exists

b P A such that pbpγq “1

papγqfor all γ P C.

A linear operator T : A Ñ A of a commutative Banach algebra A is called amultiplier if T pabq “ aT pbq holds for all a, b P A. When A is without order, thatis, for any a P A the condition aA “ t0u implies a “ 0, then every multiplier isbounded and the setMpAq of all multipliers of A is a unital commutative subalgebraof the algebra BpAq of all bounded linear operators on A, the so-called multiplieralgebra of A. For each T P MpAq, there exists a uniquely determined continuous

function pT on σpAq such that zT paqpγq “ pT pγqpapγq for all γ P σpAq. Readers arereferred for the book [41,151,223] for more details.

6 1. PRELIMINARIES

1.2. Locally compact groups and examples

A topological group G is a set with the structure of both a group and a topo-logical space such that the group product is a continuous map from G ˆ G intoG and the group inverse is continuous on G. The group product of x and y in Gwill be written multiplicatively as xy and the inverse of x is x´1 except in a fewspecific cases such as the group of integers or the real numbers. In general, theidentity element of G is denoted by e. If y P G is fixed, then each of the mapsRy : x Ñ xy, Ly : x Ñ y´1x and x Ñ x´1 is a homeomorphisms of G. When thetopology on G is Hausdorff and locally compact, we call G a locally compact group.

In this section we introduce basic notation and collect a number of standardresults on locally compact groups which will be used throughout the book withoutany further comment. Moreover, we present a number of examples which will bedealt with later.

Let G be any group. For subsets A and B of G, let

AB “ txy : x P A, y P Bu

and A´1 “ tx´1 : x P Au. The set A is called symmetric if A´1 “ A. Also, fork P N, let Ak “ tx1 ¨ ¨ ¨xk : xj P A, 1 ď j ď ku. Instead of txuA and Atyu, wewrite xA and Ay, respectively. We now list a number of elementary properties oftopological groups.

Proposition 1.2.1. Let G be a topological group with identity e. Let A and Bbe subsets of G.

(i) If A Ď G is closed (compact) and B Ď G is compact, then AB is closed(compact).

(ii) Every neighbourhood U of the identity e of G contains an open symmetricneighbourhood V of e such that V 2 Ď U .

(iii) If U Ď G is open, then AU and UA are open for every A Ď G.(iv) Let V be a neighbourhood basis of e in G. Then the closure A of A Ď G

equalsŞ

tAV : V P Vu.(v) Let C be a compact subset of G and U an open set containing C. Then

there exists a neighbourhood V of e such that CV Y V C Ď U .

Proposition 1.2.2. Let G be a topological group.

(i) If H is a subgroup of G, then so is the closure H of H.(ii) Every open subgroup of G is closed.(iii) Let U be a symmetric neighbourhood of e. Then H “

Ť

nPNUn is an open

and closed subgroup of G.

A topological group G is said to be compactly generated if there exists a compactsubset C of G such that

G “ teu YŤ

nPNpC Y C´1qn.

Corollary 1.2.3. Let G be a locally compact group and C a compact subsetof G. Then there exists an open compactly generated subgroup of G containing C.

For a subgroup H of a group G, G{H will always denote the set of all left cosetsxH, x P G, of H and q : G Ñ G{H will denote the quotient map x Ñ xH. If G isa topological group, then G{H is endowed with the quotient topology. Let H be aclosed subgroup of G and let G{H “ txH : x P Gu, the space of left H-cosets. If

1.2. LOCALLY COMPACT GROUPS AND EXAMPLES 7

G is locally compact, then G{H is a locally compact Hausdorff space and q is anopen as well as a continuous mapping.

Proposition 1.2.4. Let H be a closed subgroup of the locally compact group Gand C a compact subset of G{H. Then there exists a compact subset K of G suchthat qpKq “ C.

Proposition 1.2.5. Let H be a subgroup of the topological group G.

(i) The quotient map is continuous and open.(ii) If H is normal, then G{H is a topological group.(iii) If H is closed and normal and G is locally compact, then G{H is a locally

compact group.

We now formulate the topological group versions of the two isomorphism the-orems from algebraic group theory. Let G be a locally compact group, N a closednormal subgroup and H a closed subgroup of G.

Theorem 1.2.6. Suppose that H is also normal in G and N Ď H. Then thealgebraic isomorphism between G{H and pG{Nq{pH{Nq is a homeomorphism.

If H is some topological group with identity element e and φ : G Ñ H isa continuous homomorphism, then N “ tx P G : ψpxq “ eu is a closed normal

subgroup of G. There is a unique injective homomorphism rφ : G{N Ñ H such thatrφ ˝ q “ φ, and rφ is continuous since q is open. Note that rφ need not be open whenG is not σ-compact.

Lemma 1.2.7. Suppose that G and H are locally compact groups and φ : G Ñ H

is a continuous surjective homomorphism. If G is σ-compact, then rφ is a topologicalisomorphism.

The proof of the preceding lemma is a straightforward application of Baire’scategory theorem and it implies the following isomorphism theorem.

Theorem 1.2.8. Suppose that H is σ-compact and HN is closed in G. Thenthe map HN{N Ñ H{pH X Nq, hN Ñ hpH X Nq is a topological isomorphism.

For any topological group G, let G0 denote the connected component of theidentity of G. Then G0 is a closed normal subgroup of G and the quotient groupG{G0 is totally disconnected. G is said to be almost connected if G{G0 is compact.Moreover, if G is totally disconnected, then every neighbourhood of the identitycontains an open compact subgroup. A locally compact group G is called a Liegroup if G0 is open and G0 is an analytic group in the usual sense (see [26]). Inevery locally compact group G, there exists a (unique) largest solvable connectednormal subgroup, the so-called radical radpGq of G.

Let I be an index set and for every ι P I, let Gι be a topological group. Thenthe Cartesian product G “

ś

ιPI Gι become a topological group under the producttopology. Note that G is locally compact if and only if all but finitely many of theGι are compact and the remaining ones are locally compact.

Proposition 1.2.9. An almost connected locally compact group G is a projec-tive limit of Lie groups. More precisely, there exists a directed system of compactnormal subgroups Kα of G such that

Ş

α Kα “ teu and the map

G Ñź

α

G{Kα, x Ñ pxKαqα

is a topological embedding of G into the direct product groupś

α G{Kα.

8 1. PRELIMINARIES

We continue by recalling some further notions from group theory. Let G be alocally compact group. For subsets A and B of G, rA,Bs will always denote theclosed subgroup of G generated by all commutators ra, bs “ aba´1b´1, a P A, b P B.Then rG,Gs is the closed commutator subgroup of G, and the commutator seriesor derived series of G is defined by

G “ G0 Ě G1 “ rG,Gs Ě . . . Ě Gn “ rGn´1, Gn´1s Ě . . . .

G is called solvable if Gn “ teu for some n P N and n-step solvable if Gn “ teu, butGn´1 ‰ teu. In particular, G is 2-step solvable or metabelian if rG,Gs is abelianand ‰ teu. Of course, solvability of G is equivalent to the existence of a sequenceN0 “ teu Ď N1 Ď . . . Ď Nr “ G of normal subgroups of G such that Nj{Nj´1 isabelian for each 1 ď j ď r. By

teu “ Z0pGq Ď Z1pGq Ď . . . Ď ZmpGq Ď . . .

we denote the ascending central series of G. That is, Z1pGq (also denoted ZpGq) isthe centre of G and, for j ě 1, Zj`1pGq is specified byZj`1pGq{ZjpGq “ ZpG{ZjpGqq. Note that every ZmpGq is a closed normal sub-group of G. The group G is said to be n-step nilpotent if ZnpGq “ G, butZn´1pGq ‰ G and G is called nilpotent if it is n-step nilpotent for some n P N.Nilpotent groups can equally well be defined in terms of the descending centralseries G “ C0 Ě C1 Ě . . . Ě Cm Ě . . ., which is inductively defined by C1 “ rG,Gs

and Cm`1 “ rG,Cms form ě 1. ThenG ism-step nilpotent if and only if Cm “ teu,but Cm´1 ‰ teu.

We now present a number of examples, which serve the dual purpose of notonly providing an idea of the nature of locally compact groups and their varietybut also introducing some of the particular groups or classes of groups that will beused later in the text.

Example 1.2.10. The set R of real numbers with addition and the usual topol-ogy has already been mentioned as a locally compact group. The integers Z forma closed subgroup of R. Indeed, for every a P R, Za “ tka : k P Zu is a closedsubgroup of R, and any proper closed subgroup of R is of this form.

Let R˚ “ ta P R : a ‰ 0u and R` “ ta P R : a ą 0u. Equipped withmultiplication of real numbers, both R˚ and R` are locally compact groups with1 as identity and a´1 “ 1{a as the inverse of a generic element a. Note that R` isan open subgroup of R˚.

If C is the field of complex numbers, then the circle group,

T “ tz P C : |z| “ 1u,

is a compact group under multiplication. If a ą 0, then ψaptq “ expp2πit{aq,for t P R, defines a continuous homomorphism of R onto T. Since Za “ kerψa,rψa : t ` Za Ñ expp2πit{aq identifies R{Za with T.

Example 1.2.11. Let GLpn,Rq, n ě 2, denote the multiplicative group of

invertible real n ˆ n-matrices. Equipped with the topology inherited from Rn2

,GLpn,Rq becomes a Lie group. The closed subgroup H consisting of all uppertriangular matrices is n-step solvable.

Example 1.2.12. We can equip R ˆ R` with a different multiplication. Forpb1, a1q, pb2, a2q P R ˆ R` let pb1, a1qpb2, a2q “ pb1 ` a1b2, a1a2q. Notice that


p0, 1qpb, aq “ pb, aqp0, 1q “ pb, aq and

pb, aqp´a´1b, a´1q “ p´a´1b, a´1

qpb, aq “ p0, 1q.

Thus, this multiplication endows RˆR` with a group structure, and the resultinggroup will be denoted G. The operations of multiplication and inversion are clearlycontinuous for the product topology. Thus G is a locally compact group.

For pb, aq P G, define a transformation of R by pb, aq ¨ x “ ax ` b, for allx P R. This is an affine transformation of R, and every orientation-preserving affinetransformation of R arises this way. Moreover, this action is consistent with thegroup product in G. The group G is called the affine group or, often, the ax ` bgroup.

The previous example is a special case of an important technique for construct-ing new locally compact groups from given ones. This is the semidirect productconstruction, which we present in detail because many of our examples dealt withlater in the book arise in this manner. Let N and H be locally compact groups.Let AutpNq denote the group of automorphisms of N . An automorphism of N is atopological group isomorphism of N with itself. Suppose that there is a homomor-phism α : h Ñ αh of H into AutpNq such that pn, hq Ñ αhpnq is continuous fromN ˆH to N . We use these data to form a locally compact group denoted N �α H,or simply N �H if the homomorphism α is understood. As a set and topologicalspace, N �H “ N ˆ H. The group product of pn1, h1q with pn2, h2q in N �H isgiven by

pn1, h1qpn2, h2q “ pn1αh1pn2q, h1h2q.

One checks easily that this product is associative, that peN , eHq serves as the iden-tity, where eN and eH are the identities of N and H, respectively, that pn, hq´1 “

pαh´1pn´1q, h´1q, and that the group operations are continuous. In short, N �His a locally compact group, called the semidirect product of N and H.

If we define rN “ tpn, eHq : n P Nu and rH “ tpeN , hq : h P Hu, then rN and rH

are closed subgroups of N �H that satisfy rN X rH “ teu and rN rH “ N �H, andrN is normal in G. Moreover, for h P H and n P N ,

peH , hqpn, eHqpeH , hq´1

“ pαhpnq, eHq.

In general, if G is a locally compact group, we are sometimes able to find two closedsubgroups N and H such that N is normal in G,N X H “ teu and NH “ G. Insuch case we can view G as the semidirect product of N and H as follows. Forh P H, define αh on N by αhpnq “ hnh´1, for all n P N . Since N is normal, αh isan automorphism of N and h Ñ αh is a homomorphism of H into AutpNq. Alsopn, hq Ñ αhpnq is continuous from N ˆ H to N . Thus, we can form the semidirectproduct N �α H, and the map pn, hq Ñ nh is a topological isomorphism betweenN �α H and G.

We next present two important examples of semidirect product groups.

Example 1.2.13. Let n P N and let SOpnq denote the group of orthogonal nˆn-matrices of determinant one. Equipped with the topology inherited from embedding

SOpnq into Rn2

, SOpnq is a compact group. Form the semidirect product Gn “

Rn � SOpnq, where SOpnq acts on Rn by rotation. Gn is called the Euclidianmotion group of Rn.

10 1. PRELIMINARIES

Example 1.2.14. This example is known as the Heisenberg group of dimension2n ` 1 pn P Nq.

For x, y P Rn, x “ px1, . . . , xnq and y “ py1, . . . , ynq, and z P R, let rx, y, zs

denote the matrix

¨

˚

˚

˚

˚

˚

˝

1 x1 ¨ ¨ ¨ xn z0 1 . . . 0 y1...

......

...0 0 . . . 1 yn0 0 . . . 0 1

˛

‹

‹

‹

‹

‹

‚

.

Let Hn “ trx, y, zs : x, y P Rn, z P Ru. If x ¨ y “řn

i“1 xiyi, the matrix product ofrx, y, zs and rx1, y1, z1s is given by

rx, y, zsrx1, y1, z1s “ rx ` x1, y ` y1, z ` z1

` x ¨ y1s.

Clearly, Hn is a closed subgroup of GLpn ` 2,Rq.It is easy to check that N “ tr0, y, zs : y P Rn, z P Ru is a closed normal

subgroup of Hn, isomorphic to Rn`1, and that A “ trx, 0, 0s : x P Rnu is a closedsubgroup of Hn, isomorphic to Rn. Since Hn “ NA and N X A “ tr0, 0, 0su,Hn is isomorphic to the semidirect product Rn`1 �α Rn, where α is defined byαxpy, zq “ py, z ` x ¨ yq, for py, zq P Rn ˆ R and x P Rn.

Let Z “ tr0, 0, zs : z P Ru, a closed subgroup of Hn which is isomorphic to R. Itis obvious that r0, 0, zs commutes with every element of Hn. Conversely, if rx, y, zs

commutes with every element of Hn, then a quick look at the group product showsthat x “ y “ 0. Thus Z is exactly the centre of Hn. The map ψ : Hn Ñ R2n definedby ψprx, y, zsq “ px, yq, for rx, y, zs P Hn, is a continuous homomorphism onto R2n

with kerψ “ Z. So the quotient group Hn{Z is isomorphic to R2n.There is an important discrete subgroup of H1. Let

D “ trk, l,ms : k, l,m P Zu.

Clearly, D is discrete in the relative topology. It is called the discrete or integerHeisenberg group.

Within the class of totally disconnected locally compact groups the most promi-nent example is the additive group of the p-adic number field which we now brieflyintroduce. Let p be a fixed prime number. Then every nonzero rational numberx has a unique decomposition x “ pmy, where m P Z and y is a rational numberthe numerator and denominator of which are both not divisible by p. The p-adicvaluation of x is defined to be |x|p “ p´m. Setting |0|p “ 0, we then have

|x1x2|p “ |x1|p|x2|p and |x1 ` x2|p ď maxt|x1|p, |x2|pu(1.1)

for x1, x2 P Q. It follows that dpx1, x2q “ |x1 ´ x2|p defines a metric on Q withrespect to which the algebraic operations in Q are continuous. Consequently, theseoperations extend to the completion Ωp of Q, turning Ωp into a field, the p-adicnumber field. The elements of Ωp can be described explicitly as follows (see [78,Proposition 2.8]).

Proposition 1.2.15. Let n P Z and cj P t0, 1, . . . , p ´ 1u for j P Z with j ě n.Then the series

ř8

j“n cjpj converges in Ωp. Conversely, every p-adic number is the

sum of a unique such series.


For x P Ωp and r ą 0, let Bpx, rq denote the closed ball of radius r around x,i.e.

Bpx, rq “ ty P Ωp : |y ´ x|p ď ru.

Then Bpx, rq is also open. In fact, since | ¨ |p only attains the values pn, n P Z,and 0, for any r ą 0 and δ ą 0 sufficiently small, |y ´ x|p ď r is equivalent to|y ´ x|p ă r ` δ. This shows that Ωp is a totally disconnected field. Moreover, itfollows from (1.1) that Bp0, rq is an additive subgroup of Ωp for any r ą 0 and itis a subring for r ď 1. Bp0, 1q is called the ring of p-adic integers and denoted byΔp. Let K “ tx P Ωp : |x|p “ 1u. Then K is a compact multiplicative group. Letn P Z, x P Ωp and ε ą 0 be given and choose k P Z such that k ă n and pk ď ε.It is then easy to verify that the ball Bpx, pnq is the union of pn´k balls of radiuspk. This shows that the balls Bpx, pnq are totally bounded and hence, being closedin the complete metric space Ωp, they are compact. Thus Ωp is a locally compactfield and Δp is a compact open subring.

We close this section by introducing certain function spaces. For a complex-valued function f on G and x P G, the left and right translates through x aredefined by

Lxfpyq “ fpx´1yq and Rxfpyq “ fpyxq, y P G.

Then the maps x Ñ Lx and x Ñ Rx are group homomorphisms. A space E offunctions on G is said to be left (right, two-sided) translation invariant if Lxf P E(Rxf P E, Lxf P E and Rxf P E) for all f P E and x P G. Moreover, we setrfpxq “ fpx´1q and qfpxq “ fpx´1q.

A function f P CbpGq is called left (right) uniformly continuous, if }Lxf ´

f}8 Ñ 0 (}Rxf ´ f}8 Ñ 0) as x Ñ e, and f is said to be uniformly continuous ifit is both left and right uniformly continuous. The sets

LUCpGq “ tf P CbpGq : f is left uniformly continuousu,RUCpGq “ tf P CbpGq : f is right uniformly continuousu,UCpGq “ tf P CbpGq : f is uniformly continuousu

are closed translation invariant subalgebras of CbpGq. Moreover, C0pGq Ď UCpGq.These subspaces of CbpGq are of relevance in the study of amenability of G (Section1.8).

Note that functions in LUCpGq (respectively RUCpGq) are called right (re-spectively left) uniformly continuous functions in [125].

Theorem 1.2.16. [Kakutani-Kodaira] Let G be a σ-compact locally compactgroup, pUnqn a sequence of neighbourhoods of the identity and f a uniformly con-tinuous function on G. Then there exists a compact normal subgroup N of G suchthat N Ď

Ş8

n“1 Un, G{N is second countable and f is constant on cosets of N .

Let G be a locally compact group. A bounded continuous function f on G iscalled almost periodic if the set tLxRyf : x, y P Gu has compact closure in CbpGq.A function f P CbpGq is already almost periodic if either of the sets tLxf : x P Gu

or tRxf : x P Gu has compact closure. The set AP pGq of almost periodic functionson G is a self-adjoint closed subalgebra of CbpGq. The group G is called maximallyalmost periodic if the functions in AP pGq separate the points of G. The algebraAP pGq leads to a compactification of G, the Bohr compactification bpGq, as follows.Let bpGq denote the spectrum of the commutative C˚-algebra AP pGq, and let

12 1. PRELIMINARIES

φ : G Ñ bpGq be the mapping defined by φpxqpfq “ fpxq for f P AP pGq. Then φis continuous and φpGq is dense in bpGq because AP pGq Ď CpbpGqq. The kernel

N “ tx P G : fpxq “ fpeq for all f P AP pGqu

of φ is a closed normal subgroup of G, and φ´1pφpxqq “ xN for every x P G.

Proposition 1.2.17. The group operations on φpGq Ď bpGq, obtained by trans-ferring the ones from G{N , can be extended to all of bpGq, turning bpGq into a com-pact topological group. Note that when G is abelian, bpGq is topologically isomorphicto the dual group of Gd.

Reader are referred to [78,125] for more details.

1.3. Haar measure and group algebra

If X is a locally compact Hausdorff space, a positive Borel measure μ on X iscalled regular if, for any Borel subset E of X,

μpEq “ inftμpUq : E Ď U, U openu

“ suptμpKq : K Ď E, K compactu.

A complex Borel measure ν on X is regular if its total variation |ν| is regular.A Radon measure on X is a positive Borel measure on X such that μpKq ă 8

for any compact set K Ď X,

μpEq “ inftμpUq : E Ď U, U openu,

for any Borel subset E of X, and, for every open subset U of X,

μpUq “ suptμpKq : K Ď U, Kcompactu.

If μ is a σ-finite Radon measure, then μ is regular.Now let X be a locally compact group G. The existence of a translation in-

variant Radon measure on G is of fundamental importance. A Borel measure μ onG is called left invariant (respectively, right invariant) if μpxEq “ μpEq (respec-tively, μpExq “ μpEq), for any x P G and Borel subset E of G. The followingtheorem states the existence and essential uniqueness of a nonzero left invariantRadon measure on G.

Theorem 1.3.1. Let G be a locally compact group. Then there exists a nonzeroleft invariant Radon measure μG on G. It satisfies μGpUq ą 0 for any nonemptyopen subset U of G. If ν is another nonzero left invariant Radon measure on G,then there is a constant c ą 0 such that ν “ c μG.

Such a measure μG is called left Haar measure on G. It is understood that achoice has been made out of the family tcμG : c ą 0u. Usually this choice is notmade explicit, but if there is a distinguished compact neighbourhood V of e, wemay assume that μpV q “ 1. For example, if G is compact itself, we may assumeμGpGq “ 1. If G is an infinite group equipped with the discrete topology, we mayassume that μGpteuq “ 1. Then the left invariance of μG implies that μG is simplythe counting measure (use the left invariance).

Of course there also exists a right invariant Radon measure on G with the samekind of uniqueness, right Haar measure. In fact, the homeomorphism x Ñ x´1

interchanges left and right Haar measures νpEq “ μGpE´1q, for all Borel subsetsE of G, defines a right Haar measure on G). We shall consistently use left Haar

1.3. HAAR MEASURE AND GROUP ALGEBRA 13

measures and always assume without mentioning it that a left Haar measure ischosen. Actually, we will rarely use the notation μG. If A is a measurable subsetof G, then |A| will denote μGpAq.

If f P CcpGq, then f is integrable with respect to μG and we usually writeş

GfpxqdμGpxq simply as

ş

Gfpxqdx. Indeed, we use the same simplification for

any kind of function f on G (non-negative measurable, Haar integrable, or evenvector-valued versions of integrability) for which

ş

Gfpxqdx makes sense. The left

invariance of μG implies thatş

Gfpyxqdx “

ş

Gfpxqdx, for any y P G. Note that if

f P C`c pGq “ tg P CcpGq : g ě 0u and f ‰ 0, then

ş

Gfpxqdx ą 0.

On the other hand, let us fix a y P G and define a new measure ν by νpEq “

μGpEyq, for all Borel subsets E of G. Then ν is a left invariant Radon measure onG that is positive on nonempty open sets. Thus, there is a positive constant ΔGpyq

so that μGpEyq “ νpEq “ ΔGpyqμGpEq, for every Borel subset E of G. This givesa change of variables formula to use in integrals:

ż

G

fpxqdx “ ΔGpyq

ż

G

fpxyqdx,

for any function f where the integral makes sense and for any y P G.Letting y vary, y Ñ ΔGpyq is a continuous homomorphism of G into R`, the

multiplicative group of positive real numbers. It is called the modular function ofG. The modular function enables a change of variables by inversion:

ż

G

fpx´1qdx “

ż

G

fpxqΔGpx´1qdx.

If ΔG ” 1 on G, that is if every left Haar measure is also right invariant, then Gis called unimodular. Of course, if G is abelian, then right and left translations arethe same, and if G has the discrete topology, then counting measure is both left andright invariant. Also, if G is compact, then ΔGpGq is a compact subgroup of R`, soit must be trivial. Thus, each of these classes of groups, abelian, discrete or compact,is contained in the class of all unimodular groups. Nevertheless, one frequentlyencounters nonunimodular groups, and the modular function and functions relatedto it play an important role. For unimodular groups, left Haar measure is also rightinvariant and we usually just refer to Haar measure rather than left Haar measure.

Recall that for 1 ď p ă 8, LppG,μGq is the Banach space of all (equiva-lence classes of) Borel measurable functions f on G for which the norm }f}p “`ş

G|fpxq|dμGpxq

˘1{pis finite, and that L8pG,μGq is the space of all μG-essentially

bounded functions, endowed with the corresponding supremum norm. The Lebesguespace LppG,μGq will simply be denoted LppGq for 1 ď p ď 8. Note that, if1 ď p ă 8, then CcpGq is dense in LppGq and that L2pGq is a Hilbert space, theinner product being given by

xf, gy “

ż

G

fpxqgpxqdx.

For 1 ď p ă 8, left and right translations of Lp-functions are continuous: Givenf P LppGq and ε ą 0, there exists a neighbourhood U of the identity e in G suchthat }Lyf ´ f}p ă ε and }Ryf ´ f}p ă ε for all y P U .

14 1. PRELIMINARIES

Proposition 1.3.2. Let 1 ď p ď 8 and suppose that f P L1pGq and g P LppGq.Then the integral

f ˚ gpxq “

ż

G

fpyqgpy´1xqdy “

ż

G

fpxyqgpy´1qdy

is absolutely convergent for almost all x P G and we have f ˚ g P LppGq and}f ˚ g}p ď }f}1}g}p. Furthermore, suppose that p “ 8. Then f ˚ g is continuous.

The resulting measurable function, f ˚ g, is called the convolution of f and g.When p “ 1, this implies that L1pGq, equipped with convolution as multiplication,is a Banach algebra. For f P L1pGq and x P G, let

f˚pxq “ ΔGpx´1

qfpx´1q.

Then ||f˚||1 “ ||f ||1 and f Ñ f˚ is an involution on L1pGq.

Remark 1.3.3. Suppose that G possesses a compact neighbourhood V of theidentity such that x´1V x “ V for all x P G. Then G is unimodular. Indeed,|V | ą 0 and

|V | “

ż

G

1x´1V xpyqdy “

ż

G

LxpRx1V qpyqdy “

ż

G

Rx1V pyqdy “ Δpx´1q|V |

for every x P G.

Proposition 1.3.4. Let G be a locally compact group. Then L1pGq is a Banach˚-algebra when equipped with the convolution and involution defined above and thenorm: ||f˚||1 “ ||f ||1, for each f P L1pGq.

If G is nondiscrete, L1pGq does not have an identity. However, it always pos-sesses a two-sided (bounded) approximate identity in CcpGq, which can be con-structed as follows. Let U be a neighbourhood basis at e in G, and for each U P Uchoose fU P C`

c pGq such that fpx´1q “ fpxq for all x P G,ş

Gfpxqdx “ 1 and

with compact support contained in U . Then, for any g P LppGq, 1 ď p ă 8,}fU ˚ g ´ g}p Ñ 0 and }g ˚ fU ´ g}p Ñ 0 as U Ñ teu.

Using the existence of an approximate identity, it is straightforward to showthe following

Proposition 1.3.5. Let I be a closed linear subspace of L1pGq. Then I is anideal in L1pGq if and only if I is closed under left and right translations.

If X is a locally compact Hausdorff space, then MpXq denotes the space ofregular complex Borel measures on X equipped with the total variation norm andthe pairing pg, μq “

ş

Xgptq dμptq, g P C0pXq, μ P MpXq, identifies MpXq with

C0pXq˚, the Banach space dual of C0pXq. If X “ G, a locally compact group, thenMpGq can be equipped with a convolution product. For μ, ν P MpGq, there is aunique μ ˚ ν P MpGq such that

ż

G

ϕpxq dpμ ˚ νqpxq “

ż

G

ż

G

ϕpxyq dμpxq dνpyq,

for all ϕ P C0pGq, and we have }μ ˚ ν} ď }μ} ¨ }ν}. If δx denotes the point mass atx P G, then

ż

G

ϕpyqdpμ ˚ δxqpyq “

ż

G

Rxϕpyqdμpyq


and, similarly,ż

G

ϕpyqdpδx ˚ μqpyq “

ż

G

Lx´1ϕpyqdμpyq.

Moreover, for μ P MpGq, define μ˚ P MpGq such that μ˚pEq “ μpE´1q, for anyBorel E Ď G. Then μ Ñ μ˚ is an involution on MpGq and MpGq is a Banach˚-algebra with identity δe, called the measure algebra of G. For each f P L1pGq,there is a measure μf P MpGq such that dμf pxq “ fpxq dx. This embeds L1pGq asa closed two-sided ideal in MpGq. Indeed, if ν P MpGq and f P LppGq, then thefunction

ν ˚ f : y Ñ

ż

G

fpx´1yqdνpxq

belongs to LppGq and }ν ˚ f}p ď }ν} ¨ }f}p. When p “ 1, we haveż

G

ϕpyqdpν ˚ μf qpyq “

ż

G

ϕpyqpν ˚ fqpyqdy,

for all ϕ P CcpGq. Similarly, f ˚ ν is defined and μf ˚ ν “ f ˚ ν, for f P L1pGq andν P MpGq.

Let G be a locally compact group and H a closed subgroup of G. The group Gacts continuously on G{H by G ˆ G{H Ñ G{H, px, yHq Ñ xyH. A measure μ onG{H is called G-invariant if μpxEq “ μpEq for every Borel subset E of G{H andx P G. The existence of a G-invariant measure on G{H is clarified by the following

Proposition 1.3.6. There exists a nonzero positive G´ invariant regular Borelmeasure on G{H if and only if ΔGphq “ ΔHphq for all h P H. When this is thecase, the G´invariant measure is unique up to multiplication by a positive constant.

Proposition 1.3.7. Let H be a closed subgroup of G. For f P CcpGq, defineTHf on G{H by

THfp 9xq “

ż

H

fpxhq dh, 9x “ xH.

Then THf P CcpG{Hq and the map TH : CcpGq Ñ CcpG{Hq is surjective. More-over, given ϕ P CcpG{Hq (resp., ϕ P C`

c pG{Hq), there exists f P CcpGq (resp.,f P C`

c pGq) such that THf “ ϕ and qpsupp fq “ suppϕ.

Remark 1.3.8. Let μ be a (left) Haar measure on G and H an open subgroupof G. Then the restriction of μ to Borel subsets of H is a (left) Haar measure onH. If E is a Borel subset of G and there are countably many distinct left cosetsxjH of H, j P N, such that E Ď

Ť8

j“1 xjE, then μpEq “ř8

j“1 μpE X xjHq.

Let μ be such an invariant measure on G{H. Then Haar measures on G andon H can be normalized in such a way that Weil’s formula

ż

G

fpxqdx “

ż

G{H

ˆż

H

fpxhqdh

˙

dμp 9xq

holds for all f P CcpGq. Actually, the map TH extends to all of L1pGq. Moreprecisely, given f P L1pGq, the integral

ş

Hfpxhqdh exists for μ-almost all 9x P

G{H, the function 9x Ñş

Hfpxhqdh is μ-integrable on G{H and the above Weil’s

formula holds for all f P L1pGq. Finally, }THf}1 ď }f}1, and TH maps L1pGq ontoL1pG{H,μq. Furthermore, TH is a ˚-homomorphism.

16 1. PRELIMINARIES

Example 1.3.9. (1) The Haar measure μR of R with μRpr0, 1rq “ 1 is Lebesguemeasure and Haar measure on T is given by μTpEq “ μRpψ´1

1 pEq X r0, 1qq, for anyBorel subset E of T. That is, for any nonnegative measurable function f on T,

ż

T

fpzqdz “

ż

r0,1q

fpexpp2πitqqdt,

where the integral on the right is Lebesgue integral.(2) It is easiest to describe Haar measure on the multiplicative group R˚ “

Rzt0u by showing the formula for invariant integration. If f is a nonnegative mea-surable function on R˚, then

ş

R˚ fpaqda|a|

satisfies

ż

R˚fpbaq

da

|a|“

ż

R˚fpaq

da

|a|,

for all b P R˚. Therefore,ş8

0fpaq

daa is integration with respect to Haar measure

on R`.(3) Haar measure on the multiplicative group Czt0u is given by

dμpx ` iyq “dxdy

x2 ` y2, x, y P R.

We will often specify left Haar measure on a given locally compact group Gby writing out the expression for integration with respect to the measure in aconvenient parametrization of the elements of G.

Since ΔG|N “ ΔN , a G-invariant measure on the quotient group G{N existsand is nothing but a Haar measure. Given any of the three Haar measures on G,N and G{N , the other two can be normalized so that

ż

G

fpxqdx “

ż

G{N

ˆż

N

fpxnqdn

˙

dpxNq

holds for all f P L1pGq. Furthermore, the map TN : L1pGq Ñ L1pG{Nq is a˚-homomorphism.

Example 1.3.10. Let n P N and G1, G2, . . . , Gn be locally compact groups.Then the Cartesian product, G1 ˆ G2 ˆ . . . ˆ Gn, also denoted Πn

j“1Gj , is a group

when given the coordinatewise operations. With the product topology,nś

j“1

Gj is a

locally compact group called the product group. The left Haar measure onnś

j“1

Gj is

the product of the left Haar measures on the groups Gj . Thus, if f is a nonnegative

measurable function on G “nś

j“1

Gj ,

ż

G

fpxqdx “

ż

G1

ż

G2

. . .

ż

Gn

fpx1, x2, . . . , xnqdxn . . . dx2dx1.

If Gj “ H, a fixed locally compact group, for j “ 1, . . . , n, thennś

j“1

Gj is denoted

Hn. Thus, we have Rn,Zn,Tn and combinations Rk ˆZl ˆTm, for any nonnegativeintegers k, l and m.


Example 1.3.11. We identify the left Haar measure of the ax ` b-group Gby providing a left invariant integration formula. For a nonnegative measurablefunction f on G,

ż

G

fpzqdz “

ż

R

ż

R`fpb, aqa´2da db.

One can check that, for any pb1, a1q P G,ż

R

ż

R

fppb1, a1qpb, aqqa´2da db “

ż

R

ż

R

fpb, aqa´2da db.

On the other hand,ż

R

ż

R`fppb, aqpb1, a1

qqa´2da db “

ż

R

ż

R`fpb ` ab1, aa1

qa´2da db

“ a1

ż

R

ż

R`fpb, aqa´2da db.

Thus, G is nonunimodular and ΔGpb, aq “ a´1.

To find the left Haar integral on a semi-direct product group N �H, one usesthose on N and H together with a factor which records the amount by which αh

scales the left Haar integral of N . More precisely, fix h P H and define a measureμhN on N by μh

N pEq “ μN pαhpEqq, for Borel subsets E of N . Since

μhN pnEq “ μN pαhpnEqq “ μN pαhpnqαhpEqq “ μN pαhpEqq “ μh

N pEq,

μhN is a left Haar measure on N . Thus, there exists δphq ą 0 so that μh

N pEq “

δphqμNpEq. Then δ : H Ñ R` is a continuous homomorphism andż

N

fpxqdx “ δphq

ż

N

fpαhpxqqdx,

for any nonnegative measurable function f on N . Now the left Haar integral of anynonnegative measurable function f on N �H is given by

ż

N�H

fpn, hqdpn, hq “

ż

H

ż

N

fpn, hqδphq´1dndh.

The reader should check the left invariance. We will compute the modular functionΔN�H . Let pm, kq P N �H. Thenż

N�H

fppn, hqpm, kqqdpn, hq “

ż

H

ż

N

fpnαhpmq, hkqδphq´1dndh

“

ż

H

ż

N

fpαhpαh´1pnqmq, hkqδphq´1dndh

“

ż

H

ż

N

fpαhpnmq, hkqdndh

“

ż

H

ż

N

fpαhk´1pnq, hqΔNpm´1qΔHpk´1

qdndh

“

ż

H

ż

N

fpn, hqδpkh´1qΔN pm´1

qΔHpk´1qdndh

“ δpkqpΔN pmqΔHpkqq´1

ż

N�H

fpn, hqdpn, hq.

18 1. PRELIMINARIES

This shows that, for pm, kq P N �H,

ΔN�Hpm, kq “ ΔN pmqΔHpkqδpkq´1.

Using the above formula for the modular function of a semidirect product group,the reader can easily check that the motion group Gn is unimodular (see [125] fordetails).

Example 1.3.12. Easy calculations show that Hn is unimodular with Haarintegral given by

ż

Hn

fprx, y, zsqdrx, y, zs “

ż

Rn

ż

Rn

ż

R

fprx, y, zsq dz dy dx.

Theorem 1.3.13. If G is a locally compact group and T P BpL1pGqq is suchthat T pf ˚ gq “ f ˚ T pgq for all f, g P L1pGq, then there exists μ P MpGq such thatT pfq “ f ˚ μ for all f P L1pGq. Moreover, }T } “ }μ}.

Readers are referred to [78,125,288,289] for more details.

1.4. Unitary representations and positive definite functions

Let H be a Hilbert space and UpHq the group of unitary operators on H. IfG is a locally compact group, a continuous unitary representation of G is a pairpπ,Hpπqq, where Hpπq is a Hilbert space and π is a homomorphism of G intoUpHpπqq such that, for any ξ, η P Hpπq, the function ϕξ,η : x Ñ xπpxqξ, ηy iscontinuous on G. Thus the map π : G Ñ UpHpπqq is required to be continuouswhen UpHpπqq is equipped with the weak operator topology. Because the weak andthe strong operator topologies coincide on UpHpπqq, this is equivalent to continuityof the map x Ñ πpxqξ from G into Hpπq for every ξ P Hpπq. Throughout thisbook, we shall simply use the word representation to mean a continuous unitaryrepresentation of G and write π with the associated Hilbert space Hpπq.

Example 1.4.1. (1) Let χ : G Ñ T be a continuous homomorphism and notethat T can be identified with the unitary group of the 1-dimensional Hilbert spaceC. In particular, the so-called trivial representation of G is given by χpxq “ 1 forall x P G, and every character of an abelian group is a representation.

(2) The left regular representation λG of G on the Hilbert space L2pGq is definedby λGpxqfpyq “ fpx´1yq for f P L2pGq and almost all y P G.

(3) For x P G and f P L2pGq, define ρGpxqf by ρGpxqfpyq “ ΔGpxq1{2fpyxq

for almost all y P G. Sinceż

G

|ρGpxqfpyq|2dy “

ż

G

Δpxq|fpyxq|2dy “

ż

G

|fpyq|2dy,

each ρGpxq is a unitary operator on L2pGq. The representation ρG : x Ñ ρGpxq iscalled the right regular representation of G.

Two representations π and σ of G are said to be (unitarily) equivalent if thereexists a unitary map U : Hpπq Ñ Hpσq such that Uπpxq “ σpxqU for all x P G. Wethen write π – σ and call U an intertwining operator for π and σ.

Example 1.4.2. The left and right regular representations of G are equivalent.In fact, for f P L2pGq, define the function Uf by Ufpyq “ ΔGpy´1q1{2fpy´1q foralmost all y P G. Then the map f Ñ Uf is a unitary map from L2pGq onto

1.4. UNITARY REPRESENTATIONS & POSITIVE DEFINITE FUNCTIONS 19

itself, and it is easily verified that pλGpxqUqpfq “ pUρGpxqqpfq for all x P G andf P L2pGq.

A linear subspace V of Hpπq is called π-invariant if πpxqξ P V for all x P Gwhenever ξ P V . The representation π is called irreducible if Hpπq and t0u are theonly closed π-invariant subspaces of Hpπq.

Given a subset δ Ď Hpπq, let

δ1“

T P B pHpπqq such that TS “ ST for all S P δ(

Proposition 1.4.3. (See [125].) For a representation π of G, the followingare equivalent.

(i) π is irreducible.(ii) πpGq1 “ CI, where I denotes the identity operator on Hpπq.(iii) For ξ, η P Hpπq, ξ ‰ 0 and η ‰ 0 imply that ϕξ,η ‰ 0.

If G is abelian and π is any representation of G, then all the operators πpxq,x P G, commute with each other. It follows from Proposition 1.4.3 that if π isirreducible, then πpxq is a scalar multiple of the identity operator on Hpπq. Thusevery 1-dimensional subspace is π-invariant and so dimHpπq “ 1. Thus the irre-ducible representations of G are in one-to-one correspondence with the charactersof G. The following theorem tells us that, for any locally compact group G, thereare enough irreducible representations to separate the elements of G.

Theorem 1.4.4. [Gelfand-Raikov theorem](See [125].) Let G be a locally com-pact group, and let x, y P G, x ‰ y. Then there exists an irreducible representationπ of G such that πpxq ‰ πpyq.

If V is π-invariant and η P V K, the orthogonal complement of V , then for allξ P V and x P G, xπpxqη, ξy “ xη, πpx´1qξy “ 0, whence πpxqη P V K. Then, if V isclosed in Hpπq, π is the direct sum of the two subrepresentations x Ñ πpxq|V andx Ñ πpxq|V K . More generally, we have the following definition.

Definition 1.4.5. Let I be a nonempty index set. For each ι P I, let Hι be aHilbert space and πι a representation of G withHpπιq “ Hι. For ξ “ pξιqι P ‘ιPIHι,the Hilbert space direct sum of the Hι, let

p‘ιPIπιq pxqξ “ pπιpxqξιqιPI , x P G.

This defines a representation ‘ιPIπι of G, the so-called direct sum of the represen-tation πι. If all the πι are identical, say πι “ σ and m denotes the cardinality of I,then ‘ιPIπι is denoted simply mσ simply.

Proposition 1.4.6. Let π be a representation of G, and let tHι : ι P Iu be afamily of closed π-invariant subspaces of Hpπq such that Hι K Hλ for ι ‰ λ andthe linear span of

Ť

ιPI Hι is dense in Hpπq. Then π is equivalent to the direct sum‘ιPIπι, where πι is defined by πιpxq “ πpxq|Hι

: Hι Ñ Hι.

An element ξ P Hpπq is called a cyclic vector for the representation π if the settπpxqξ : x P Gu is total in Hpπq. If there exists such a cyclic vector for π, then π iscalled a cyclic representation.

Proposition 1.4.7. Every representation π of G is equivalent to a direct sumof cyclic representations.

20 1. PRELIMINARIES

Definition 1.4.8. LetG andH be locally compact groups with representationsπ and ρ. Then the outer tensor product π ˆ ρ of π and ρ is the representation ofG ˆ H acting on the Hilbert space tensor product Hpπq b Hpρq defined by

pπ ˆ ρqpx, yqpξ b ηq “ πpxqξ b ρpyqη,

for x P G, y P H, ξ P Hpπq and η P Hpρq. If π and ρ are both representations ofthe same group G, then the (inner) tensor product π b ρ is the representation of Gon defined on Hpπq b Hpρq defined by

pπ b ρqpxq “ pπ ˆ ρqpx, xq, x P G.

For a representation π of G, the conjugate or contragredient representation π of πis defined on the conjugate Hilbert space Hpπq by πpxq “ πpxq, that is

xπpxqξ, ηy “ xπpxqξ, ηy “ xπpx´1ηq, ξy

for x P G and ξ, η P Hpπq.

Every representation π of G determines a nondegenerate ˚-representation rπ ofL1pGq in the following manner. For f P L1pGq, define the bounded linear operatorrπpfq on Hpπq by

xrπpfqξ, ηy “

ż

G

fpxqxπpxqξ, ηydx, ξ, η P Hpπq.

We then write rπpfq “ş

Gfpxqπpxqdx, an operator-valued integral. It is clear that

rπ is a representation of L1pGq and }rπpfq} ď }f}1 for f P L1pGq. Observe that

πpxqrπpfq “ rπpLxfq and rπpfqπpx´1q “ ΔpxqrπpRxfq

for x P G and f P L1pGq, where Δ is the modular function for G.

Theorem 1.4.9. For each representation π of G, let rπ denote the associated˚-representation of L1pGq. Then the assignment π Ñ rπ has the following properties.

(i) π is irreducible if and only if rπ is irreducible.(ii) π is cyclic if and only if rπ is cyclic.(iii) π1 ” π2 if and only if Ăπ1 – Ăπ2.(iv) For each non-degenerate representation σ of L1pGq, there exists a unique

representation π of G such that σ “ rπ.

Apart from forming tensor products of representations and restricting repre-sentations to subgroups, there is another important procedure of constructing newrepresentations from given ones. We briefly indicate the construction, referring thereader to [158] for details. Let H be a closed subgroup of G, and let π be a unitaryrepresentation of H.

Let FpG, πq denote the space of continuous mappings ξ : G Ñ Hpπq such thatqpsupp ξq is compact in G{H and

ξpxhq “ ΔHphq1{2ΔGphq

´1{2ξpxq

for all x P G and h P H. An inner product can be defined on FpG, πq by

xξ, ηy “

ż

G

ψpxqxξpxq, ηpxqy dx,


where ψ P CcpGq is such that THψ “ 1 on the compact set qpsupp ξq Y qpsupp ηq.

To each x P G, one can associate a unitary operator, denoted indGH πpxq, on thecompletion of FpG, πq defined, for ξ P FpG, πq, by

´

indGH πpxqξ¯

pyq “ ξpx´1yq, y P G.

This representation is called the representation of G induced by π and denotedindGH π . We list a few properties of induced representations, which will be usedlater in the book. A very important result is the following induction in stagestheorem.

Theorem 1.4.10. Let K and H be closed subgroups of the locally compactgroup G such that K Ď H, and let π be a unitary representation of K. Then therepresentations indGK π and indGHpindHK πq are unitarily equivalent.

Theorem 1.4.11. Let H be a closed subgroup of G. Then, for arbitrary repre-sentations ρ of G and π of H,

ρ b indGH π – indGHpρ|H b πq.

It follows from the preceding theorem that if H is a closed subgroup of G, thenindGH λH is equivalent to λG.

Let N be an abelian closed normal subgroup of G. Then G acts on the dual

group pN of N by px, χq Ñ x ¨ χ, where x ¨ χpnq “ χpx´1nxq for χ P pN , n P N andx P G. Then Gχ “ tx P G : x ¨ χ “ χu is a closed subgroup of G, the stabilizer ofχ, and Gpχq “ tx ¨ χ : x P Gu is called the G-orbit of χ.

We are going to describe a procedure for determining the irreducible represen-

tations of G (up to unitary equivalence) in terms of pN , the action of G on pN and

the irreducible representations of the stabilizer groups Gχ, χ P pN . This procedurewas developed by Mackey. It is often referred to as the Mackey machine and it isbased on the so-called imprimitivity theorem, which is far beyond the scope of thisbook. In order for the procedure to work, certain technical conditions have to beplaced on the manner in which N is embedded in G. These conditions are fulfilled

at least when G is second countable and the G-orbits in pN are locally compact(equivalently, open in their closures).

Theorem 1.4.12. Suppose that G is second countable and that Gpχq is locally

compact for every χ P pN . Let π be an irreducible representation of G. Then there

exist χ P pN and an irreducible representation σ of Gχ such that σpnq “ χpnqIHpσq

for all n P N and π is unitarily equivalent to the induced representation indGGχσ.

In the preceding theorem, for a given π, the orbit in pN is uniquely determined,but the choice of χ in this orbit is arbitrary. If γ “ x¨χ P Gpχq, then Gγ “ xGχx

´1,and the assignment σ Ñ x ¨ σ, where x ¨ σpyq “ σpx´1yxq for y P Gγ , defines abijection between the representations of Gχ and those of Gγ , and the induced

representations indGGχσ and indGGγ

px ¨ σq are equivalent.

Theorem 1.4.13. Suppose that G is second countable and that Gpχq is locally

compact for every χ P pN . Let χ P pN and let σ be an irreducible representation ofGχ such that σpnq “ χpnqIHpσq for every n P N . Then indGGχ

σ is irreducible. If

τ is another irreducible representation of Gχ such that τ pnq “ χpnqIHpσq for all

n P N and indGGχσ and indGGχ

τ are equivalent, then σ and τ are equivalent.

22 1. PRELIMINARIES

Now suppose that G is a semidirect product G “ N �H, where N is abelian,

and for χ P pN , let Hχ “ Gχ X N . Then, if σ is an irreducible representations ofHχ, pχˆ σqpn, hq “ χpnqσphq, n P N , h P Hχ, defines an irreducible representationof Gχ with Hpχ ˆ σq “ Hpσq. Conversely, every irreducible representation ρ of Gχ

with ρpnq “ χpnqIHpρq for all n P N is of this form. Moreover, χ ˆ σ is equivalentto χ ˆ τ if and only if σ and τ are equivalent.

Combing Theorems 1.4.12 and 1.4.13, one obtains the following description ofthe irreducible representations of G in terms of the characters χ of N and theirreducible representations of their stability groups Hχ.

Theorem 1.4.14. Let G be a semidirect product G “ N � H, where N is

abelian, and suppose that G is second countable and the H-orbits in pN are locallycompact.

(i) If χ P pN and σ is an irreducible representation of the stability group Hχ,

then the induced representation indGGχpχ ˆ σq of G is irreducible.

(ii) Every irreducible representation of G is equivalent to one of this form.

(iii) If also γ P pN and τ is an irreducible representation of Hγ , then indGGχpχˆ

σq and indGGγpγ ˆ τ q are equivalent if and only if there exists x P H such

that γ “ x ¨ χ and the two representations σ and h Ñ τ px´1hxq of Hχ

are equivalent.

Let H be a closed subgroup of G, π a unitary representation of H and x P G.Then x ¨ π denotes the representation of xHx´1 on Hpπq defined by x ¨ πpyq “

πpx´1yxq. The two induced representations indGxHx´1px ¨ πq and indGH π are equiv-alent.

Definition 1.4.15. A complex-valued function ϕ on a group G is positivedefinite if the inequality

nÿ

j“1

nÿ

k“1

λjλkϕpx´1j xkq ě 0

holds for any finitely many x1, . . . , xn P G and λ1, . . . , λn P C.

Proposition 1.4.16. Let ϕ be a positive definite function on G such thatϕpeq “ 1. Then

(i) |ϕpxq| ď 1 for all x P G.

(ii) ϕpx´1q “ ϕpxq for all x P G.(iii) |ϕpxq ´ ϕpyq|2 ď 2r1 ´ Repϕpx´1yqqs for all x, y P G. In particular,

ϕpxyq “ ϕpxq whenever ϕpyq “ 1.(iv) |ϕpy´1zq ´ ϕpy´1xqϕpx´1zq|2 ď r1 ´ |ϕpx´1yq|2sr1 ´ |ϕpx´1zq|2s for all

x, y, z P G.(v) If y P G is such that |ϕpyq| “ 1, then |ϕpxyq| “ |ϕpxq| for all x P G.

Proposition 1.4.17. (See [78].) Let ϕ be a bounded continuous function onthe locally compact group G. Then the following are equivalent.

(i) ϕ is positive definite.(ii)

ş

Gpf˚ ˚ fqpxqϕpxqdx ě 0 for all f P L1pGq.

(iii) There exist a unitary representation π of G and a vector ξ P Hpπq suchthat ϕpxq “ xπpxqξ, ξy for all x P G.


Definition 1.4.18. Let P pGq denote the set of all continuous positive definitefunctions on G and let P1pGq “ tϕ P P pGq : ϕpeq “ 1u and Pď1pGq “ tϕ P P pGq :ϕpeq ď 1u.

Since }ϕ}8 “ ϕpeq for any ϕ P P pGq, both P1pGq and Pď1pGq are boundedconvex sets. Let expP1pGqq and expPď1pGqq denote the set of all extreme points ofP1pGq and Pď1pGq, respectively.

Lemma 1.4.19. (See [126].) Let ϕ and ψ be positive definite functions on G.Then ϕ, qϕ and ϕψ are also positive definite.

Pď1pGq is a w˚-closed convex subset of the unit ball of L8pGq, hence w˚-compact. Assertion (i) of the following theorem is then a consequence of the Krein-Milman theorem.

Theorem 1.4.20. Let G be a locally compact group.

(i) Pď1pGq is the w˚-closure of the convex hull of expP1pGqq.(ii) expPď1pGqq “ expP1pGqq Y t0u.(iii) The convex hull of expP1pGqq is w˚-dense in P1pGq.

Theorem 1.4.21. The w˚-topology on P1pGq Ď L8pGq coincides with the topol-ogy of uniform convergence on compact subsets of G.

Let H be an open subgroup of G and ϕ P P pHq. Then the trivial extension 9ϕof ϕ to all of G, defined by 9ϕpxq “ ϕpxq for x P H and ϕpxq “ 0 for x P GzH, isalso positive definite.

Corollary 1.4.22. Let ϕ be a positive definite function on G such that ϕpeq “

1, and let H “ tx P G : |ϕpxq| “ 1u. Then H is a subgroup of G and

ϕpxyq “ ϕpyxq “ ϕpxqϕpyq

for all x P H and y P G. In particular, ϕ|H is a character of H.

For ξ, η P Hpπq, the coefficient function ϕξ,η on G is defined by ϕξ,ηpxq “

xπpxqξ, ηy. Then ϕξ,η is a bounded continuous function on G.Note that if χ is a 1-dimensional representation of G, then Hpχq “ C and hence

Hpχq b Hpπq “ Hpπq for any representation π of G. Thus pχ b πqpxq “ χpxqπpxq

and xpχ b πqpxqξ, ηy “ χpxqxπpxqξ, ηy for all x P G and ξ, η P Hpπq.

Proposition 1.4.23. Suppose that π and σ are cyclic representations of G withcyclic vectors ξ and η. If xπpxqξ, ξy “ xσpxqη, ηy for all x P G, then there exists aunitary map U from Hpπq onto Hpσq intertwining π and σ such that Uξ “ η. Inparticular, π and σ are equivalent.

We close this section by mentioning a result which will be used in Chapter 5. Acomplex-valued function ϕ on a group G is called negative definite if ϕpx´1q “ ϕpxq

for all x P G and, for any finitely many x1, . . . , xn P G and c1, . . . , cn P C, n ě 2,such that

řnj“1 cj “ 0,

nÿ

i,j“1

cicjϕpx´1j xiq ď 0.

The remarkable part of the following theorem is the ‘only if’ assertion.

Theorem 1.4.24. [Schoenberg’s theorem] A complex-valued function ϕ on agroup is negative definite if and only if e´αϕ is positive definite for every α ą 0.


24 1. PRELIMINARIES

1.5. Abelian locally compact groups

We shall now specialize to abelian groups where a rich theory generalizingclassical Fourier analysis emerges. As pointed out at the outset of this chapter, weshall only present the notation and basic results that are necessary for our laterchapters.

Let G be a locally compact abelian group. A character of G is a continuous

homomorphism of G into the circle group T. Let pG denote the set of all suchcharacters. Clearly, the pointwise product of two characters is again a character,

and if χ P pG, then χ´1 P pG, where χ´1pxq “ χpxq for x P G. Thus pG is an abelian

group. We endow pG with the topology of uniform convergence on compact subsets

of G. Then a neighbourhood basis of χ0 P pG is formed by the sets

V pχ0, C, εq “ tχ P pG : |χpxq ´ χ0pxq| ă ε for all x P Cu,

where ε ą 0 and C is a compact subset of G. This turns pG into a topologicalgroup, the dual group of G. The following proposition is a major step towardsthe Pontryagin duality theorem (Theorem 1.5.2 below) and the development ofharmonic analysis on locally compact abelian groups.

Proposition 1.5.1. The dual group pG is a locally compact group and it sepa-

rates the points of G, that is, if x, y P G are such that χpxq “ χpyq for all χ P pG,then x “ y.

Elements of a locally compact abelian group may be considered as characterson the dual group. That is, for a locally compact abelian group G and x P G, the

map χ Ñ χpxq is a continuous homomorphism of pG into T.

Theorem 1.5.2. [Pontryagin duality theorem] Let G be a locally compact abelian

group. For each x P G, let αxpχq “ χpxq, for χ P pG. Then the map x Ñ αx is a

topological group isomorphism of G with the dual group of pG.

The Pontryagin duality theorem allows us to derive a number of further im-portant results. The first one is a duality between quotient groups of a locally

compact abelian group G and subgroups of pG. For a closed subgroup H of G, let

HK “ tχ P pG : χ|H ” 1u, the annihilator of H in pG. This is a closed subgroup ofpG.

Lemma 1.5.3. Let q : G Ñ G{H denote the quotient homomorphism. Then the

map ω Ñ ω ˝ q is a topological isomorphism between zG{H and HK.

Identifying G with the dual group of pG, we have pHKqK “ H. Using this, it iseasy to show the following.

Proposition 1.5.4. (See [125].) Let H be a closed subgroup of G and de-

fine a map ψ : pG{HK Ñ pH by ψpχHKq “ χ|H . Then ψ is a topological groupisomorphism. In particular, every character of H extends to a character of G.

The second application of the duality theorem concerns the structure of com-pactly generated abelian groups. Let N0 denote the set of non-negative integers.

Theorem 1.5.5. (See [125].) Let G be a locally compact abelian group.

(i) G contains an open subgroup of the form Rk ˆ K, where k P N0 and Kis a compact group.

1.5. ABELIAN LOCALLY COMPACT GROUPS 25

(ii) If G is compactly generated, then G “ Rk ˆ Zl ˆ K, where k, l P N0 andK is a compact group. In particular, if G is connected, then G is thedirect product of a vector group and a compact connected group.

Finally, Proposition 1.5.4 together with Theorem 1.5.2 yield

Proposition 1.5.6. Let G be a locally compact abelian group. Then

(i) G is discrete if and only if pG is compact.

(ii) G is compact if and only if pG is discrete.

We now present a number of examples.

Example 1.5.7. Let G “ R. For each γ P R define χγ : R Ñ T by χγptq “ eiγt,

for all t P R. Then γ Ñ χγ is a topological group isomorphism of R with pR. Note

that it will often be convenient for us to parametrize pR by scaling by 2π. That is,by considering χγptq “ e2πiγt, t P R, for each γ P R.

Example 1.5.8. Let G “ T. For each n P Z define ψnpzq “ zn, for all z P T.Then n Ñ ψn identifies Z with pT. Let qptq “ e2πit, for t P R. Then q is acontinuous homomorphism of R onto T with kernel Z. This identifies T with R{Z.

For ψn P pT “ yR{Z, note that ψn ˝ qptq “ e2πint, for t P R. This embeds yR{Z as a

closed subset of pR as in Proposition 1.5.4.

Example 1.5.9. (i) Let G “ Z. For each z P T, let σzpnq “ zn, for all n P Z.Then z Ñ σz identifies T with pZ.

(ii) Let n P N, n ą 1 and G “ Z{nZ. By Proposition 1.5.4,

{Z{nZ “ tσz : z P Tnu,

where Tn “ tz P T : zn “ 1u, the group of nth roots of unity.

Many of the frequently occurring abelian locally compact groups are productsof finitely many of the above examples, so the following identification is helpful toquickly determine dual groups for such products.

Example 1.5.10. Let G1, ¨ ¨ ¨ , Gn be abelian locally compact groups and formśn

i“1 Gi andśn

i“1xGi, the product groups. For pχ1, . . . , χnq P

śni“1

xGi and px1, . . . ,xnq P

śni“1 Gi, let

pχ1, . . . , χnqpx1, . . . , xnq “ χ1px1q . . . χnpxnq.

Thus, we can naturally consider pχ1, . . . , χnq as a character ofśn

i“1 Gi. All char-

acters ofśn

i“1 Gi arise this way. Therefore, {

śni“1 Gi “

śni“1

xGi.

Example 1.5.11. Let p be a prime number and consider the additive groupΩp of the p-adic number field. Recall that every p-adic number x has a uniquerepresentation x “

ř8

n“´8cnp

n, where cn P t0, . . . , p ´ 1u and cn ‰ 0 only forfinitely many n P ´N. Moreover x P Δp if and only if cn “ 0 for all n ă 0. We candefine a character γ1 of Ωp by setting

γ1

˜

8ÿ

n“´8

cnpn

¸

“ exp

˜

2πi8ÿ

n“´8

cnpn

¸

.

Continuity of γ1 follows from the fact that kerpγ1q “ Δp and Δp is open in Ωp.For an arbitrary y P Ωp, define γy on Ωp by γypxq “ γ1pyxq, x P Ωp. Then γy is a

26 1. PRELIMINARIES

character of Ωp with kernel equal to tx P Ωp : |x|p ď |y|´1p u. It is less easy to show

that every character of Ωp is of this form. It is then clear that the map y Ñ γy is a

continuous group isomorphism from Ωp onto xΩp. That this map is also open, canbe seen as follows. The sets

Upj, kq “ tx P Ωp : |χpxq ´ 1| ă 1{j for all x with |x|p ď pku,

j P N, k P Z, form a neighbourhood basis of 1Ωpin xΩp. Moreover, χy P Upj, kq if

and only if |y|p ď p´k. Thus the map y Ñ χy is a topological isomorphism betweenΩp and its dual group. For all this, see [78] or [125].

We now identify pG with the spectrum of the commutative Banach algebra

L1pGq. For χ P pG, define γχ : L1pGq Ñ C by

γχpfq “

ż

fpxqχp´xqdx.

It is easily verified that γχpf ˚gq “ γχpfqγχpgq and γχpf˚q “ γχpfq for f, g P L1pGq.

Proposition 1.5.12. The map χ Ñ γχ is a homeomorphism from pG ontoσpL1pGqq, the latter space equipped with the Gelfand topology.

Definition 1.5.13. For f P L1pGq, the Fourier transform of f is the functionpf : pG Ñ C defined by

pfpχq “ γχpfq “

ż

G

fpxqχpxq dx,

for all χ P pG. The Fourier transformation extends to complex Radon measures on

G. If μ P MpGq, then its Fourier-Stieltjes transform pμ is the function on pG definedby

pμpχq “

ż

G

χpxqdμpxq, χ P pG.

Many of the properties of the Fourier transform are summarized in the fol-

lowing statement. We view C0p pGq as a commutative C˚-algebra under pointwiseoperations and equipped with the norm } ¨ }8.

Theorem 1.5.14. Let G be a locally compact abelian group.

i) The map f Ñ pf is an injective ˚-homomorphism of L1pGq into C0p pGq

and the image of L1pGq under the Fourier transform is dense in C0p pGq.

(ii) Given a compact subset C of pG and a neighbourhood U of C in pG, there

exists f P L1pGq such that pf “ 1 on C and pf “ 0 on pGzU . Thus L1pGq

is regular.(iii) Given f P L1pGq and ε ą 0, there exists g P L1pGq such that pg has

compact support and }f ´ g}1 ď ε. Thus L1pGq is Tauberian.

One can show that the Gelfand homomorphism Γ : f Ñ pf from L1pGq into

C0p pGq is surjective only when G is finite.

Theorem 1.5.15. [Plancherel theorem] Let G be a locally compact abelian

group. Then Haar measures on G and pG can be simultaneously chosen so that

} pf}2 “ }f}2, for any f P L1pGq X L2pGq. Moreover, t pf : f P L1pGq X L2pGqu

is dense in L2p pGq and so there is a unitary map F : L2pGq Ñ L2p pGq such that

Fpfq “ pf , for all f P L1pGq X L2pGq.

1.5. ABELIAN LOCALLY COMPACT GROUPS 27

Unless otherwise indicated, we will always assume that Haar measures arescaled so that the Plancherel equality holds. The unitary map F in Theorem 1.5.15is called the Plancherel transform. The Plancherel formula and linearization imply

Corollary 1.5.16. For f, g P L2pGq the Parseval identityż

G

fpxqgpxqdx “

ż

pG

pfpχqpgpχqdχ

holds.

Example 1.5.17. When G “ R, suppose pR is parametrized by tχγ : γ P Ru,where χγptq “ e2πiγt, for t P R. Then we can write

pfpγq “

ż

R

fptqe2πiγtdt,

for γ P R, where we are writing γ in place of χγ . Thenş

pR| pfpγq|2dγ “

ş

R|fptq|2dt,

for all f P L1pRq XL2pRq. Therefore, both R and pR can be equipped with unscaledLebesgue measure and the Plancherel equality holds.

Definition 1.5.18. For ξ P L1p pGq, define the inverse Fourier transform ξ_ of

ξ on G by ξ_pxq “ş

pGξpχqχpxqdχ, for x P G. More generally, for μ P Mp pGq, the in-

verse Fourier-Stieltjes transform μ_ of μ on G is defined by μ_pxq “ş

pGχpxqdμpχq.

The inverse Fourier transform terminology is justified by the following inversionresult.

Theorem 1.5.19. [Bochner’s theorem] Let G be an abelian locally compact

group and ϕ a continuous function on pG. Then ϕ is positive definite if and only ifthere exists a (unique) non-negative μ P MpGq such that

ϕpγq “ pμpγq “

ż

G

γpxqdμpxq

for all γ P pG. Thus the set tpμ : μ P MpGqu coincides with the linear span of P p pGq.

An alternative characterization of the functions in the linear span of P p pGq isgiven by the following theorem.

Theorem 1.5.20. For a continuous function ϕ on the dual group pG, the fol-lowing conditions are equivalent.

(i) ϕ is a linear combination of positive definite functions and }ϕ} ď C.(ii) For every trigonometric polynomial f on G of the form fpxq“

řnj“1 cjγjpxq,

cj P C, γj P pG, we haveˇ

ˇ

ˇ

ˇ

ˇ

nÿ

j“1

cjϕpγjq

ˇ

ˇ

ˇ

ˇ

ˇ

ď C ¨ }f}8.

Theorem 1.5.21. [Inversion theorem](See [247].) Let G be a locally compact

abelian group. Let f P L1pGq XBpGq, where BpGq “ xP pGqy. Then pf P L1p pGq and

fpxq “

ż

pG

pfpχqχpxqdχ,

for all x P G.

28 1. PRELIMINARIES

In view of the Pontryagin duality theorem, Bochner’s theorem can be reformu-lated as follows. A function ϕ on G is a finite linear combination of continuouspositive definite functions if and only if it equals the inverse Fourier-Stieltjes trans-

form of some ν P Mp pGq.Readers are referred to [78,247] for more details.

1.6. Representations and positive definite functionals

Let A be a normed ˚-algebra. A ˚-representation of A is a pair pπ,Hpπqq,where Hpπq is a Hilbert space and π is a homomorphism of A into BpHpπqq suchthat πpa˚q “ πpaq˚ for all a P A. If A is complete, then such a π is automaticallycontinuous. More precisely, }πpaq} ď }a} for all a P A. If A is not assumed tobe complete and π is continuous, then π extends uniquely to a ˚-representation ofthe completion of A and hence }πpaq} ď }a} holds. The representation π is callednondegenerate if the closed subspace

N “ tξ P Hpπq : πpaqξ “ 0 for all a P Au

of Hpπq is trivial. Equivalently, Hpπq is the closed linear span of the set

tπpaqξ : a P A, ξ P Hpπqu.

The definitions of equivalence, subrepresentation, irreducibility, direct sum, cyclic-ity etc. for ˚-representations are the obvious analogues to those for unitary repre-sentations of locally compact groups.

Every unitary representation π of G determines a nondegenerate ˚-representa-tion rπ of L1pGq in the following manner. For f P L1pGq, define the bounded linearoperator rπpfq on Hpπq by

xrπpfqξ, ηy “

ż

G

fpxqxπpxqξ, ηydx, ξ, η P Hpπq.

We then write rπpfq “ş

Gfpxqπpxqdx, an operator-valued integral. It is clear that

rπ is a ˚´representation of L1pGq and }rπpfq} ď }f}1 for f P L1pGq. Observe that

πpxqrπpfq “ rπpLxfq and rπpfqπpx´1q “ ΔpxqrπpRxfq

for x P G and f P L1pGq.If λG is the left regular representation of G, then an elementary calculation

shows that, for f P L1pGq and ξ, η P L2pGq,

xλGpfqξ, ηy “

ż

G

fpxqxλGpxqξ, ηy dx “ xf ˚ ξ, ηy.

Thus λGpfq is just the operator of left convolution of f on L2pGq.A linear functional f on a Banach ˚-algebra A is said to be positive if fpa˚aq ě 0

for every a P A. Then, for a, b P A,

fpb˚aq “ fpa˚bq and |fpb˚aq|2

ď fpa˚aqfpb˚bq.

Moreover, if f is continuous and A has an approximate identity, then

fpa˚q “ fpaq, |fpb˚aq| ď }a}fpb˚bq and |fpaq|

2ď }f}fpa˚aq

for all a, b P A.

1.6. REPRESENTATIONS AND POSITIVE DEFINITE FUNCTIONALS 29

Let π be a unitary representation of a locally compact group G on the Hilbertspace Hpπq and let ξ, η P Hpπq. Then the coefficient function x Ñ ϕξ,ηpxq “

xπpxqξ, ηy is clearly continuous and it is positive definite when η “ ξ since

nÿ

j“1

nÿ

k“1

λjλkϕξ,ξpx´1j xkq “

›

›

›

›

›

nÿ

k“1

λkπpxkqξ

›

›

›

›

›

2

for any x1, . . . , xn P G and λ1, . . . , λn P C. As an example, taking f P L2pGq, thefunction

ϕpxq “ pf ˚ rfqpxq “ xλGpx´1qf, fy,

x P G, is a continuous positive definite function. Moreover, given a neighbourhoodU of e in G, choosing a symmetric neighbourhood V of e such that V 2 Ď U andtaking f “ |V |´1{21V , the resulting function ϕ satisfies ϕpeq “ 1 and suppϕ Ď U .

Theorem 1.6.1. Let φ be a positive linear functional on L1pGq. Then thereexists a unique ϕ P P pGq such that

φpfq “

ż

G

fpxqϕpxqdx

for all f P L1pGq.

Proposition 1.6.2. If ϕ P P1pGq, then ϕ P expP1pGqq if and only if the GNS-representation πϕ associated with ϕ is irreducible.

We now indicate a fundamental method for constructing cyclic representationsof G. This construction is usually referred to as the GNS-construction, named afterGelfand, Naimark and Segal.

A function of positive type on G is a function ϕ P L8pGq which defines apositive linear functional on L1pGq, that is, which satisfies

ż

G

pf˚˚ fqpxqϕpxqdx ě 0

for every f P L1pGq. An easy calculation shows that ϕ P L8pGq is of positive typeif and only if

ż

G

ż

G

fpxqfpyqϕpy´1xq dxdy ě 0

for all f P L1pGq. Thus such a function ϕ defines a positive semidefinite sesquilinearform on L1pGq by

xf, gyϕ “

ż

G

ż

G

fpxqgpyqϕpy´1xq dxdy,

which satisfies |xf, gyϕ| ď }ϕ}8}f}1}g}1. Let Nϕ “ tf P L1pGq : xf, fyϕ “ 0u. Itfollows from the Cauchy-Schwarz inequality that xf, fyϕ “ 0 if and only if xf, gyϕ “

0 for all g P L1pGq. Thus Nϕ is a linear subspace of L1pGq, and x¨, ¨yϕ defines aninner product on the quotient space L1pGq{Nϕ. Let Hϕ denote the Hilbert spacecompletion of L1pGq{Nϕ. It is easily verified that xLxf, Lxgyϕ “ xf, gyϕ for allx P G and f, g P L1pGq. In particular, LxpNϕq Ď Nϕ, and hence for each x P G wecan define a unitary operator πϕpxq on Hϕ by

πϕpxqpf ` Nϕq “ Lx´1f ` Nϕ, f P L1pGq.

It is easily verified that πϕ is a unitary representation of G, the so-called GNS-representation associated with ϕ.

30 1. PRELIMINARIES

Theorem 1.6.3. Let ϕ be a function of positive type on G and let πϕ be theunitary representation constructed above. Then there exists a cyclic vector ξϕ P Hϕ

such that

xπϕpxqξϕ, ξϕy “ ϕpxq

for locally almost all x P G.

Corollary 1.6.4. If π is a cyclic representation of G with cyclic vector ξ andif ϕpxq “ xπpxqξ, ξy for all x P G, then π – πϕ.

Remark 1.6.5. (1) Since the function x Ñ xπϕpxqξϕ, ξϕy is continuous, The-orem 1.6.3 in particular shows that every function of positive type agrees locallyalmost everywhere with a continuous function.

(2) The ˚-representation Ăπϕ of L1pGq associated with πϕ is given by

Ăπϕpfqrgs “ rf ˚ gs, g P L1pGq,

where rhs “ h ` Nϕ for h P L1pGq.

Readers are referred to [78,126] for more details.

1.7. Weak containment of representations

Let A be a C˚-algebra. We start by introducing the notion of weak containmentfor representations of A. For such a representation π of A, the ideal

kerπ “ ta P A : πpaq “ 0u

is called the kernel of π. Note that a representation of C˚-algebra is always assumeto be a ˚-representation.

Let pA denote the set of equivalence classes of irreducible representations of A.If π and σ are equivalent representations, then kerpπq “ kerpσq. Therefore, one can

abuse notation safely and write kerpπq for π P pA and consider the canonical map

k : pA Ñ PrimpAq, π Ñ kerpπq.

Definition 1.7.1. The set pA equipped with the pull-back of the hull-kernel

topology on PrimpAq is called the dual space of A. Thus a subset U of pA is open

in pA if and only if U “ k´1pV q for some open subset V of PrimpAq. This topology

on pA is called the dual space topology or Fell topology.

Definition 1.7.2. Let S and T be two sets of representations of A. Then S isweakly contained in T (S ă T ) if

Ş

τPT ker τ ĎŞ

σPS kerσ. If S is a singleton, saytπu, we simply write π ă T . We say that S and T are weakly equivalent (S „ T )if S ă T and T ă S. For an arbitrary representation π of A, the support of π is

defined to be the set of all σ P pA such that σ ă π.

If π is a representation of A on Hpπq, then for every ξ P Hpπq, the functionϕ defined by ϕpaq “ xπpaqξ, ξy, is a positive linear functional on A associatedwith π. We also say that a positive linear functional ϕ is associated with a setS of representations of A if ϕ is associated with some π P S. It is important tocharacterize weak containment in terms of positive functionals.

Theorem 1.7.3. Let S Ď pA and π P pA. Then the following conditions areequivalent.

(i) π is weakly contained in S.

1.7. WEAK CONTAINMENT OF REPRESENTATIONS 31

(ii) Every nonzero positive functional associated with π is the w˚-limit offinite linear combinations of positive functionals associated with S.

(iii) Every nonzero positive functional ϕ associated with π is the w˚-limitof finite sums ψ of positive functionals associated with S which satisfy}ψ} ď }ϕ}.

Suppose that π and all the representations in S are irreducible. Then Theorem1.7.3 can be sharpened as follows.

Theorem 1.7.4. Let S Ď pA and π P pA. Then the following conditions areequivalent.

(i) π is weakly contained in S.(ii) Some nonzero positive functional associated with π is the w˚-limit of

positive functionals associated with S.(iii) Every nonzero positive functional ϕ associated with π is the w˚-limit of

positive functionals ψ associated with S such that }ψ} ď }ϕ}.

Definition 1.7.5. Let A be a C˚-algebra. An ideal I of A is called a primitiveideal if I “ kerpπq for some irreducible representation π of A, and the collectionPrimpAq of all primitive ideals of A is called the primitive ideal space of A. PrimpAq

is endowed with the hull-kernel or Jacobson topology, which can most easily bedefined by describing the closure operation. For any subset I of PrimpAq, let

I “ tP P PrimpAq : P Ě

č

IPIIu.

Then the assignment I Ñ I is a closure operation, and hence there is a uniquetopology on PrimpAq such that I equals the closure of I.

Let G be a locally compact group. Given any positive linear functional φon C˚pGq, there is a unique ϕ P P pGq such that φpfq “

ş

Gfpxqϕpxqdx for all

f P L1pGq. Conversely, if ϕ P P pGq, then the functional on L1pGq given by

f Ñ

ż

G

fpxqϕpxqdx

extends uniquely to a positive linear functional on C˚pGq. Since there is a bijectionbetween (equivalence classes of) unitary representations of a locally compact groupG and (equivalence classes of) nondegenerate ˚-representations of the group C˚-algebra C˚pGq, the notion of weak containment can simply be transferred from theC˚-algebra context to the group context.

Since the w˚-topology on P1pGq coincides with the topology of uniform con-vergence on compact sets, weak containment can be expressed as follows.

Proposition 1.7.6. Let S be a set of representations of G and π a representa-tion of G. Then π ă S if and only if, for any positive definite function ϕ associatedwith π, there exists a net pψαqα, where each ψα is a linear combination of posi-tive definite functions associated with S, such that ψα Ñ ϕ uniformly on compactsubsets of G.

Theorem 1.7.4 can now be reformulated as follows

Corollary 1.7.7. For S Ď pG and π P pG, the following are equivalent.

(i) π is weakly contained in S.

32 1. PRELIMINARIES

(ii) At least one nonzero function of positive type associated with π is theuniform limit on compacta of a net of functions of positive type associatedwith S.

(iii) Every nonzero function of positive type associated with π is the uniformlimit on compacta of a net of functions of positive type associated withS.

Definition 1.7.8. Let G be a locally compact group and let pG denote the setof unitary equivalence classes of irreducible representations of G. The dual space

topology or Fell topology on pG is the topology which makes the bijection pG Ñ {C˚pGq

a homeomorphism. Endowed with this topology, pG is called the dual space of G.

The set of all π P pG such that π ă λG is called the reduced dual of G and denotedpGr. This is a closed subset of pG. The support of a representation π of G is the set

of all ρ P pG which are weakly contained in π.

Weak containment of group representations is inherited by restricting represen-tations to subgroups and by forming tensor products. More precisely, we have thefollowing.

(1) If S and T are sets of unitary representations of G such that S ă T andH is a closed subgroup of G, then

S|H “ tσ|H : σ P Su ă T |H “ tτ |H : τ P T u.

(2) If S ă T and S1 ă T 1, then

S b S1“ tσ b σ1 : σ P S, σ1

P S1u ă T b T 1

“ tτ b τ 1u.

For f P L1pGq, let }f}˚ “ supt}πpfq} : π P pGu. Since we always have }σpfq} ď

}f}1 for any representation σ, }f}˚ ď }f}1. Moreover,

}f˚˚ f}˚ “ supt}πpfq}

2 : π P pGu “ }f}2˚

as well as}f ˚ g}˚ ď }f}˚}g}˚ and }f˚

}˚ “ }f}˚

for f, g P L1pGq. Thus } ¨ }˚ defines a C˚-norm on L1pGq. The completion ofpL1pGq, } ¨ }˚q is called the group C˚-algebra of G and denoted C˚pGq.

The norm } ¨ }˚ on L1pGq is designed so that every representation π of L1pGq

extends uniquely to a representation, also denoted π, of C˚pGq. The conceptsof irreducibility and equivalence of representations mean the same whether one isconsidering the representations as being of G or of C˚pGq. For a representation πof G, πpC˚pGqq is a norm closed ˚-subalgebra of BpHpπqq. Indeed,

πpC˚pGqq “ πpL1pGqq “ spanpπpGqq,

where the closure is in the operator norm.Another norm on the Banach ˚-algebra L1pGq satisfying the C˚-condition is

given by f Ñ }λGpfq}, where λG denotes the left regular representation of L1pGq

on L2pGq. The completion of L1pGq with respect to this norm is called the reducedgroup C˚-algebra.

Let N be a closed normal subgroup of G, q : G Ñ G{N the quotient homomor-phism and TN : C˚pGq Ñ C˚pG{Nq the canonical homomorphism of C˚-algebras.Then the map π Ñ π ˝ q is a bijection between the unitary representations of G{Nand the unitary representations σ of G such that σpNq “ tIHσ

u. Equivalently,σ “ π ˝ q if and only if kerσ Ě kerTN .

1.8. AMENABLE LOCALLY COMPACT GROUPS 33

Proposition 1.7.9. The map π Ñ π ˝ q gives a homeomorphism from zG{N

and the closed subset of all σ P pG which annihilate N .

Note that if 1G ă λG, then π ă λG for every representation π of G and

hence pGr “ pG. In fact, since weak containment is preserved under forming tensorproducts,

π – π b 1G ă π b λG “ π b indGteu 1teu

“ indGteupπ|teu b 1teuq „ indGteu “ λG.

We shall see in Section 1.8 that pGr equals pG if and only if G is amenable.Then, for any representation π of G{N , ξ, η P Hpπq and f P L1pGq,

xπ ˝ qpfqξ, ηy “

ż

G

fpxqxπ ˝ qpxqξ, ηydx

“

ż

G{N

ˆż

N

fpxnqxπpxNqξ, ηydn

˙

dpxNq

“

ż

G{N

TNfpxNqxπpxNqξ, ηydpxNq

“ xπpTN pfqξ, ηy.

This implies that }TNf}C˚pG{Nq ď }f}C˚pGq and hence TN : L1pGq Ñ L1pG{Nq

extends to a ˚-homomorphism, also denoted TN , from C˚pGq onto C˚pG{Nq.Readers are referred to [60,74,158] for more details.

1.8. Amenable locally compact groups

Amenability of a locally compact group can also be characterized through theexistence of left invariant means on certain, much smaller, subspaces of L8pGq

which have been introduced in Section 1.2.

Definition 1.8.1. Let E be a linear subspace of L8pGq containing the constantfunctions. A mean on E is an element of E˚ satisfying xm, 1y “ }m} “ 1. Supposethat E is left (right) translation invariant. Then a mean on E is called left invariant(right invariant) if mpLxfq “ mpfq (mpRxfq “ mpfq) for all f P E and x P G.

Definition 1.8.2. A locally compact group is amenable if there exists a leftinvariant mean on L8pGq.

Proposition 1.8.3. For a locally compact group G, the following are equiva-lent.

(i) G is amenable.(ii) There exists a left invariant mean on CbpGq.(iii) There exists a left invariant mean on LUCpGq.(iv) There exists a left invariant mean on RUCpGq.(v) There exists a left invariant mean on UCpGq.

Remark 1.8.4. Let Gd denote a locally compact group G with the discretetopology. If Gd is amenable, then so is G.

Proposition 1.8.5. For a locally compact group G, the following are equiva-lent.

(i) G is amenable.

34 1. PRELIMINARIES

(ii) There exists a right invariant mean on L8pGq.(iii) There exists a two-sided invariant mean on L8pGq.

Proposition 1.8.6. Amenability of a locally compact group G is equivalent tothe so-called Folner’s condition: Given any compact subset C of G and ε ą 0, thereexists a compact subset U of G such that C Ď U and

|pxUzUq Y pUzxUq| ď ε|U |, for all x P C.

Theorem 1.8.7 (Day’s fixed point theorem). For a locally compact group G,the following are equivalent.

(i) G is amenable.(ii) If G acts affinely on a non-empty compact, convex subset K of a separated

locally convex vector space E, i.e., the map G ˆ K Ñ K, pg, xq Ñ g ¨ xsatisfies

g ¨ ptx ` p1 ´ tqyq “ tpg ¨ xq ` p1 ´ tqg ¨ y, x, y P E, t P r0, 1s,

and such that pg, xq Ñ g ¨ x is separately continuous, then there existssome x P E such that g ¨ x “ x for all g P G.

The Markov-Kakutani fixed point theorem asserts that if G is commutative,then G has the fixed point properties (ii). In particular, G is amenable.

Theorem 1.8.8. Let G be a locally compact group and N a closed normalsubgroup. If N and G{N are amenable, then so is G.

Theorem 1.8.9. Let G be an amenable locally compact group. Then everyclosed subgroup of G is amenable.

Proposition 1.8.10. Let G and H be locally compact groups and ϕ : G Ñ H acontinuous homomorphism with dense range. If G is amenable, then H is amenableas well. In particular, every quotient group of an amenable group is amenable.

Proposition 1.8.11. Let pGαqα be an upwards directed family of closed sub-groups of G such that

Ť

α Gα is dense in G and each Gα is amenable. Then G isamenable.

Definition 1.8.12. Let pgαqα be a net in L1pGq such that gα ě 0 and }gα}1 “ 1for each α. Then this net is strongly convergent to left invariance if }Lxpgαq´gα}1 Ñ

0 for each x P G.

Example 1.8.13. (1) Let G be a compact group and let μ denote nor-malized Haar measure on G. Then xm, fy “

ş

Gfpxqdμpxq, f P L8pGq Ď

L1pGq, defines an invariant mean on L8pGq.(2) The free group F2 on two generators a and b is not amenable. Towards a

contradiction, assume that there exists a left invariant mean on �8pF2q.Write every element of F2 as a reduced word in ta, b, a´1, b´1u, and forx P ta, b, a´1, b´1u, let Ex denote the set of all elements of F2 beginningwith x. Then

1 “ mp1Gq “ mpδeq ` mp1Eaq ` mp1Eb

q ` mp1Ea´1 q ` mp1Eb´1 q.

On the other hand, since F2 is the disjoint union of the sets Ea and a´1Ea

as well as Eb and b´1Eb,

1 “ mp1Eaq ` mp1a´1Ea

q “ mp1Ebq ` mp1b´1Eb

q.

Obviously, left invariance of m now leads to a contradiction.

1.8. AMENABLE LOCALLY COMPACT GROUPS 35

(3) By (1), the orthogonal groups SOpNq are all amenable. However, forN ě 3, SOpNq contains a subgroup isomorphic to F2. Thus SOpNqd isnot amenable. This shows that the converse of Remark 1.8.4 does nothold.

It follows from Theorem 1.8.9 that any locally compact group, which containsF2 as a closed subgroup, cannot be amenable. This implies that all the groupsSLpn,Rq, SLpn,Cq, GLpn,Rq and GLpn,Cq are not amenable. If G is a locally com-pact group, then radical of G (denoted by radpGq) is the largest solvable connectednormal subgroup in G (see [107, A52]). More generally, the following theoremholds:

Theorem 1.8.14. Let G be an almost connected locally compact group. ThenG is amenable if and only if G modulo its radical, the maximal connected solvablenormal subgroup of G, is compact.

Proposition 1.8.15. A locally compact group G is amenable if and only ifthere exists a net pgαqα in L1pGq such that gα ě 0, }gα}1 “ 1 for each α and pgαqα

is strongly convergent to left invariance.

Since abelian groups are amenable, it follows from Theorem 1.8.8 that solvablelocally compact groups are amenable. In particular, every nilpotent locally compactgroup is amenable. A group G is locally finite if every finite subset of G generatesa finite subgroup. Every locally finite group is amenable.

Suppose that G contains an abelian closed normal subgroup N such that G{Nis compact. Then G is amenable (Theorem 1.8.8). For example, the classicalEuclidean motion groups Rn � SOpnq are amenable.

Proposition 1.8.16. Suppose that E is a subspace of L8pGq which containsthe constant functions and is closed under complex conjugation. Let m be a linearfunctional on E such that xm, 1y “ 1. Then m is a mean on E if and only if m ispositive, i.e. xf,my ě 0 for every f P E, f ě 0.

Theorem 1.8.17. For a locally compact group, the following are equivalent.

(i) G is amenable.(ii) 1G ă π b π for every unitary representation π of G.(iii) 1G ă π b π for every irreducible unitary representation π of G.

A locally compact group G is said to satisfy Reiter’s condition (P1) if given acompact subset K of G and ε ą 0, there exists u P L1pGq with u ě 0, }u}1 “ 1 and}Lxu ´ u}1 ď ε for all x P K. We shall need in the next chapter that amenablegroups share Reiter’s condition (P1). Actually, the two properties are equivalent(see [59,158,231] for details).

Theorem 1.8.18. Let G be a locally compact group. The following are equiva-lent:

(i) G is amenable.(ii) G satisfies Reiter’s condition pP1q.(iii) The trivial representation of G is weakly contained in λG.

Proposition 1.8.19. Let G be an amenable locally compact group and H aclosed subgroup of G. Then, for any unitary representation π of G,

π ă indGHpπ|Hq.

36 1. PRELIMINARIES

Proposition 1.8.20. Suppose that }λGpfq} “ }f}1 for all nonnegative f P

L1pGq. Then the trivial representation of G is weakly contained in λG.

Remark 1.8.21. Let C˚λG

pGq or C˚λ pGq denote the C˚ subalgebra of BpL2pGqq

generated by tλGpfq : f P L1pGqu, and V NpGq denote the von Neumann algebragenerated by tλGpfq : f P L1pGqu in BpL2pGqq. When G is abelian, then

C˚λ pGq “ C0p pGq and V NpGq “ L8

p pGq “ L1p pGq

˚.


CHAPTER 2

Basic Theory of Fourier andFourier-Stieltjes Algebras

In this chapter the Fourier and Fourier-Stieltjes algebras, ApGq and BpGq,associated to a locally compact group G, are introduced and studied almost tothe extent of Eymard’s fundamental paper [73]. In particular, BpGq is identifiedas the Banach space dual of the group C˚-algebra C˚pGq and a fair number ofbasic functorial properties are presented. Similarly, for the Fourier algebra ApGq,the elements of which are shown to be precisely the convolution products of L2-functions on G.

Given a commutative Banach algebra A, immediate problems arising are todetermine the spectrum (or Gelfand space) of A and to check whether the range ofthe Gelfand transform is a regular function algebra. As we show in Section 2.3, thespectrum σpApGqq turns out to be homeomorphic to G and the Gelfand homomor-phism is then nothing but the identity mapping. Moreover, ApGq is regular.

We next identify, following Eymard’s seminal paper [73], the Banach spacedual of ApGq as the von Neumann subalgebra V NpGq of BpL2pGqq generated bythe left regular representation of G. The fact that ApGq is the predual of a vonNeumann algebra will prove to be of great importance. For instance, it allows usto equip ApGq with a natural operator space structure and employing the theoryof operator spaces has led to significant progress, as will be shown in Chapters 4and 6.

In Section 2.5 the very important notion of support of an operator in V NpGq

is introduced and several properties of these supports, which are extremely usefullater on, are shown. An immediate consequence of one of the results about thesupport is that singletons in G are sets of synthesis for ApGq.

Let H be a closed subgroup of the locally compact group G. A challengingproblem is whether functions in ApHq and BpHq extend to functions in ApGq andBpGq, respectively. For the Fourier algebras there is a very satisfactory solution tothe effect that every function in ApHq extends to a function in ApGq of the samenorm (Section 2.6). For Fourier-Stieltjes algebras, however, the problem is consid-erably more difficult and its investigation will cover a major portion of Chapter 7.

If A is a nonunital Banach algebra, then often the existence of a boundedapproximate identity in A proves useful. In Section 2.7 we present Leptin’s the-orem [191] saying that ApGq has a bounded approximate identity precisely whenthe group G is amenable. The proof uses several different characterizations ofamenability of a locally compact group.

The notion of Fourier algebra has been generalized by Arsac [5]. He associatedto any unitary representation π of G a closed subspace AπpGq of BpGq and studiedthese spaces extensively. When π is the left regular representation of G, then

37

38 2. BASIC THEORY OF FOURIER AND FOURIER-STIELTJES ALGEBRAS

AπpGq equals ApGq. We present in Section 2.8 those results from [5] which eitherwill be needed in Chapter 3 or used in Section 2.9 to show that for certain examplesof locally compact groups the Fourier-Stieltjes algebra BpGq decomposes into thedirect sum of ApGq and BpG{Nq for a large normal subgroup N of G.

2.1. The Fourier-Stieltjes algebra BpGq

Let G be a locally compact group. In this section we introduce the Fourier-Stieltjes algebra BpGq and its subspaces BSpGq, where S is a collection of (equiva-lence classes of) unitary representations of G, and prove a number of basic resultson these spaces.

Let Σ denote the equivalence classes of continuous unitary representations ofG. For S Ď Σ and μ P MpGq, let }μ}S “ supt}πpμq} : π P Su. Then the assignmentμ Ñ }μ}S is a semi-norm on MpGq, and for μ, ν P MpGq, f P L1pGq and x, y P G,we have

(1) }μ}S ď }μ} and }μ ˚ ν}S ď }μ}S}ν}S(2) }μ˚}S “ }μ}S and }μ ˚ μ˚}S “ }μ˚}2S(3) }Lxf}S “ }f}S and }Ryf}S “ Δpy´1q}f}S .

Let NS “ tf P L1pGq : πpfq “ 0 for all π P Su. Then NS is a closed ˚-ideal

of L1pGq, and if 9f “ f ` NS P L1pGq{NS , then } 9f}S “ }f}S defines a norm onL1pGq{NS . It is clear that L1pGq{NS becomes a normed ˚-algebra, and the norm

satisfies } 9f ˚ 9f˚}S “ } 9f}2S . Let C˚SpGq denote the completion of L1pGq{NS . The

group G acts on L1pGq{NS , and we have

}Lx9f}S “ } 9f}S “ Δpyq}Ry

9f}S

for all x, y P G. These actions extend uniquely to the C˚-algebra C˚SpGq.

Lemma 2.1.1. The mapping f Ñ 9f “ f ` kpSq from L1pGq onto pL1pGq `

kpSqq{kpSq extends to a homomorphism from C˚pGq onto C˚SpGq with kernel kpSq,

where kpSq denotes for all g P S such that πpgq “ 0.

Proof. Since } 9f}S “ }f}S ď }f}C˚ for f P L1pGq, the map f Ñ 9f ex-tends uniquely to a ˚-homomorphism φ : C˚pGq Ñ C˚

SpGq, and }φ} ď 1. SinceφpC˚pGqq Ě φpL1pGqq and a homomorphism between C˚-algebras with dense rangeis surjective, φpC˚pGqq “ C˚

SpGq. It remains to show that kerφ “ kpSq.Let g P C˚pGq such that φpgq “ 0. Then there exist fn P L1pGq, n P N, such

that

supπPS

}πpgq ´ πpfnq} ď }g ´ fn}C˚ Ñ 0

and } 9fn}S “ }fn}S Ñ 0. This implies that supπPS }πpgq} “ 0, i.e. g P kpSq.Conversely, let g P kpSq and let fn P L1pGq such that }g ´ fn}C˚ Ñ 0. ThenlimnÑ8 }fn}S “ 0, and hence φpgq “ limnÑ8 φpfnq “ 0. �

Let G be a locally compact abelian group with dual group pG and let S Ď pG.

Then, for f P L1pGq, f P kpSq if and only if pf vanishes on S Ď pG. Thus NS “ t0u

if and only if S is dense in pG, and in this case }f}S “ } pf}8, and the Fourier

transform is an isometric isomorphism between C˚SpGq “ C˚pGq and C0p pGq. If S

is not dense in pG, then L1pGq{NS is isometrically isomorphic to the subalgebra

of C0pSq consisting of all pf |S , f P L1pGq, equipped with the uniform norm, andC˚

SpGq identifies with C0pSq.

2.1. THE FOURIER-STIELTJES ALGEBRA BpGq 39

Lemma 2.1.2. For S Ď Σ and u P P pGq, the following are equivalent.

(i) πu ă S.(ii) There exists a positive linear functional ϕ on C˚

SpGq such that, for eachf P L1pGq,

ϕpf ` kpSqq“

ż

G

fpxqupxqdx.

Proof. Let upxq “ xπupxqξ, ξy, where ξ is a cyclic vector for πu. Then πu ă Sis equivalent to xπupgqξ, ξy “ 0 for all g P kpSq. In fact, the sufficiency of this lattercondition is immediate from the facts that kpSq is a two-sided ideal of C˚pGq andthat ξ is a cyclic vector. Now, by Lemma 2.1.1, the positive linear functionals onC˚

SpGq are exactly the positive linear functionals on C˚pGq which are zero on kpSq.It follows that (i) and (ii) are equivalent. �

From now on, PSpGq will denote the set of all u P P pGq which satisfy any ofthe equivalent conditions in Lemma 2.1.2.

Lemma 2.1.3. For collections S and T of representations of G, the followingconditions are equivalent.

(i) S ă T .(ii) For every μ P MpGq, }μ}S ď }μ}T .(iii) For every f P L1pGq, }f}S ď }f}T .

Proof. First, assume (iii) and let u P PSpGq. Then, for any f P L1pGq,ˇ

ˇ

ˇ

ˇ

ż

G

fpxqupxqdx

ˇ

ˇ

ˇ

ˇ

ď }f}Supeq ď }f}T upeq.

Thus u defines a positive linear functional on L1Gq{NT , which extends to a positivelinear functional on C˚

T pGq “ C˚pGq{kpT q. This implies that πu ă T . Since thisholds for every u P PSpGq, (i) follows.

Now suppose that (i) holds. Since kpT q Ď kpSq, we get

}f}S “ inft}f ` g}C˚pGq : g P kpSqu ď inft}f ` g}C˚pGq : g P kpT qu “ }f}T

for every f P L1pGq. From this inequality we are going to deduce }μ}S ď }μ}T forμ P MpGq. Let V be a neighbourhood basis of the identity, and for each V P V ,choose a nonnegative continuous function gV with supp gV Ď V , }gV }1 “ 1 and letfV “ μ ˚ gV . Then the bounded net pfV qV in L1pGq converges to μ in the weaktopology σpMpGq, CbpGqq of G. Let π be an arbitrary unitary representation andξ, η P Hpπq. Then

xπpμqξ, ηy “

ż

G

xπpxqξ, ηydμpxq “ limV

ż

G

xπpxqξ, ηyfV pxqdx

“ limV

xπpfV qξ, ηy,

and hence the net of operators πpfV q converges to πpμq in the weak operator topol-ogy of BpHpπqq of G. Since }πpfV q} ď }πpμq} ¨ }πpgV q} ď }πpμq} and the ball ofradius }πpμq} in BpHpπqq is weakly closed and the weak topology agrees with theultraweak topology, it follows that }πpμq} “ supV PV }πpfV q}. Since S ă T , for


π P S we get

}πpμq} “ supV PV

}πpfV q} ď supV PV

}πpfV q}S

ď supV PV

}πpfV q}T “ supσPT

ˆ

supV PV

}σpfV q}

˙

“ }μ}T .

Since π P S was arbitrary, we conclude that }μ}S ď }μ}T . This completes theproof. �

Lemma 2.1.4. For a function u on G and a collection S of unitary representa-tions of G the following assertions are equivalent.

(i) u is a finite linear combination of functions in PSpGq.(ii) There exist a unitary representation π of G, which is weakly contained in

the S, and vectors ξ, η P Hpπq such that upxq “ xπpxqξ, ηy for all x P G.(iii) u is a bounded continuous function and

supfPL1pGq,}f}Sď1

ˇ

ˇ

ˇ

ˇ

ż

G

fpxqupxqdx

ˇ

ˇ

ˇ

ˇ

ă 8.

Proof. (i) ñ (ii) Since every u P PSpGq can be represented as in (ii), itsuffices to observe that the functions in (ii) form a linear space. So let π1 and π2 berepresentations that are both weakly contained in S and let ξj , ηj P Hpπjq, j “ 1, 2.Then π1 ‘ π2 is weakly contained in S and

xpπ1 ‘ π2qpξ1 ‘ ξ2q, η1 ‘ η2y “

2ÿ

j“1

xπjpxqξj, ηjy.

(ii) ñ (iii) Since π ă S, for each f P L1pGq,ˇ

ˇ

ˇ

ˇ

ż

G

fpxqupxqdx

ˇ

ˇ

ˇ

ˇ

“ |xπpfqξ, ηy| ď }πpfq} ¨ }ξ} ¨ }η} ď }f}S}ξ} ¨ }η}.

(iii) ñ (i) Condition (iii) implies that u defines a bounded linear functional onC˚

SpGq, which then can be written as a linear combination of positive functionalson C˚

SpGq. By Lemma 2.1.2(iii), each of the latter functionals is given by a functionin PSpGq. �

Definition 2.1.5. Let BSpGq denote the set of functions satisfying the equiv-alent conditions of Lemma 2.1.4. Of course BSpGq is translation invariant. Wesimply write BpGq for BΣpGqpGq. Thus BpGq consists of all finite linear combina-tions of continuous positive definite functions and hence equals the collection of allcoefficient functions xπp¨qξ, ηy, where π P ΣpGq and ξ, η P Hpπq.

It follows from Lemma 2.1.4 that BpGq identifies with the Banach space dualof C˚pGq through the pairing

(2.1) xf, uy “

ż

G

fpxqupxq dx, f P L1pGq, u P BpGq.

The norm on BpGq is then given by

(2.2) }u} “ sup

"ˇ

ˇ

ˇ

ˇ

ż

G

fpxqupxqdx

ˇ

ˇ

ˇ

ˇ

: f P L1pGq, }f}C˚ ď 1

*

.

Note that if up¨q “ xπp¨qξ, ηy, then xg, uy “ xπpgqξ, ηy for all g P C˚pGq.


Remark 2.1.6. (1) For any S Ď ΣpGq, C˚SpGq is isometrically isomorphic to a

quotient of C˚pGq and hence BSpGq “ C˚SpGq˚ is a closed subspace of BpGq. Thus

the norms of u, considered as an element of C˚SpGq and of C˚pGq, respectively, are

equal. In particular, if u P BλpGq, then

}u} “ sup

"ˇ

ˇ

ˇ

ˇ

ż

G

fpxqupxqdx

ˇ

ˇ

ˇ

ˇ

: f P CcpGq, }λGpfq} ď 1

*

.

(2) If u P BSpGq, then u admits Jordan decompositions

(2.3) u “ u`´ u´, u`, u´

P P pGq, }u} “ u`peq ` u´

peq

as an element of BpGq, and

u “ u`S ´ u´

S , u`S , u

´S P PSpGq, }u} “ u`

S peq ` u´S peq

as an element of BSpGq. It follows from the uniqueness of the decomposition (2.3)that u`

S “ u` and u´S “ u´. Similarly, for u P BSpGq “ C˚

SpGq˚, the absolutevalue |u| does not depend on S. In fact, this follows from the uniqueness of thepolar decomposition. This in particular shows that if u P BSpGq, then |u| P PSpGq

and if u “ ru, then also u` and u´ belong to PSpGq.(3) Let ω denote the universal representation of G (see [270, page 122]). Then

C˚pGq˚˚ is the von Neumann subalgebra of BpHpωqq generated by either the op-erators ωpfq, f P L1pGq, or the operators ωpxq, x P G, since the sets ωpL1pGqq

and ωpGq have the same commutant in BpHpωqq. Let μ P MpGq and let V be aneighbourhood basis of e. For each V P V , let gV and fV “ μ ˚ gV be as in theproof of Lemma 2.1.3. Then ωpμq is the ultraweak limit of the net pωpfV qqV andhence ωpμq P C˚pGq˚˚. Since BpGq “ C˚pGq˚ and

xωpfV q, uy “

ż

G

fV pxqupxqdx,

for all V , passing to the limit, by duality of BpGq and C˚pGq˚˚, we conclude that

(2.4) xωpμq, uy “

ż

G

upxqdμpxq,

and in particular

(2.5) xωpxq, uy “ upxq

for u P BpGq and x P G.(4) Let u P BpGq and let π be a representation of G and ξ, η P Hpπq such that

upxq “ xπpxqξ, ηy for all x P G. Then, for any g P C˚pGq,

xωpgqξ, ηy “ xπpgqξ, ηy “ xπ2pωpgqqξ, ηy.

Again, passing to the ultraweak limit, we get

(2.6) xT, uy “ xπ2pT qξ, ηy

for all T P C˚pGq˚˚.

Lemma 2.1.7. Let u “ ru P BpGq. Then the functions u`, u´ and |u| areuniform limits on G of finite linear combinations of right translates of u.

Proof. Let u “ V |u| be the polar decomposition of u. Then, since u` “12 p|u| ` uq and u´ “

12 p|u| ´ uq, it suffices to prove the assertion for |u|. Let A

denote the ˚-subalgebra of C˚pGq˚˚ generated by the operators ωpxq, x P G. SinceV ˚ is a partial isometry, V ˚ is contained in the unit ball of C˚pGq˚˚. Then, by


Kaplansky’s density theorem, V ˚ is the limit in the strong operator topology of anet pSαqα in A such that }Sα} ď 1 for all α. Each of the functions

x Ñ Sαupxq “ xωpxq, Sαuy “ xωpxqSα, uy

is a linear combination of right translates of u. Now, using that u “ ru, formula(2.5) and the estimate |xT˚, uy|2 ď }u}xT˚T, |u|y for every T P C˚pGq˚˚, we obtainfor x P G

||u|pxq ´ Sαupxq|2

“ |xωpxq, |u|y ´ xωpxqSα, uy|2

“ |xωpxq, V ˚uy ´ xωpxqSα, uy|2

“ |xωpxqpV ˚´ Sαq, uy|

2

“ |xpV ˚´ Sαq

˚ωpx´1q, uy|

2

ď }u}xpV ˚´ Sαq

˚ωpx´1qωpxqpV ˚

´ Sαq, |u|y

“ }u}xpV ˚´ Sαq

˚pV ˚

´ Sαq, |u|y.

Now Sα Ñ V ˚ in the strong operator topology, and therefore the bounded netrpV ˚ ´ Sαq˚pV ˚ ´ Sαqsα in C˚pGq˚˚ converges in the weak operator topology andhence ultraweakly to zero. Since the weak operator topology on C˚pGq˚˚ coincideswith the topology σpC˚pGq˚˚, BpGqq on bounded sets, the above estimate impliesthat the net pSαuqα converges uniformly on G to |u|. �

The next two lemmas provide additional expressions for the norms of elementsin BSpGq.

Lemma 2.1.8. Let S Ď ΣpGq and u P BSpGq. Then

}u} “ sup

$

&

%

ˇ

ˇ

ˇ

ˇ

ˇ

nÿ

j“1

cjupxjq

ˇ

ˇ

ˇ

ˇ

ˇ

: xi P G, ci P C, 1 ď i ď n, n P N,

›

›

›

›

›

nÿ

j“1

cjδxj

›

›

›

›

›

S

ď 1

,

.

-

.

Proof. Suppose first that S “ ΣpGq and let A denote the C˚-subalgebra ofC˚pGq˚˚ generated by ωpGq. Then, by Kaplansky’s density theorem, the unit ballof A is ultraweakly dense in the unit ball of C˚pGq˚˚. By (2.5) this implies

}u} “ sup t|xT, uy| : T P C˚pGq

˚˚, }T } ď 1u

“ sup t|xS, uy| : S P A, }S} ď 1u

“ sup

#ˇ

ˇ

ˇ

ˇ

ˇ

nÿ

j“1

cjupxjq

ˇ

ˇ

ˇ

ˇ

ˇ

+

,

where the supremum extends over all x1, . . . , xn P G, c1, . . . , cn P C such that›

›

›

řnj“1 cjδj

›

›

›

Sď 1.

Now let S be arbitrary and consider Gd, the group G made discrete. SincePSpGq Ď PSpGdq, u P BSpGdq and Remark 2.1.6(1), applied to Gd, shows that

sup

$

&

%

ˇ

ˇ

ˇ

ˇ

ˇ

nÿ

j“1

cjupxjq

ˇ

ˇ

ˇ

ˇ

ˇ

:

›

›

›

›

›

nÿ

j“1

cjδj

›

›

›

›

›

S

ď 1

,

.

-

“ sup

$

&

%

ˇ

ˇ

ˇ

ˇ

ˇ

nÿ

j“1

cjupxjq

ˇ

ˇ

ˇ

ˇ

ˇ

:

›

›

›

›

›

nÿ

j“1

cjδj

›

›

›

›

›

ΣpGq

ď 1

,

.

-

.

The statement of the lemma now follows from the first part of the proof. �


Lemma 2.1.9. Let u P BpGq and upxq “ xπpxqξ, ηy, x P G. Then }u} ď

}ξ} ¨ }η}. Conversely, if u P BSpGq then there exist a representation π which isweakly contained in S and ξ, η P Hpπq such that

upxq “ xπpxqξ, ηy, x P G, and }u} “ }ξ} ¨ }η}.

More precisely, if u “ V |u|, V P C˚pGq˚˚, denotes the polar decomposition of u, itsuffices to take π and η such that |u|pxq “ xπpxqη, ηy, where η is a cyclic vector inHpπq, and then put ξ “ π2pV qη.

Proof. The first statement follows from

}u} “ sup

"ˇ

ˇ

ˇ

ˇ

ż

G

fpxqupxqdx

ˇ

ˇ

ˇ

ˇ


*

“ sup

|xπpfqξ, ηy| : f P L1pGq, }f}C˚ ď 1

(

ď }ξ} ¨ }η}.

Now choose π, η and ξ as announced. Then, for every x P G,

upxq “ xωpxq, uy “ xωpxq, V |u|y “ xωpxqV, |u|y

“ xπ2pωpxqV qη, ηy “ xπpxqπ2

pV qη, ηy

“ xπpxqξ, ηy.

Since |u| P PSpGq, π is weakly contained in S (compare Lemma 2.1.2). So it onlyremains to show that }u} ě }ξ} ¨ }η}. Now, since V is a partial isometry, }V } “ 1and hence }π2pV q} ď 1 and }ξ} “ }π2pV qη} ď }η}. Consequently,

}u} “ }|u|} “ |u|peq “ }η}2

ě }ξ} ¨ }η},

as required. �

Remark 2.1.10. (1) For each u P BpGq, we have }u}8 ď }u}. In fact, since}f}C˚ ď }f}1 for every f P L1pGq, (2.2) implies

}u} “ sup

"ˇ

ˇ

ˇ

ˇ

ż

G

fpxqupxqdx

ˇ

ˇ

ˇ

ˇ


*

ě sup

"ˇ

ˇ

ˇ

ˇ

ż

G

fpxqupxqdx

ˇ

ˇ

ˇ

ˇ

: f P L1pGq, }f}1 ď 1

*

“ }u}8.

(2) Let μ P MpGq. Then, for any S Ďř

pGq,

}μ}S “ sup

"ˇ

ˇ

ˇ

ˇ

ż

G

upxqdμpxq

ˇ

ˇ

ˇ

ˇ

: u P BSpGq, }u} ď 1

*

.

To see this, let φ denote the continuous linear functional on BSpGq defined byxφ, uy “

ş

Gupxqdμpxq and let upxq “ xπpxqξ, ηy, where π is weakly contained in S

and ξ, η P Hpπq satisfy }u} “ }ξ} ¨ }η} (Lemma 2.1.9). Then

|xφ, uy| “ |xπpμqξ, ηy| ď }πpμq} ¨ }ξ} ¨ }η} ď }μ}S}u}

and hence }μ}S ě }φ}. Conversely, consider any π P S and ξ, η P Hpπq with }ξ} ď 1and }η} ď 1 and let vpxq “ xπpxqξ, ηy. Then v P BSpGq and

|xπpμqξ, ηy| “

ˇ

ˇ

ˇ

ˇ

ż

G

vpxqdμpxq

ˇ

ˇ

ˇ

ˇ

ď }φ} ¨ }v} ď }φ} ¨ }ξ} ¨ } η} ď }φ}.

Since }μ}S is the supremum of all such values |xπpμqξ, ηy|, it follows that }μ}S ď }φ}.


(3) Let u P BpGq, (respectively, u P BλpGq). Then the functions ru, u and u allbelong to BpGq (respectively, BλpGq) and

}u} “ }ru} “ }u} “ }u}.

The first statement follows from the fact that it holds for P pGq (respectively,PλpGq). For the equality of the norms, let upxq “ xπpxqξ, ηy with }u} “ }ξ} ¨ }η}.Then

rupxq “ xπpxqη, ξy and upxq “ xη, πpxqξy

and therefore }ru} ď }η} ¨ }ξ} “ }u} and }u} ď }u}. Since pruq„ “ u and puq´ “ u,the reverse norm inequalities follow. Finally, as u “ puq„, all four norms have tobe equal.

Theorem 2.1.11. Let G be a locally compact group. Then BpGq, equipped withpointwise multiplication and the norm

}u} “ sup

"ˇ

ˇ

ˇ

ˇ

ż

G

fpxqupxqdx

ˇ

ˇ

ˇ

ˇ


*

,

is a unital commutative Banach algebra, called the Fourier-Stieltjes algebra of G,and BpGq contains BλpGq as a closed ideal.

Proof. Let u and v be elements of BpGq. By Lemma 2.1.9 u and v admitrepresentations

upxq “ xπpxqξ, ηy and vpxq “ xπ1pxqξ1, η1

y,

where }u} “ }ξ} ¨ }η} and }v} “ }ξ1} ¨ }η1}. Thus

upxqvpxq “ xpπ b π1qpxqpξ b ξ1

q, η b η1y

and hence uv P BpGq. Also,

}uv} ď }ξ b ξ1} ¨ }η b η1

} “ }ξ} ¨ }η} ¨ }ξ1} ¨ }η1

}.

So the norm } ¨ } on BpGq is submultiplicative.Finally, BλpGq is a closed linear subspace of BpGq. To see that BλpGq is an

ideal in BpGq, it suffices to verify that P pGqPλpGq Ď PλpGq because BpGq andBλpGq are spanned by P pGq and PλpGq, respectively. Now, P pGqPλpGq Ď PλpGq

follows from the fact that for any representation π of G, π b λ „ λ (see Section1.6). �

Proposition 2.1.12. Let T and S be collections of unitary representations ofG such that PT pGqPSpGq Ď PSpGq. If u P BT pGq and μ P MpGq, then uμ P BSpGq

and}uμ}T ď }u} ¨ }μ}S .

In particular, }uμ}ΣpGq ď }u} ¨ }μ}ΣpGq for any u P BpGq, and if u P BλpGq, then}uμ}ΣpGq ď }u} ¨ }λpμq}.

Proof. The hypothesis that PT pGqPSpGq Ď PSpGq implies BT pGqBSpGq Ď

BSpGq. Then, by Remark 2.1.10(2),

}uμ}T “ sup

"ˇ

ˇ

ˇ

ˇ

ż

G

upxqvpxqdμpxq

ˇ

ˇ

ˇ

ˇ

: v P BT pGq, }v} ď 1

*

ď sup t}uv} ¨ }μ}S : v P BT pGq, }v} ď 1u

ď }u} ¨ }μ}S .

The remaining statements follow since BλpGq is an ideal in BpGq. �


Lemma 2.1.13. Let S be a collection of equivalence classes of unitary represen-tations of G and let u P BSpGq.

(i) If μ P MpGq, then μ ˚ u P BSpGq and

}μ ˚ u} ď }u} ¨ supt}σpμq} : σ P Su.

(ii) If Δ´1μ P MpGq, then u ˚ μ P BSpGq and

}u ˚ μ} ď }u} ¨ supt}σpΔ´1μq} : σ P Su.

Proof. By Lemma 2.1.9, there exist a unitary representation π of G which isweakly contained in S and ξ, η P Hpπq such that upxq “ xπpxqξ, ηy for all x P Gand }u} “ }ξ} ¨ }η}. Then

μ ˚ upxq “

ż

G

upy´1xqdμpyq “

ż

G

xπpxqξ, πpyqηydμpyq

“ xπpxqξ, πpμqηy.

This shows that μ˚u P BSpGq and }μ˚u} ď }ξ} ¨}η} ¨}πpμq}. Since π is subordinateto S, (i) follows.

The proof of (ii) is of course similar. In fact,

u ˚ μpxq “

ż

G

upxy´1qΔpy´1

qdμpyq

“

ż

G

xπpxy´1qξ, ηyΔ´1

pyqdμpyq

“ xπpxqπpΔ´1μq˚ξ, ηy.

This implies that u ˚ μ P BSpGq and

}u ˚ μ} ď }ξ} ¨ }η} ¨ }πpΔ´1μq},

so that (ii) follows. �

Taking μ “ δx in (i) and μ “ Δδx´1 in (ii) of Lemma 2.1.13, respectively, weget

Corollary 2.1.14. Let u P BSpGq and x P G. Then Lxu P BSpGq andRxu P BSpGq and }Lxu} “ }u} “ }Rxu}.

Remark 2.1.15. Let G be a locally compact abelian group and pG its dual

group. We claim that Bp pGq is isometrically isomorphic to the measure algebraMpGq via the Fourier-Stieltjes transform μ Ñ pμ, μ P MpGq. In fact, this can beseen as follows.

Given u P Bp pGq, by Bochner’s theorem there exists μ P MpGq such that

upχq “ pμpχq “

ż

G

χpxqdμpxq


for all χ P pG. Then, using the inversion formula,

}u} “ sup

"ˇ

ˇ

ˇ

ˇ

ż

pG

upχqfpχqdχ

ˇ

ˇ

ˇ

ˇ

: f P Ccp pGq, }f}C˚p pGqď 1

*

“ sup

"ˇ

ˇ

ˇ

ˇ

ż

pG

upχqpgpχqdχ

ˇ

ˇ

ˇ

ˇ

: g P CcpGq, }g}8 ď 1

*

“ sup

"ˇ

ˇ

ˇ

ˇ

ż

G

ˆż

pG

pgpχqχpxq

˙

dμpxq

ˇ

ˇ

ˇ

ˇ

: g P CcpGq, }g}8 ď 1

*

“ sup

"ˇ

ˇ

ˇ

ˇ

ż

G

gpxqdμpxq

ˇ

ˇ

ˇ

ˇ

: g P CcpGq, }g}8 “ 1

*

“ }μ}.

Recall that by the Pontryagin duality theorem, a locally compact abelian group G

is topologically isomorphic to the dual group of pG. Therefore, the above can berestated as follows. For any locally compact abelian group G, BpGq is isometrically

isomorphic to the measure algebra Mp pGq, the isomorphism being performed by the

inverse Fourier-Stieltjes transform μ Ñ μ, μ P Mp pGq.

2.2. Functorial properties of BpGq

Let G and H be locally compact groups and φ : H Ñ G a continuous ho-momorphism. If π is a unitary representation of G, then π ˝ φ is a unitary rep-resentation of H in the same Hilbert space. If u is a positive definite functionassociated with π, that is, upxq “ xπpxqξ, ξy for some ξ P Hpπq and all x P G, thenu ˝ φpyq “ xπ ˝ φpyqξ, ξy defines a positive definite function of H associated withπ ˝ φ. Moreover, if S is a set of equivalence classes of representations of G, thenwe denote by S ˝ φ the set of equivalence classes of representations π ˝ φ, π P S.Note that if π and π1 are equivalent representations of G then π ˝ φ and π1 ˝ φ areequivalent.

Theorem 2.2.1. Let φ be a continuous homomorphism from H into G.

(i) The map j : u Ñ u ˝ φ is a norm decreasing homomorphism from BpGq

into BpHq, and for any S,jpPSpGqq Ď PS˝φpHq and jpBSpGqq Ď BS˝φpHq.

(ii) Suppose that φpHq is dense in G. Then j is isometric and

(2.7) jpBpGqq “ BpHq X jpCpGqq “ BΣpHq˝φpHq X jpCpGqq.

Moreover, if u “ ru P BpGq, then

pu ˝ φq`

“ u`˝ φ and pu ˝ φq

´“ u´

˝ φ.

(iii) Suppose that φ is surjective and that given any compact subset K of G,there exists a compact subset C of H such that φpCq “ K. Then

(2.8) jpBSpGqq “ BS˝φpHq X jpCpGqq

for any subset S of ΣpGq.

Proof. (i) It is clear that if u is positive definite, then so is u˝φ. Now assumethat u P PSpGq. To show that u ˝ φ P PS˝φpHq, let a compact subset C of Hand ε ą 0 be given. Since K “ φpCq is compact, there exist elements u1, . . . , un

2.2. FUNCTORIAL PROPERTIES OF BpGq 47

of P pGq associated with representations π1, . . . , πn in S, respectively, such that|upxq ´

řnj“1 ujpxq| ď ε for all x P K. Thus

ˇ

ˇ

ˇ

ˇ

ˇ

u ˝ φpyq ´

nÿ

j“1

uj ˝ φpyq

ˇ

ˇ

ˇ

ˇ

ˇ

ď ε

for all y P C. Since uj ˝ φ is associated to πj ˝ φ, it follows that u ˝ φ P PS˝φpHq.By linearity, we obtain that jpBSpGqq Ď BS˝φpHq.

Taking S “ ΣpGq, we conclude that j is a homomorphism from BpGq intoBpHq. To show that j is norm decreasing, let u P BpGq such that upxq “ xπpxqξ, ηy

with }u} “ }ξ}¨}η}. Then u˝φpyq “ xπ˝φpyqξ, ηy and hence }u˝φ} ď }ξ}¨}η} “ }u}.(ii) Suppose that φpHq is dense in G. Let ω denote the universal representation

of G and A the subalgebra of C˚pGq˚˚ consisting of all finite linear combinationsof operators ωpxq, x P φpHq. Since the mapping x Ñ ωpxq from G into BpHpωqq

is continuous with respect to the strong operator topology, A is strongly dense inC˚pGq˚˚. Then the Kaplansky density theorem assures that the unit ball of A isultra-weakly dense in the unit ball of C˚pGq˚˚.

Now, let u P BpGq “ BωpGq. Then u ˝ φ P Bω˝φpHq by (i), and hence, byLemma 2.1.8,

}u ˝ φ} “ sup

#ˇ

ˇ

ˇ

ˇ

ˇ

nÿ

j“1

cjpu ˝ φqpyjq

ˇ

ˇ

ˇ

ˇ

ˇ

: yj P H, cj P C,

›

›

›

›

›

nÿ

j“1

cjpω ˝ φqpyjq

›

›

›

›

›

ď 1

+

“ sup

#ˇ

ˇ

ˇ

ˇ

ˇ

nÿ

j“1

cjupxjq

ˇ

ˇ

ˇ

ˇ

ˇ

: xj P φpHq, cj P C,

›

›

›

›

›

nÿ

j“1

cjωpxjq

›

›

›

›

›

ď 1

+

“ supt|xT, uy| : T P A, }T } ď 1u

“ supt|xT, uy| : T P C˚pGq

˚˚, }T } ď 1u

“ }u}.

This shows that j is isometric.Let u “ ru P BpGq. Then

u ˝ φ “ pu ˝ φq„

“ u`˝ φ ´ u´

˝ φ.

On the other hand, since j is isometric,

}u ˝ φ} “ }u} “ u`peGq ` u´

peGq “ u`˝ φpeHq ´ u´

˝ φpeHq.

These two equations imply that pu ˝ φq` “ u` ˝ φ and pu ˝ φq´ “ u´ ˝ φ. Since, by(i),

jpBpGqq Ď BΣpHq˝φpHq X jpCpGqq Ď BpHq X jpCpGqq,

to finish the proof of (ii), it remains to show that if v “ u ˝ φ, where u P CpGq

and v P BpHq, then u P BpGq. Obviously, we can assume that v “ rv. By Lemma2.1.7, v` and v´ are uniform limits on H of linear combinations of right translatesof v “ u ˝ φ. Therefore, given m P N, there exist y1, . . . , yn P H and c1, . . . , cn P Csuch that, for every y P H,

ˇ

ˇ

ˇ

ˇ

ˇ

v`pyq ´

nÿ

j“1

cjRyjpu ˝ φqpyq

ˇ

ˇ

ˇ

ˇ

ˇ

ď 1{m.


Setting vm “řn

j“1 cjRφpyjqu and observing that

nÿ

j“1

cjRyjpu ˝ φq “

nÿ

j“1

cjpRφpyjquq ˝ φ “

˜

nÿ

j“1

cjRφpyjqu

¸

˝ φ “ vm ˝ φ,

we see that for any m P N, there exists vm, a linear combination of translates of u,such that

(2.9) |v`pyq ´ vm ˝ φpyq| ď 1{m, y P H.

By (2.9) the sequence of functions vm, m P N, converges uniformly on φpHq andhence, by density, uniformly on G, to a function w` on G. Being a linear com-bination of translates of u, vm is a continuous function on G and therefore w` iscontinuous. It follows from (2.9) that w` satisfies v` “ w` ˝φ. Moreover, since v`

is positive definite on H, w` is positive definite on φpHq and hence positive definiteon G by continuity. So we have shown that v` “ w` ˝ φ for some w` P P pGq.

Similarly, it is shown that there exists w´ P P pGq such that v´ “ w´ ˝φ. Since

u ˝ φ “ v “ v`´ v´

“ w`˝ φ ´ w´

˝ φ “ pw`´ w´

q ˝ φ,

we get u “ w` ´ w´ P BpGq. This shows (2.7).(iii) Suppose that φ is onto and satisfies the condition on compact sets in (iii).

Let v P BS˝φpHq, so that v` and v´ belong to PS˝φpHq. Let K be a compact subsetof G and ε ą 0, and choose a compact subset C of H such that φpCq “ K. Sincev` and v´ can be uniformly approximated on C, up to ε, by sums of continuouspositive definite functions associated to π˝φ, where π P S, w` and w´ (see (ii)) canbe uniformly approximated on K, up to ε, by continuous positive definite functionsassociated with representation π, π P S. Since K and ε ą 0 are arbitrary, it followsthat w`, w´ P PSpGq and hence u “ w` ´ w´ P BSpGq. This shows that

BS˝φ X jpCpHqq Ď jpBSpHqq,

and hence (2.8) follows. �

As an immediate consequence of the preceding theorem we obtain the followingcorollary. It extends, to arbitrary locally compact groups, a theorem which is due toBochner and Schoenberg for R and to Eberlein for general locally compact abeliangroups G and characterizes the Fourier-Stieltjes transforms of measures in MpGq.

Corollary 2.2.2. Let G be a locally compact group and Gd the same groupequipped with the discrete topology. A function u on G belongs to BpGq if and onlyif u is continuous and u P BpGdq. In that case, the norms of u in BpGq and inBpGdq are equal.

Corollary 2.2.3. The unit ball of BpGq is closed in CpGq with respect to thetopology of pointwise convergence.

Proof. If v P BpGq is such that }v} ď 1, then by Lemma 2.1.8ˇ

ˇ

ˇ

ˇ

ˇ

nÿ

j“1

cjvpxjq

ˇ

ˇ

ˇ

ˇ

ˇ

ď

›

›

›

›

›

nÿ

j“1

cjδxj

›

›

›

›

›

ΣpGq

for any finitely many x1, . . . , xn P G and c1, . . . , cn P C.

2.2. FUNCTORIAL PROPERTIES OF BpGq 49

Now, let u P CpGq be a pointwise limit of such functions v. Then the sameinequality holds for u. It then follows for any f P l1pGdq that

ˇ

ˇ

ˇ

ˇ

ˇ

ÿ

xPG

fpxqupxq

ˇ

ˇ

ˇ

ˇ

ˇ

ď }f}ΣpGq

and hence u P BpGdq. Since u is continuous, Corollary 2.2.2 shows that u P

BpGq. �

Corollary 2.2.4. Let G be a locally compact group, N a closed normal sub-group of G and q : G Ñ G{N the quotient homomorphism. Then the map u Ñ u˝qis an isometric isomorphism from BpG{Nq onto the subspace of BpGq consisting ofall functions in BpGq which are constant on cosets of N . For each S Ď ΣpG{Nq,the image of BSpG{Nq under this map is BS˝qpGq.

Proof. The homomorphism q satisfies the hypotheses of Theorem 2.2.1(iii).Moreover, a function u on G{N is continuous if and only if u ˝ q is continuous onG. The statement now follows from Theorem 2.2.1(iii). �

Alternatively, Corollary 2.2.4 can be obtained more directly by using dual-ity arguments for Banach spaces as follows. Let TN denote the canonical ˚-homomorphism from C˚pGq onto C˚pG{Nq. Then the dual map, T˚

N , from BpG{Nq

into BpGq is given by

xT˚N puq, fy “ xu, TNfy

“

ż

G{N

upxNq

ˆż

N

fpxnqdn

˙

dpxNq

“

ż

G

upqpxqqfpxqdx

“ xu ˝ q, fy,

so that T˚N equals j, where j denotes the map u Ñ u ˝ q.

Corollary 2.2.5. Let G be a locally compact group, bG its Bohr compactifi-cation and φ : G Ñ bG the canonical homomorphism. Let AP pGq denote the spaceof almost periodic functions on G. Then the map j : u Ñ u ˝φ is an isometry fromBpbGq onto BpGq X AP pGq.

Proof. Since φpGq is dense in bG, by Theorem 2.2.1(ii) j is an isometry andjpBpbGqq “ BpGq X jpCpbGqq. Now, just note that jpCpbGqq “ AP pGq. �

Lemma 2.2.6. Let H be an open subgroup of the locally compact group G,

and for a function f on H, let˝

f denote the trivial extension of f to all of G.

Then the map f Ñ˝

f from CcpHq into CcpGq extends uniquely to an isometric ˚-homomorphism of C˚

λ pHq into C˚λ pGq. Its adjoint map, which is the restriction map

u Ñ u|H , is a norm decreasing algebra homomorphism from BλpGq onto BλpHq

and it maps PλGpGq onto PλH

pHq.


Proof. The map f Ñ˝

f is clearly an injective ˚-homomorphism of CcpHq intoCcpGq, and

}λHpfq} “ supt}f ˚ g}2 : g P CcpHq, }g}2 ď 1u

“ supt}˝

f ˚˝g}2 : g P CcpHq, }g}2 ď 1u

ď supt}˝

f ˚ h}2 : h P CcpGq, }h}2 ď 1u

“ }λGp˝

fq}.

Conversely, since BλpGq|H Ď BλpHq and }v|H}BpHq ď }v}BpGq for any v P BpGq,

}λGp˝

fq} “ sup

"ˇ

ˇ

ˇ

ˇ

ż

G

˝

fpxqvpxqdx

ˇ

ˇ

ˇ

ˇ

: v P BλpGq, }v}BpGq ď 1

*

“ sup

"ˇ

ˇ

ˇ

ˇ

ż

H

fpxqv|Hpxqdx

ˇ

ˇ

ˇ

ˇ

: v P BλpGq, }v}BpGq ď 1

*

ď sup

"ˇ

ˇ

ˇ

ˇ

ż

H

fpxqupxqdx

ˇ

ˇ

ˇ

ˇ

: u P BλpHq, }u}BpHq ď 1

*

“ }λHpfq}.

So }λHpfq} “ }λGp˝

fq} for f P CcpHq, and hence the map f Ñ˝

f extends uniquelyto an isometric ˚-homomorphism φ from C˚

λ pHq into C˚λ pGq. Since

ż

G

˝

fpxqvpxqdx “

ż

H

fpxqv|Hpxqdx

for f P CcpHq and v P BλpGq, the adjoint map of φ is simply the map v Ñ v|H .Finally, φ˚ is surjective since φ is an isometry. �

2.3. The Fourier algebra ApGq, its spectrum and its dual space

Let G be a locally compact group. In this section we first introduce the Fourieralgebra of G and then identify its Gelfand spectrum and its Banach space dual.

Lemma 2.3.1. Let f, g P L2pGq. Then f ˚rg P BλpGq [Section 2.1] and }f ˚rg} ď

}f}2}g}2.

Proof. For x P G, we have

pf ˚ rgqpxq “

ż

G

fpxyqgpyqdy “ xλGpx´1qf, gy.

The statement now follows from Lemma 2.1.9 and Remark 2.1.10. �

Proposition 2.3.2. Let G be a locally compact group, C a compact subset ofG and U an open subset of G such that C Ď U . Then there exists a function u onG which is a finite linear combination of functions in P pGq X CcpGq and satisfies

0 ď u ď 1, u|C “ 1 and u|GzU “ 0.

Proof. Since C is compact there exists a compact symmetric neighbourhoodV of the identity such that CV 2 Ď U . Let

upxq “ |V |´1

p1CV ˚ 1V qpxq “ |V |´1

¨ |xV X CV |,

2.3. THE FOURIER ALGEBRA ApGq, ITS SPECTRUM & ITS DUAL SPACE 51

x P G. Then 0 ď upxq ď 1 for all x P G. If x P C then |xV X CV | “ |xV | “ |V |

and hence upxq “ 1, whereas if x R CV 2, then xV X CV “ H and so upxq “ 0. Inparticular, supp u Ď CV 2, which is compact, and upxq “ 0 for x P GzU .

Finally, the identity

4pf ˚ g˚q “ pf ` gq ˚ pf ` gq

˚´ pf ´ gq ˚ pf ´ gq

˚

` ipf ` igq ˚ pf ` igq˚

´ ipf ´ igq ˚ pf ´ igq˚,

f, g P L1pGq, ensures that u is a finite linear combination of functions of the formh ˚ h˚, where h P L8pGq and h has compact support and hence h ˚ h˚ P P pGq X

CcpGq. �

Proposition 2.3.3. For 1 ď j ď 10, define a subset Mj of BpGq as follows:M1 “ tf ˚ rg : f, g P CcpGqu;

M2 “ th ˚ rh : h P CcpGqu;M3 “ tf ˚ rg : f, g P L8pGqwith compact supportu;

M4 “ th ˚ rh : h P L8pGqwith compact supportu;M5 “ BpGq X CcpGq;M6 “ P pGq X CcpGq;M7 “ tu P P pGq : Δ´1{2u P L1pGqu;M8 “ P pGq X L2pGq;

M9 “ th ˚ rh : h P L2pGqu;M10 “ tf ˚ rg : f, g P L2pGqu.

Let Ej denote the linear span of Mj, 1 ď j ď 10. Then

E1 “ E2 Ď E3 “ E4 Ď E5 “ E6 Ď E7 Ď E8 Ď E9 “ E10 Ď BλpGq,

and all these subspaces of BλpGq have the same closure, denoted ApGq, in BλpGq.Moreover, ApGq is an ideal in BpGq.

Proof. The equalities E1 “ E2, E3 “ E4 and E9 “ E10 all follow from thepolar identity. The inclusions E2 Ď E3, E4 Ď E5 and E6 Ď E7 are evident.

To see that E5 Ď E6, let v P BpGq XCcpGq and v “ v1 ´ v2 ` ipv3 ´ v4q, wherevk P P pGq, 1 ď k ď 4. By Proposition 2.3.2 there exists a function u of the formu “

řnj“1 cjuj , where cj P C and uj P P pGq X CcpGq, such that u “ 1 on supp v.

Then

v “ uv “

nÿ

j“1

cjujrv1 ´ v2 ` ipv3 ´ v4qs,

which shows that v P E6. The inclusion E7 Ď E8 is a consequence of [100, Propo-sition 12], and E8 Ď E9 follows from the fact that every u P P pGq X L2pGq can be

written in the form u “ f ˚ rf with f P L2pGq [60, Theorem 13.8.6].Finally, all these subspaces will have the same closure once we have shown that

E1 is dense in E10. To this end, let f, g P L2pGq and ε ą 0 be given. Then thereexist functions h, k P CcpGq such that }f ´ h}2 ď ε and }g ´ k}2 ď ε. Now, byLemma 2.3.1,

}f ˚ rg ´ h ˚ rk}BpGq “ }pf ´ hq ˚ rg ` h ˚ prg ´ rkq}BpGq

ď }pf ´ hq ˚ rg}BpGq ` }h ˚ prg ´ rkq}BpGq

ď }f ´ h}2}g}2 ` }h}2}g ´ k}2

ď εp}g}2 ` }f}2 ` εq.

This shows that M10 Ď M1 and hence E10 Ď E1. �


Definition 2.3.4. The algebra ApGq, defined in Proposition 2.3.3, is called theFourier algebra of the locally compact group G.

Corollary 2.3.5. Let G be a locally compact group.

(i) Every u P ApGq vanishes at infinity.(ii) ApGq is uniformly dense in C0pGq.

Proof. (i) follows from }u}8 ď }u} and the density of the subspace BpGq X

CcpGq “ ApGq X CcpGq in ApGq (Proposition 2.3.3).(ii) ApGq X CcpGq is a self-adjoint subalgebra of C0pGq which, by Proposition

2.3.2, strongly separates the points of G. Thus ApGq is uniformly dense in C0pGq

by the Stone-Weierstrass theorem. �

Lemma 2.3.6. Let u P ApGq and x, y P G. Then the functions Lxu, Ryu, u, quand ru all belong to ApGq.

Proof. By Remark 2.1.10(3) and Corollary 2.1.14, all the linear maps of BpGq

into itself in question are continuous. Since they clearly map BpGq X CcpGq intoitself and BpGq X CcpGq is dense in ApGq, the statements of the lemma follow. �

Our next purpose in this section is to show that the spectrum of ApGq can becanonically identified with G and that ApGq is a regular algebra of functions on G.To that end, we need the following lemma.

Lemma 2.3.7. Let a P G and f P ApGq such that fpaq “ 0. Then, givenε ą 0, there exists h P ApGq X CcpGq vanishing in a neighbourhood of a such that}h ´ f}ApGq ď ε.

Proof. Notice first that, since ApGq X CcpGq is dense in ApGq, without lossof generality we can assume that f ‰ 0, f has compact support and ε ď }f}8 andε ă 1. Let

W “ ty P G : }f ´ Ryf}ApGq ď εu.

Then W is a compact neighbourhood of e in G. Choose an open neighbourhood Vof e such that V Ď W and supt|fpayq| : y P V u ď ε. By regularity of Haar measure,there exists a compact neighbourhood U of e such that U Ď V and |U | ě |V |p1´εq.Now, define functions u, g and h on G by setting u “ |U |´11U , g “ 1aV f and

h “ pf ´ gq ˚ qu P ApGq.

Then h has compact support since W is compact and f has compact support. Forany x P G,

hpxq “ |U |´1

ż

U

fpxyqr1 ´ 1aV pxyqsdy.

It follows that, if x P G satisfies a´1xU Ď V, then hpxq “ 0. Thus h vanishes in aneighbourhood of a. Moreover,

}u}2 “ |U |´1{2

ď |V |´1{2

ˆ

1

1 ´ ε

˙1{2

,

}g}2 “

ˆż

aV

|fpyq|2dy

˙1{2

ď ε|V |1{2,


and

}f ´ f ˚ qu}ApGq “

›

›

›

›

f ´ |U |´1

ż

U

pRyfqdy

›

›

›

›

ApGq

ď supyPU

}f ´ Ryf}ApGq ď ε.

Combining all these estimates, we obtain

}f ´ h}ApGq ď }f ´ f ˚ qu}ApGq ` }g}2}qu}2 ď ε ` ε

ˆ

1

1 ´ ε

˙1{2

.

This finishes the proof. �

Theorem 2.3.8. Let G be a locally compact group. For each x P G, let

ϕx : ApGq Ñ C, u Ñ upxq.

Then the map x Ñ ϕx is a homeomorphism from G onto σpApGqq. Moreover, ApGq

is regular.

Proof. It is obvious that ϕx P σpApGqq and that the map x Ñ ϕx is injective.Now let ϕ P σpApGqq be given and suppose that ϕ ‰ ϕx for all x P G. Then, foreach x P G there exists fx P ApGq such that ϕpfxq “ 1, but ϕxpfxq “ 0. By Lemma2.3.6, every g P ApGq vanishing at x is the limit of a sequence pgnqn in ApGq withthe property that each gn vanishes in a neighbourhood of x. Therefore we canassume that fx vanishes in a neighbourhood Vx of x.

Since ApGq XCcpGq is dense in ApGq, there exists f0 P CcpGq XApGq such thatϕpf0q “ 1. Choose x1, . . . , xn P supp f0 such that

supp f0 ĎŤn

j“1 Vxj

and let

f “ f0fx1. . . fxn

P ApGq.

Then fpxq “ 0 for every x P G, whereas

ϕpfq “ ϕpf0qśn

j“1 ϕpfxjq “ 1.

This contradiction shows that ϕ “ ϕx for some x P G.Now, since the subalgebra ApGq of C0pGq is uniformly dense in C0pGq, the

topology on G coincides with the weak topology defined by the set of functionsx Ñ fpxq “ ϕxpfq, f P ApGq. Thus the map x Ñ ϕx from G to σpApGqq is ahomeomorphism.

Finally, Proposition 2.3.2 implies that ApGq is a regular algebra of functionson G. �

Of course, after identifying σpApGqq with G, the Gelfand homomorphism ofApGq is nothing but the identity mapping. In particular, ApGq is a semisimplecommutative Banach algebra.

We remind the reader that if G is a locally compact abelian group with dual

group pG, then ApGq˚ “ L1p pGq˚ “ L8p pGq and that V NpGq “ L8p pGq (see Re-mark 1.8.21). The following theorem shows that V NpGq is isometrically isomorphicto the dual space of ApGq for general locally compact groups G.


Theorem 2.3.9. Let G be a locally compact group. For any ϕ P ApGq˚ thereexists a unique operator Tϕ P V NpGq such that

xTϕpfq, gy2 “ xϕ, g ˚ qf y “ xϕ, pf ˚ rgqqy

for all f, g P L2pGq. The mapping ϕ Ñ Tϕ from ApGq˚ to V NpGq is a surjectivelinear isometry and has the following additional properties.

(i) If u “ř8

j“1pgj ˚ qfjq, fj , gj P L2pGq, withř8

j“1 }fj}2}gj}2 ă 8, then

xϕ, uy “

8ÿ

j“1

xTϕpfjq, gjy2.

(ii) If μ P MpGq and ϕμ is the element of ApGq˚ defined by

xϕμ, uy “

ż

G

upxqdμpxq, u P ApGq,

then Tϕμ“ λGpμq.

(iii) ϕ Ñ Tϕ is a homeomorphism for the w˚-topology on ApGq˚ and theultraweak topology on V NpGq.

Proof. If ϕ P ApGq˚, then for any f, g P L2pGq,

|xϕ, g ˚ qfy| ď }ϕ} ¨ }g ˚ qf}ApGq ď }ϕ} ¨ }g}2}f}2.

Thus, for each f P L2pGq, the assignment g Ñ xϕ, g ˚ qfy defines a conjugatelinear functional on L2pGq. Hence there exists a unique fϕ P L2pGq such that

xϕ, g ˚ qfy “ xfϕ, gy2 for all g P L2pGq. Define Tϕ : L2pGq Ñ L2pGq by Tϕpfq “ fϕ.Then Tϕ is linear and }Tϕpfq}2 ď }ϕ} ¨ }f}2 for all f P L2pGq, so that }Tϕ} ď }ϕ}.For any f, g P L2pGq and h P L1pGq, we have

xTϕpλGphqfq, gy2 “ xTϕpf ˚ hq, gy2 “ xϕ, g ˚ pf ˚ hqqy

“ xϕ, g ˚ qh ˚ qfy “ xϕ, g ˚ h˚ ˚ qfy

“ xTϕpfq, g ˚ h˚y2 “ xTϕpfq, λGphq

˚gy2

“ xλGphqTϕpfq, gy2.

So Tϕ commutes with the right regular representation operators and therefore Tϕ P

V NpGq. If u is as in (i), the seriesř8

j“1pgj ˚ qfjq is absolutely convergent and hence

xϕ, uy “

8ÿ

j“1

xϕ, gj ˚ qfjy “

8ÿ

j“1

xTϕpfjq, gjy2

and |xϕ, uy| ď }Tϕ} ¨ř8

j“1 }fj}2}gj}2. Since this holds for all such representations

of u P ApGq, we conclude that |xϕ, uy| ď }Tϕ} ¨ }u}ApGq. It follows that ϕ Ñ Tϕ isan isometry, and the above equation yields statement (iii). If μ P MpGq, then forall f, g P L2pGq,

xTϕμpfq, gy2 “ xϕμ, g ˚ qfy “

ż

G

ż

G

gpxyqfpyqdydμpxq

“

ż

G

ż

G

gpyqfpx´1yqdμpxqdy

“

ż

G

pμ ˚ fqpyqgpyqdy “ xλGpμqf, gy2,

so that Tϕμ“ λGpμq.


It remains to show that every T P V NpGq is of the form Tϕ for some ϕ P

ApGq˚. Thus, let T P V NpGq and note first that given f, g P CcpGq, the functionT puq “ T pf ˚ rgq “ T pfq ˚ rg is continuous. Therefore, for u P E1, we can putxϕT , uy “ T puqpeq, and this definition does not depend on the representation of u.To show that ϕT is a bounded linear functional on E1, recall that by Kaplansky’sdensity theorem there exists a net phαqα in CcpGq such that }λGphαq} ď }T } for allα and }Tg´λGphαqg} Ñ 0 for every g P L2pGq. For u of the form u “

řnj“1pfj˚g˚

j q,

where fj , gj P C2c pGq, it follows that

xϕT , uy “ Tupeq “

nÿ

j“1

xTfj , gjy “ limα

nÿ

j“1

xλGphαqfj , gjy “ limα

xϕλGphαq, uy

and hence

|xϕT , uy| “ limα

|xϕλGphαq, uy| ď }u} ¨ supα

}λGphαq} ď }u} ¨ }T }.

Since E1 is dense in ApGq, ϕT extends uniquely to a bounded linear functional onApGq, also denoted ϕT , of norm ď }T }. By definition of ϕT ,

xTϕTpfq, gy2 “ xfϕT

, gy2 “ xϕT , g ˚ qfy

“ xϕT , pf ˚ rgqqy “ T pf ˚ rgqpeq

“ pT pfq ˚ rgqpeq “ xT pfq, gy2

and hence TϕT“ T . �

Given T P V NpGq, we let qT denote the operator in V NpGq defined by xϕqT , uy “

xϕT , quy, u P ApGq. Thus T Ñ qT is the transpose of the isometry u Ñ qu of ApGq.Then, for μ P MpGq,

xϕλGpμq

, uy “

ż

G

qupxqdμpxq “ xϕλGpqμq, uy,

where dqμpxq “ dμpx´1q. Thus λGpμq “ λGpqμq, and passing to ultraweak limits, we

deduce that the map T Ñ qT is an isometric and ultraweakly continuous involutionon V NpGq.

Definition 2.3.10. For T P V NpGq and u P ApGq, let Tu denote the uniqueelement of ApGq such that

xS, Tuy “ x qTS, uy

for all S P V NpGq.

It is easily verified that the assignment pT, uq Ñ Tu turns ApGq into a leftV NpGq-module.

Lemma 2.3.11. Let G be a locally compact group and T P V NpGq.

(i) The map u Ñ Tu is a bounded linear operator on ApGq with norm }T }.(ii) For u P ApGq and x P G, T qupxq “ xT, Lxuy.(iii) If u P ApGqXL2pGq, then Tu P ApGqXL2pGq and Tu “ T puq, the action

of T on L2pGq.

Proof. (i) For every u P ApGq, we have

}Tu} “ sup}S}ď1

|xS, Tuy| “ sup}S}ď1

|x qTS, uy| ď } qT } ¨ }u} “ }T } ¨ }u}.


So u Ñ Tu is a bounded linear operator on ApGq. By definition, its transpose is

the map S Ñ qTS which has norm } qT } “ }T }. Since the two maps have the samenorm, (i) follows.

(ii) Consider first elements u of ApGq of the form u “ pf ˚ rgqp, where f, g P

CcpGq. Then

T pquq “ T pf ˚ rgq “ T pfq ˚ rg

is a continuous function and, for each x P G,

T pquqpxq “ pT pfq ˚ rgqpxq “ xT pfq, Lxgy

“ ϕT pLxg ˚ qfq “ ϕT pLxrpf ˚ rgqqsq

“ xT, Lxuy.

Using that Spquqpeq “ xS, uy, it follows that

xT, Lxuy “ T pquqpxq “ pρpx´1qT qpquqpeq

“ xρpx´1qT, uy “ x qTρpxq, quy “ xρpxq, T quy

“ T qupxq.

By linearity, this shows that T pquq “ T qu for all u P E1, where E1 is the space definedin Proposition 2.3.3. Since E1 is dense in ApGq and the maps u Ñ qu and u Ñ Lxuare continuous, we conclude that T qupxq “ xT, Lxuy for all u P ApGq.

(iii) Let A be the ˚-subalgebra of V NpGq consisting of all operators T of theform T “ λGpμq, where μ is a finite linear combination of Dirac measures. Notethat, for u P ApGq X L2pGq and x P G, by (ii)

λGpxqupyq “ xλGpxq, Lyquy “ Lyqupxq “ upx´1yq “ Lxupyq

for all y P G. Let μ “řn

j“1 cjδxj, T “ λGpμq and f P CcpGq. Then

ˇ

ˇ

ˇ

ˇ

ż

G

Tupxqfpxqdx

ˇ

ˇ

ˇ

ˇ

“

ˇ

ˇ

ˇ

ˇ

ˇ

ż

G

˜

nÿ

j“1

cjupx´1j xqfpxq

¸

dx

ˇ

ˇ

ˇ

ˇ

ˇ

“

ˇ

ˇ

ˇ

ˇ

ˇ

ż

G

upxq

˜

nÿ

j“1

cjfpxjxq

¸

dx

ˇ

ˇ

ˇ

ˇ

ˇ

“

ˇ

ˇ

ˇ

ˇ

ż

G

pμ˚˚ fqpxqupxqdx

ˇ

ˇ

ˇ

ˇ

“

ˇ

ˇ

ˇ

ˇ

ż

G

T˚pfqpxqupxqdx

ˇ

ˇ

ˇ

ˇ

ď }T } ¨ }f}2}u}2

by Schwarz’ inequality.Now let T P V NpGq be arbitrary. By Kaplansky’s density theorem, there exists

a net pTαqα in A such that }Tα} ď }T } for all α and Tα Ñ T in the ultra weaktopology. Then

ż

G

Tαupxqfpxqdx “ xλGpfq, Tαuy “ x|TαλGpfq, uy,

and passing to the limit,ż

G

Tupxqfpxqdx “ xλGpfq, Tuy “ x qTλGpfq, uy.

2.4. FUNCTORIAL PROPERTIES AND A DESCRIPTION OF ApGq 57

The above estimate then yieldsˇ

ˇ

ˇ

ˇ

ż

G

Tupxqfpxqdx

ˇ

ˇ

ˇ

ˇ

ď }T } ¨ }u}2}f}2

for all f P CcpGq and u P ApGq X L2pGq. This shows that Tu P L2pGq and at thesame time that the mapping u Ñ Tu from ApGq X L2pGq into L2pGq is continuouswith respect to the L2-norms. On the other hand, we know that Tu “ T puq forall u P E1. Since E1 is dense in ApGq, continuity implies that Tu “ T puq for allu P ApGq X L2pGq. This finishes the proof of (iii). �

2.4. Functorial properties and a description of ApGq

Let H be an open subgroup of G and let Haar measure on H be the one

induced by Haar measure of G. As before, we denote by˝

f the trivial extensionof a function f of H to all of G. To T P V NpGq and f P L2pHq we associate the

function T |Hpfq “ T p˝

fq|H on H. Sinceż

H

|T |Hpfqpxq|2dx ď

ż

G

|T p˝

fqpxq|2dx ď }T }

2}

˝

f}22 “ }T }

2}f}

22,

the map T |H : f Ñ T |Hpfq is an operator on L2pHq with }T |H} ď }T }. Moreover,T |H P V NpHq since, for any g P CcpHq,

T |Hpf ˚ gq “ T pf ˚ gq ˝|H “ T p˝

f ˚˝gq|H “ pT p

˝

fq ˚˝gq|H

“ T p˝

fq|H ˚ g “ THpfq ˚ g.

Let H be a closed subgroup of G. Let V NHpGq denote the w˚-closure of thelinear span of the set tλGphq : h P Hu. Then V NHpGq is a von Neumann algebra.

Proposition 2.4.1. Let H be an open subgroup of the locally compact groupG.

(i) The map φ : u Ñ˝u is an isometric isomorphism of ApHq into ApGq and

φpP pHq X ApHqq Ď P pGq X ApGq. The map T Ñ T |H is the adjoint φ˚

of φ and

φ˚pV NpGqq “ V NpHq and φ˚

pC˚λ pGqq “ C˚

λ pHq.

(ii) The restriction map r : u Ñ u|H maps ApGq onto ApHq. Its ad-joint r˚ is an isomorphism of V NpHq onto the von Neumann subal-gebra of V NpGq generated by the operators λGpxq, x P H. Moreover,r˚pC˚

λ pHqq Ď C˚λ pGq.

Proof. (i) We know from Theorem 2.2.1(i) that r : u Ñ u|H is a norm de-creasing map from BpGq into BpHq. Since rpBpGq X CcpGqq Ď BpHq X CcpHq,continuity of r implies that rpApGqq Ď ApHq. Let φ : T Ñ φpT q denote the adjointof the map r|ApGq : ApGq Ñ ApHq. Then φ is a norm decreasing linear map ofV NpHq into V NpGq and it is also continuous for the ultraweak topologies. Forf P CcpHq and u P ApGq we have

xφpλHpfqq, uy “ xλHpfq, u|Hy “

ż

H

fpxqupxqdx

“ xλGp˝

fq, uy,


and hence φpλHpfqq “ λGp˝

fq. Since λHpCcpHqq is uniformly dense in C˚λ pHq, φ

maps C˚λ pHq into C˚

λ pGq. Moreover, since C˚λ pHq is ultraweakly dense in V NpHq,

it follows that φpT˚q “ φpT q˚ and φpST q “ φpSqφpT q for all S, T P V NpHq. So φ isa von Neumann algebra homomorphism from V NpHq into V NpGq. More precisely,the range of φ is contained in V NHpGq since φpλHphqq “ λGphq for all h P H.

Let f, g P CcpHq and put

u “ f ˚ rg P E1pHq Ď ApHq Ď V NpHq˚.

Then, for T P V NpGq,

xϕT ,q

uy “ xϕT , g ˚ qfy “ xϕT ,˝g ˚

q

fy

“ xT p˝

fq,˝gy2 “ xT p

˝

fq|H , gy2

“ xT |Hpfq, gy2 “ xϕT |H , g ˚ qfy

“ xϕT |H , quy.

This shows that R˚puq “ uG P E1pGq for each u P E1pHq. Since R˚ is continuousand E1pHq is dense in ApHq, it follows that R˚pApHqq Ď ApGq. Actually, R˚puq “

uG for every u P ApHq. In fact, this equation holds for all u P E1pHq, E1pHq isdense in ApHq and the map u Ñ uG is continuous for the topologies of pointwiseconvergence. In particular, this implies that uG P ApGq for every u P ApHq. Onthe other hand,

}uG} “ }R˚puq} ď }u} “ }uG|H} ď }uG},

so that }uG} “ }u}. Observe next that if u P P pHqXApHq, then uG P P pGqXApGq

since uG is hermitian and

}uG} “ }u} “ upeq “ uGpeq.

This proves (i).(ii) Because the restriction map r : v Ñ v|H is surjective, it follows from duality

theory that r˚ is a topological isomorphism for the ultraweak topologies betweenV NpHq and its range in V NpGq. Thus r˚pV NpHqq is ultraweakly closed in V NpGq

and hence coincides with V NHpGq since it contains all the operators λGphq, h P H.This completes the proof of (ii). �

Let K be a compact normal subgroup of G and let qK : G Ñ G{K denotethe quotient homomorphism and μK the normalized Haar measure of K. Thenthe map jK : f Ñ f ˝ qK is a Hilbert space isomorphism between L2pG{Kq andL2KpGq, the subspace of L2pGq consisting of all functions which are constant almost

everywhere on xK for almost every coset xK P G{K. In other words, g P L2KpGq if

and only if g P L2pGq and g “ g ˚μK . It follows that T pL2KpGqq Ď L2

KpGq for everyT P V NpGq. Therefore, to each T P V NpGq we can associate an operator TK onL2pG{Kq defined by

TKpfq “ pj˚K ˝ T ˝ jKqpfq, f P L2

pG{Kq.

It is easy to see that TK P V NpG{Kq. In addition,

pST qK “ SKTK and pT˚qK “ pTKq

˚

for any S, T P V NpGq.

Proposition 2.4.2. Let K be a compact normal subgroup of the locally compactgroup G.

2.4. FUNCTORIAL PROPERTIES AND A DESCRIPTION OF ApGq 59

(i) The map u Ñ u˝qK is an isometric isomorphism from ApG{Kq onto thesubalgebra AKpGq of ApGq consisting of all v P ApGq such that vpxkq “

vpxq for all x P G and k P K.(ii) The map T Ñ TK is the adjoint of the map u Ñ u ˝ qK and it is an

ultraweakly continuous homomorphism of V NpGq onto V NpG{Kq.

Proof. (i) By Corollary 2.2.4, φ : u Ñ u˝qK maps BpG{Kq isometrically ontoBKpGq, the algebra of all u P BpGq which are constant on cosets of K. Clearly,

φpBpG{Kq X CcpG{Kqq “ BKpGq X CcpGq.

Since ApG{Kq is the closure of BpG{KqXCcpG{Kq, it suffices to show that BKpGqX

CcpGq is dense in AKpGq. Thus let v P AKpGq and let pvnqn be a sequence inBpGq X CcpGq converging to v. Then vn ˚ μK P BKpGq X CcpGq and

}vn ˚ μK ´ v}BpGq “ }pvn ´ vq ˚ μK}BpGq ď }μK} ¨ }vn ´ v}BpGq Ñ 0,

whence v P BKpGq X CcpGq.(ii) The map T Ñ TK is the adjoint of the map u Ñ u ˝ qK . In fact, for

T P V NpGq and u P ApG{Kq X L2pG{Kq we have

xT, u ˝ qKy “ T ppu ˝ qKqqpeq “ T pqu ˝ qKqpeq

“ T pjKpquqqpeq “ jKpTKpquqqpeq

“ TKpquqpeq “ xTK , uy.

The second statement in (ii) then follows from duality theory of Banach spaces. �Theorem 2.4.3. Let G be a locally compact group. Then ApGq is precisely the

set of all functions f ˚ rg, where f, g P L2pGq.

Proof. Let us first assume that G is second countable. Then the space L2pGq

is separable and hence the von Neumann algebra V NpGq is countably generated.It follows from [60, Proposition 14.5.1] that every normal positive linear functionalon V NpGq is of the form T Ñ xTf, fy, where f P L2pGq. This in turn implies thatevery ultraweakly continuous linear functional on V NpGq is of the form

T Ñ xTf, gy “ ϕT pg ˚ fq, f, g P L2pGq.

On the other hand, as we have seen in Theorem 2.3.9, the ultraweakly continuouslinear functionals on V NpGq are exactly given by T Ñ ϕT puq, where u P ApGq.This establishes the theorem for second countable G.

Now suppose that G is σ-compact, and let u P ApGq. Since u is continuous andvanishes at infinity, u is uniformly continuous. Then, by a theorem of Kakutaniand Kodaira (Theorem 1.2.16), there exists a compact normal subgroup K of Gsuch that G{K is second countable and u is constant on cosets of K. Then u isof the form u “ v ˝ q, where v P ApG{Kq and q : G Ñ G{K denotes the quotienthomomorphism. The first part of the proof shows that there exist f, g P L2pG{Kq

such that v “ f ˚ rg. However, this implies

u “ v ˝ q “ pf ˝ qq ˚ pg ˝ qq„,

where f ˝ q, g ˝ q P L2pGq.Finally, let G be an arbitrary locally compact group. Since u P C0pGq, we find a

sequence pCnqn of compact subsets ofG such that, for all n, Cn Ď Cn`1, Cn containsa neighbourhood of e in G and also the set of all x P G such that |upxq| ě 1{n. LetHn denote the open subgroup generated by Cn and let H “

Ť8

n“1 Hn. Then H is


an open subgroup of G since Hn Ď Hn`1, and H is σ-compact because each Hn isσ-compact. Moreover, u “ 0 on GzH. Since u|H P ApHq, by the second part of theproof there exist f, g P L2pHq such that u|H “ f ˚ rg. Denoting by f1 and g1 thetrivial extensions of f and g to all of G, it follows that u “ f1 ˚ rg1. This finishesthe proof. �

Since pf ˚rgqpxq “ş

Gfpxyqgpyqdy “ xλGpx´1qf, gy, the preceding theorem tells

us that ApGq coincides with the collection of coefficient functions of the left regularrepresentation.

Corollary 2.4.4. On V NpGq Ď BpL2pGqq the weak and the ultraweak opera-tor topologies coincide.

Proof. Let pfnqn and pgnqn be sequences in L2pGq such thatř8

n“1 }fn}2}gn}2

ă 8. Then, by Theorem 2.4.3, there exist f, g P L2pGq such that

8ÿ

n“1

xλGpxqfn, gny “ xλGpxqf, gy

for all x P G, and hence8ÿ

n“1

xTfn, gny “ xTf, gy

for all T P V NpGq. This implies the statement. �

Remark 2.4.5. Let G be a locally compact abelian group. Then

Ap pGq “ {L1pGq “ t pf : f P L1pGqu

and } pf}ApGq “ }f}1. Since }pμ}BpGq “ }μ} for all μ P MpGq (Remark 2.1.15), we

only have to verify that Ap pGq “ {L1pGq. Clearly, {L1pGq Ď Ap pGq since the set of

all f P L1pGq such that pf P Ccp pGq is dense in L1pGq and Bp pGq X Ccp pGq Ď Ap pGq.

Conversely, u P Ap pGq is of the form u “ ξ ˚ qη, where ξ, η P L2p pGq. By the

Plancherel theorem, ξ “ pf and η “ pg for certain f, g P L2pGq. Then fqg P L1pGq

and xfqg “ pf ˚ p

qg “ ξ ˚ qη “ u.

2.5. The support of operators in V NpGq

The main theme of this section is to associate to each T P V NpGq a closedsubset of G, the so-called support of T . This notion, which turns out to be a majortool in the sequel, will be studied thoroughly. We start by defining an action ofBpGq on V NpGq. Recall that, for u P BpGq and T P V NpGq, the assignmentv Ñ xT, uvy defines a bounded linear functional on ApGq.

Definition 2.5.1. Let u¨T denote the operator in V NpGq defined by xu¨T, vy “

xT, uvy for v P ApGq.

It is clear that }u ¨ T } ď }u} ¨ }T } and that with this action V NpGq becomesa left BpGq-module. Note that if u P BpGq and v P ApGq, then by Lemma 2.3.11,

2.5. THE SUPPORT OF OPERATORS IN V NpGq 61

pu ¨ T qv P ApGq and, for every x P G,

rpu ¨ T qvspxq “ ϕu¨T pLxqv q

“ ϕu¨T ppRxvqqq “ ϕT ppRxvqquq

“ ϕT prpRxvqqusqsq

“ T rvRx´1pquqspxq.

Remark 2.5.2. (1) If u P ApGq and T P V NpGq, the reader should be carefulto not mix up the operator u ¨ T in V NpGq with the function Tu P ApGq.

(2) Let T “ λGpμq for some μ P MpGq. Then u ¨λGpμq “ λGpuμq for u P BpGq,where uμ denotes the product of the function u and the measure μ in the usualsense. Indeed, for any x P G and v P ApGq, by the above formula

ru ¨ λGpμqsvpxq “ λGpμqrvRx´1pquqspxq

“ μ ˚ rvRx´1pquqspxq “

ż

G

qupy´1xqvpy´1xqdμpyq

“

ż

G

vpy´1xqupyqdμpyq “ ruμ ˚ vspxq

“ rλGpuμqvspxq.

Let T P C˚λ pGq Ď V NpGq and T “ Tϕ, ϕ P ApGq˚. Then the proof of Theorem

2.3.9 shows that xu, T y “ xϕ, uy for all u P ApGq Ď BλpGq, where xu, T y refers tothe duality C˚

λ pGq˚ “ BλpGq.

Proposition 2.5.3. Let T P V NpGq and a P G. Then the following threeconditions are equivalent.

(i) The operator λGpaq is the w˚-limit in V NpGq of operators of the formv ¨ T , where v P ApGq.

(ii) For every neighbourhood V of a in G, there exists v P ApGq such thatsupp v Ď V and xT, vy ‰ 0.

(iii) If u P ApGq is such that u ¨ T “ 0, then upaq “ 0.

Proof. (i) ñ (ii) Let λGpaq be the w˚-limit of a net pvα ¨T qα, vα P ApGq, andlet V be a neighbourhood of a in G. Since ApGq is regular (Lemma 2.3.7), thereexists w P ApGq such that suppw Ď V and wpaq ‰ 0. Then

xvα ¨ T,wy “ xT, vαwy Ñ xλGpaq, wy “ wpaq ‰ 0,

and hence xT, vαwy ‰ 0 eventually.(ii) ñ (iii) Towards a contradiction, assume that there exists u P ApGq with

u ¨ T “ 0, but upaq ‰ 0. Then we can find δ ą 0 and a compact neighbourhood Vof a such that |upxq| ě δ for all x P V . Since ApGq is regular, by Theorem 1.1.19there exists w P ApGq such that wpxq “

1upxq

for all x P V . Now, by (ii), there

exists v P ApGq with supp v Ď V and xT, vy ‰ 0. Then v “ vwu since supp v Ď V .It follows that

xT, vy “ xu ¨ T, vwy “ 0.

This contradiction proves (iii).(iii) ñ (i) Let I “ tu P ApGq : u ¨ T “ 0u. Then I is a closed ideal in ApGq

since xpuvq ¨ T,wy “ xu ¨ T,wvy for all v, w P ApGq and u P I. On the other hand,since xu ¨T, vy “ xv ¨T, uy for v P ApGq, I is the annihilator in ApGq of the subspaceApGq ¨ T of V NpGq “ ApGq˚. Consequently, TK is the w˚-closure of ApGq ¨ T


in V NpGq. Now, if a P G satisfies (iii) then λGpaq P IK and hence λGpaq is thew˚-limit of operators of the form v ¨ T , v P ApGq. �

Definition 2.5.4. Let T P V NpGq. Then the support of T, supp T , is the setof all elements a P G satisfying one and hence all three conditions in Proposition2.5.3.

It is clear that supp 0 “ H. On the other hand, we have

Lemma 2.5.5. If T P V NpGq, T ‰ 0, then suppT is a nonempty closed subsetof G.

Proof. Let a P suppT and let V be an open neighbourhood of a. ThenV X suppT ‰ H and hence by Proposition 2.5.3(ii) there exists v P ApGq withsupp v Ď V and xT, vy ‰ 0. Thus a P suppT .

Now, assume that T ‰ 0. Since ApGq X CcpGq is dense in ApGq, there existsv P ApGqXCcpGq with xT, vy ‰ 0. Towards a contradiction, suppose that supp T “

H. Then for each a P G there exists ua P ApGq such that ua ¨T “ 0, but uapaq ‰ 0.Since supp v is compact, we find a1, . . . , an P supp v such that uai

¨ T “ 0 andupxq “

řni“1 uai

pxq2 ą 0 for all x P supp v. There exists w P ApGq such thatwpxq “

1upxq

for all x P supp v (Theorem 1.1.19). Then v “ vwu and hence

xT, vy “

nÿ

i“1

xuai¨ T, uai

wvy “ 0.

This contradiction finishes the proof. �

We now collect a number of useful facts about the supports of operators inV NpGq.

Proposition 2.5.6. Let G be a locally compact group and T P V NpGq.

(i) For v P BpGq, we have

supppv ¨ T q Ď suppT X supp v.

In particular, v ¨ T “ 0 whenever v vanishes in some neighbourhood ofsuppT .

(ii) suppT is the smallest closed subset C of G with the following property:If v P ApGq X CcpGq vanishes in a neighbourhood of C, then xT, vy “ 0.

(iii) suppT is the smallest closed subset C of G with the following property:Given any closed neighbourhood V of C such that GzV is relatively com-pact, the operator T is a w˚-limit in V NpGq of finite linear combinationsof operators λGpxq, where x P V .

(iv) Let F be a closed subset of G. Suppose that pTαqα is a net in V NpGq

converging to T in the w˚-topology and satisfying suppTα Ď F for all α.Then suppT Ď F .

(v) Let T, T1, T2 P V NpGq and λ P C, λ ‰ 0. Then(1) supppλT q “ suppT ;(2) supp T˚ “ psuppT q´1;(3) supppT1 ` T2q Ď suppT1 Y suppT2, and equality holds whenever

suppT1 X suppT2 “ H.(4) supppT1T2q Ď psuppT1qpsuppT2q provided that one of suppT1

and suppT2 is compact.


Proof. (i) The inclusion supppv ¨ T q Ď suppT follows immediately from thedescription of the support in Proposition 2.5.3(i). In order to show that supppv¨T q Ď

supp v, let a P Gz supp v and choose a neighbourhood V of a such that v|V “ 0.Then uv “ 0 for every u P ApGq with supp u Ď V and hence xv ¨T, uy “ xT, vuy “ 0for every such u. Proposition 2.5.3(ii) implies that a R supp v ¨ T .

If v vanishes in a neighbourhood of suppT , then supp v X suppT “ H by (i)and therefore supppv ¨ T q “ H. By Lemma 2.5.5, v ¨ T “ 0.

(ii) Let v P AcpGq and suppose that v vanishes in a neighbourhood of suppT .There exists w P ApGq such that w|supp v “ 1 and w “ 0 in a neighbourhood ofsuppT . Then w ¨ T “ 0 by (i). On the other hand, wv “ v and hence

xT, vy “ xT,wvy “ xw ¨ T, vy “ 0.

This shows that supp T has the indicated property.Now, let C be any closed subset of G satisfying the condition in (ii) and let

a P GzC. Then there exist neighbourhoods V of a and U of C such that V iscompact and V XU “ H. If v P ApGq is such that supp v Ď V , then v P AcpGq andv vanishes in a neighbourhood of C. It follows that xT, vy “ 0 and consequentlya R suppT by Proposition 2.5.3(ii).

(iii) Let C be a closed subset of G and V a closed neighbourhood of C suchthat GzV is relatively compact, and suppose that T is a w˚-limit of finite linearcombinations of operators λGpxq, x P V . If v belongs to the ideal IpV q “ tu P

ApGq : u|V “ 0u, then xλGpxq, vy “ vpxq “ 0 for all x P V and by w˚-continuitythis implies xT, vy “ 0. It follows now from (ii) that suppT Ď C, as required.

(iv) Let u P ApGq X CcpGq be such that u vanishes in a neighbourhood of F .Then, for all α, xTα, uy “ 0 since suppTα Ď F . Since Tα Ñ T in the w˚-topology,xT, uy “ 0, and this implies suppT Ď F by (ii).

(v) (1) is evident and (2) follows from (iii) taking into account that the mapT Ñ T˚ is weakly continuous and that λGpxq˚ “ λGpx´1q.

To show (3) we apply (ii). Thus let v P AcpGq vanish in a neighbourhood ofsuppT1 Y suppT2. Then, by (ii),

xT1 ` T2, vy “ xT1, vy ` xT2, vy “ 0

and hence, by (ii) again,

supppT1 ` T2q Ď suppT1 Y suppT2.

Suppose that in addition suppT1 X suppT2 “ H. Let a P suppT1 and let v P ApGq

such that v ¨ pT1 ` T2q “ 0. Then v ¨ T1 “ ´v ¨ T2 “ S, say. By (i),

suppS Ď suppT1 X suppT2 “ H.

This implies S “ 0 by Lemma 2.5.5. Thus v ¨ T1 “ 0 and therefore vpaq “ 0. So

suppT1 Ď supppT1 ` T2q

by Proposition 2.5.3(iii). Similarly, suppT2 Ď supppT1 ` T2q.For (4), suppose that supp T2 is compact. It is a straightforward consequence

of (iii) that

supppλGpxqT2q “ psupppλGpxqqpsuppT2q

for any x P G. Hence, by (1) and (3), we get

supppT1T2q Ď psuppT1qpsuppT2q


whenever T1 is a finite linear combination of operators λGpxq. If T1 is arbitrary, thengiven any closed neighbourhood V of suppT1 with relatively compact complement,by (iii) there exists a net pSαqα in V NpGq such that T1 “ w˚-limα Sα and each Sα

is a finite linear combination of operators λGpxq, x P V . Then

supppSαT2q Ď V ¨ suppT2

for all α, and since V ¨ suppT2 is closed and T1T2 “ w˚-limαpSαT2q, it follows from(iv) that

supppT1T2q Ď V ¨ suppT2.

Now let V denote the collection of all closed neighbourhoods of suppT1 with rela-tively compact complement. Then

Ş

tV : V P Vu “ suppT1, and since suppT2 iscompact, it is easily verified that

Ş

V PVpV ¨ suppT2q “ pŞ

V PV V q ¨ suppT2.

This implies supppT1T2q Ď psuppT1qpsuppT2q. �

Lemma 2.5.7. Let T P V NpGq and u P ApGq X CcpGq. Then

supppTuq Ď psuppT qpsuppuq.

Proof. Let a R psuppT qpsuppuq and hence apsupp uq´1 X suppT “ H. Sincesuppu is compact, there exist closed neighbourhoods V of a and U of suppT suchthat V psuppuq´1XU “ H and GzU is relatively compact. By Proposition 2.5.6(iii)there exists a net pSαqα in V NpGq such that T “ w˚-limα Sα and each Sα is a finitelinear combination of operators λGpxq, x P U .

Fix α and let Sα “řn

j“1 cjλGpxjq, xj P U . Then, for any x P G,

xϕSα, pRxuqqy “

nÿ

j“1

cjxλGpxjq, pRxuqqy “

nÿ

j“1

cjupx´1j xq.

Now, if x P V then x´1j x P U´1V and hence x´1

j x R supp u. Thus xϕSα, pRxuq y “ 0

for all x P V and all α. Passing to the w˚-limit, it follows that

Tupxq “ xϕT , pRxuqqy “ 0

for all x P V . This shows that a R supppTuq and hence proves the statement of thelemma. �

The formula T qupxq “ xT, Lxuy for u P ApGq and x P G in Lemma 2.3.11(ii)implies for any μ P MpGq

rλpμquspxq “ xϕλpμq, pRxuqqy “

ż

G

prxuqqpyqdμpyq

“

ż

G

upy´1xqdμpyq “ pμ ˚ uqpxq

for all x P G, so that λpμqu “ μ ˚ u.

Lemma 2.5.8. Let T P V NpGq and suppose that supppTuq Ď suppu for allu P ApGq X CcpGq. Then T “ λI for some λ P C.

Proof. The proof is divided into several steps.We first show if U is a relatively compact open subset of G, then for any

u P ApGq X CcpGq such that u is constant on U , Tu is also constant on U . Toverify this, fix any two points a and b in U and choose an open neighbourhood V


of the identity such that V a Y V b Ď U . If x P V b, then x P U and xb´1a P U . Sothe function u´ Rb´1au P ApGq XCcpGq vanishes on V b and hence, by hypothesis,T pu ´ Rb´1auq also vanishes on V b. In particular,

Tupbq “ TRb´1aupbq “ Rb´1aTupbq “ Tupaq,

since T commutes with right translations.Next we prove the existence of a constant λ, depending only on T , with the

following property: For any relatively compact open subset U of G and any u P

ApGqXCcpGq which is identically 1 on U , the function Tu is identically λ on U . Bywhat we have seen above, for every pair pU, uq there exists a constant λpU, uq suchthat Tupxq “ λpU, uq for all x P U . Now, fix U and let u1, u2 P ApGq X CcpGq suchthat u1|U “ u2|U “ 1. Then u1 ´ u2 vanishes on U and hence so does Tu1 ´ Tu2

by hypothesis. Thus λpU, u1q “ λpU, u2q. Finally, let U1 and U2 be two relativelycompact open subsets of G and let u P ApGqXCcpGq be such that u “ 1 on U1YU2.Then

λpUj , uq “ Tupxq “ λpU1 Y U2, uq

for x P Uj , j “ 1, 2, and hence λpU1, uq “ λpU2, uq. This shows that λpU, uq doesneither depend on U nor on u.

Let λ denote the constant associated to T by the preceding discussion. Weproceed to show that T p1Cq “ λ1C for every compact subset of G with the propertythat the boundary of C has measure zero. Fix such a set C. Since Haar measureis regular, there exists an ascending sequence pVnqn of open subsets of G such thatVn Ď C˝ and

|CzVn | “ |C˝zVn | ď 1{n

for all n. For each n, we find un P ApGq X CcpGq such that 0 ď un ď 1, un “ 1 onVn and un “ 0 on GzC. Then the bounded continuous function Tun vanishes onGzC and takes the value λ on Vn. Since un Ñ 1C in L2pGq, it follows that

T p1Cq “ limnÑ8

Tun “ λ1C .

Since the characteristic functions 1C , where C is an arbitrary compact subset of G,form a total set in L2pGq, to finish the proof it suffices to show that the conclusionof the preceding paragraph also holds if the hypothesis that the boundary of C beof measure zero is dropped.

Thus let C be an arbitrary compact subset of G. Then there exists a sequencepUnqn of relatively compact open sets Un in G such that, for each n, C Ď Un,|Un| ď |C| `

1n and the boundary of Un is of measure zero. Then 1C “ limnÑ8 1Un

in L2pGq and hence T p1Cq “ limnÑ8 T p1Unq. Since we already know that T p1Un

q “

λ1Un, we conclude that T p1Cq “ λ1C , as was to be shown. �

Corollary 2.5.9. Let G be a locally compact group and a P G. If suppT “

tau, then T “ αλGpaq for some α P C.

Proof. Since, by Proposition 2.5.6(v)(4),

supppλGpa´1qT q Ď suppλGpa´1

q ¨ suppT “ teu,

we can assume that a “ e. Then supppTuq Ď supp u for all u P ApGq by Lemma2.5.7 since suppT “ teu. Lemma 2.5.8 now shows that T “ α ¨ I “ αλGpeq, asrequired. �


The preceding corollary allows a quick application to the ideal theory of Fourieralgebras. Recall that an ideal I of a commutative Banach algebra A is calledprimary if the hull of I in σpAq is a singleton.

Corollary 2.5.10. Every closed primary ideal of ApGq is maximal.

Proof. Let I be a closed ideal of ApGq such that hpIq “ txu for some x P G “

ApGq. The annihilator IK of I in V NpGq “ ApGq˚ is weakly closed in V NpGq andinvariant under the transformations T Ñ u ¨ T , u P ApGq, because I is an ideal.For T P V NpGq, let JT “ tu P ApGq : xT, uy “ 0u. Then, for T P IK, JT Ě Iand hence hpJT q Ď txu. Thus, if T P IK and T ‰ 0, then hpJT q “ txu and sosuppT “ txu. Corollary 2.5.9 implies that T is a multiple of λGpxq. This showsthat IK is 1-dimensional and therefore I is maximal and I “ kpxq. �

Corollary 2.5.11. Let x P G and u P ApGq such that upxq “ 0. Then thereexists a sequence punqn in ApGq such that }un ´ u}ApGq Ñ 0 and un vanishes onsome neighbourhood of x.

Proof. By Corollary 2.5.10, kpxq is the only closed ideal in ApGq with hull txu.The statement follows now from Section 1.1 since ApGq is regular and semisimple.

�Corollary 2.5.12. Let u P ApGq be such that upeq “ 0 and let ε ą 0. Then

there exists w P P 1pGq X CcpGq such that }uw} ď ε.

Proof. By Corollary 2.5.11 there exists v P ApGq such that }v ´ u} ď ε andv “ 0 in a neighbourhood U of e in G. Choose a compact symmetric neighbourhoodV of e such that V 2 Ď U and set

w “ }1V }´22 p1V ˚ r1V q P P pGq X CcpGq.

Then wpeq “ 1, suppw Ď V 2 and

}wu ´ wv} ď }w} ¨ }u ´ v} “ }u ´ v} ď ε.

Since suppw X supp v “ H, wv “ 0 and hence }wu} ď ε. �

2.6. The restriction map from ApGq onto ApHq

Let H be a closed subgroup of the locally compact group G. In the study ofFourier algebras, a natural question arising is whether functions in ApHq extend tofunctions in ApGq. In this section we show that this question admits an affirmativeanswer. The corresponding problem for Fourier-Stieltjes algebras is much moreinvolved and will be investigated in Chapter 7.

In the following, let H be a closed subgroup of G and let GzH denote the spaceof all right cosets of H in G, endowed with the quotient topology defined throughthe map p : G Ñ GzH,x Ñ Hx.

Lemma 2.6.1. Retain the above notation and let C be a compact, second count-able subset of G. Then there exists a Borel subset M of C such that ppMq “ ppCq

and p is one-to-one on M .

Proof. From general topology, it is known that there exist a perfect subset Tof r0, 1s and a continuous map ϕ from T onto the metric space C. For each n P N,define Tn to be the set of all t P T with the following property: for every s P T withs ď t ´ 1{n, we have ppϕpsqq ‰ ppϕptqq. Using that T is closed in r0, 1s, it is easily

2.6. THE RESTRICTION MAP FROM ApGq ONTO ApHq 67

verified that Tn is relatively open in T . Then M “Ş8

n“1 ϕpTnq is a Borel subset ofC. To show that ppMq “ ppCq, fix x P C. Since ϕpT q “ C and T is closed in r0, 1s,the real number

t “ inf ts P T : ppϕpsqq “ ppxqu P T

satisfies ppϕptqq “ ppxq. It is also clear that t P Tn for all n and hence ϕptq P M .It remains to show that ϕ is one-to-one on M . To see this, assume that y P C

is such that y ‰ x and ppyq “ ppxq and set S “ ts P T : ϕpsq “ yu. Then, sincet P Tn for all n, S Ď rt, 1s, and since S is closed in T and y ‰ x, it even followsthat S Ď rt ` δ, 1s for some δ ą 0. For every n ą 1{δ we then have S Ď T zTn andconsequently

y P ϕpSq Ď ϕpT qzϕpTnq “ CzϕpTnq Ď CzM.

This contradiction completes the proof. �Proposition 2.6.2. Let G be a second countable locally compact group and let

H be a closed subgroup of G. Then there exists a Borel set S in G with the followingproperties.

(i) S intersects each right coset of H in exactly one point.(ii) For each compact subset C of G, HC X S has a compact closure.(iii) H X S “ teu.

Moreover, there is a closed neighbourhood V of e in G such that HV “ V and V XSis relatively compact.

Proof. We first show the existence of a set S satisfying properties (i), (ii) and(iii). Choose a compact symmetric neighbourhood V of e in G. Then L “

Ť8

n“1 Vn

is an open subgroup of G and L has at most countably many right cosets in G.Since every compact subset of G is contained in the union of finitely many cosetsof L, we can find a sequence C1 Ď C2 Ď ¨ ¨ ¨ of compact subsets of G such thatevery compact subset of G is contained in some Cj . By Lemma 2.6.1, for each jthere exists a Borel subset Sj Ď Cj such that ppSjq “ ppCjq and p is one-to-oneon Sj . Observe next that the sets Sj can be chosen so that Sj Ď Sj`1 for everyj P N. Indeed, suppose that we already have arranged for S1 Ď S2 Ď ¨ ¨ ¨ Ď Sj .Then choose any Borel set Mj`1 Ď Cj`1 such that p is one-to-one on Mj`1 andppMj`1q “ ppCj`1q and set

Sj`1 “ pMj`1z p´1pppSjqqq

Ť

Sj .

Then Sj`1 is a Borel set. To see this, since Sj and Mj`1 are Borel sets, it suffices toverify that p´1pppSjqq is a Borel set. Since Sj is a Borel set in the complete metricspace Cj and p is continuous and one-to-one on Sj , ppSjq is a Borel set [170], andhence p´1pppSjqq is a Borel set as well.

Now, set S “Ť8

j“1 Sj . Then S is a Borel set satisfying (i) and (ii). Indeed,

ppSq “

8ď

j“1

ppSjq “

8ď

j“1

ppCjq “ GzH,

and since p is one-to-one on each Sj and Sj Ď Sj`1 for each j, p is one-to-one onS. To verify (ii), let psnqn be a sequence in HC X S and choose j P N such thatC Ď Cj . Then, since ppSjq “ ppCjq, for each n there exist xn P H and tn P Sj suchthat sn “ xntn. This implies that sn “ tn and, since Sj Ď Cj , the sequence ptnqn

has a convergent subsequence. This shows that HC X S is relatively compact.Finally, translating S if necessary, we can arrange for S X H “ teu.


For the remaining statement, choose an open neighbourhood U of e in G suchthat U is compact and set V “ HU . Then V is a closed neighbourhood of e in G

such that HV “ HpHUq “ HU “ V . We also have V X S “ pHUq X S, which iscompact by (ii). �

The following proposition, the proof of which is fairly long and technical, is amajor step towards Theorem 2.6.4 below.

Proposition 2.6.3. Let G, H, S and V be as in Proposition 2.6.2 and forx P G, let βpxq be the unique element in H such that x “ βpxqs for some s P S.For any complex-valued function f on H define fV on G by

fV pxq “ fpβpxqq1V pxq, x P G.

Then the following hold.

(i) If f is a measurable function on H, then fV is measurable on G.(ii) If f has compact support, then so does fV .(iii) There exists a constant c ą 0 such that f Ñ c fV is a linear isometry of

L2pHq into L2pGq.(iv) The mapping f Ñ fV is a linear isometry of L8pHq into L8pGq.(v) If f and g are in L2pHq, then for all h P H, we have

c2pfV ˚ |gV qphq “ pf ˚ qgqphq,

the convolution on the left and on the right being over G and H, respec-tively.

Proof. In the sequel, mG and mH will denote left Haar measures on G andH, respectively.

(i) Assume first that f is real-valued. Then fV is real-valued, and for everyr P R, we have

tx P G : fpβpxqq ą ru “ tx P G : βpxq P th P H : fphq ą ruu

“ th P H : fphq ą ru ¨ S.

Since f is measurable, T “ th P H : fphq ą ru is measurable in H and hence TSis a measurable subset of G. Thus f ˝ β is measurable, and so is 1V since V isclosed in G. So fV is measurable. If f is an arbitrary complex-valued measurablefunction, then standard arguments on Ref and Imf show that fV is measurable.

(ii) If fV pxq ‰ 0, then x P V and fpβpxqq ‰ 0. This implies

supp fV Ď V X β´1psupp fq “ V X psupp fqS.

If y P V X psupp fqS, then βpyq P supp f and γpyq “ βpyq´1y P HV “ V , andtherefore y P psupp fqpS X V q. This shows that

supp fV Ď V X psupp fqS Ď psupp fq ¨ S X V ,

which is compact since both supp f and S X V are compact.(iii) It will be convenient to prove the following fact first. Let f be any real-

valued measurable function on H, let δ ą 0 and suppose that

mHpth P H : fphq ą δuq ą 0.


Then mGptx P G : fV pxq ą δuq ą 0. To see this, let C be a compact neighbourhoodof e in G such that C Ď V and set D “ th P H : fphq ą δu. Since mHpDq ą 0, wehave mGpDpCH X Sqq ą 0. Because C Ď V and D Ď V ,

DpCH X Sq Ď DpV X Sq Ď DS X V,

so that mGpV X DSq ą 0 (clearly, DS X V is a measurable set). Now note that ify P DS X V , then βpyq P D and x P V . Therefore,

DS X V Ď tx P G : fV pxq ą δu,

and hence the latter set has positive measure.We next define a positive and additive functional on C`

c pHq as follows. Iff P C`

c pHq, then fV is bounded and measurable by (i) and fV has compact supportby (ii). Thus we may define

Ipfq “

ż

G

fV pxqdx.

If f ‰ 0, then for some δ ą 0 the set th P H : fphq ą δu is nonempty and open inH. The previous paragraph then shows that mGpty P G : fV pyq ą δuq ą 0, whenceIpfq ą 0. Furthermore, since HV “ V , for h P H we have

IpLhfq “

ż

G

1V pxqpLhfqpβpxqqdx “

ż

G

1V pxqfph´1βpxqqdx

“

ż

G

1V ph´1xqfpβph´1xqqdx “

ż

G

1V pxqfpβpxqqdx

“ Ipfq.

Hence I is left invariant on C`c pHq. By the uniqueness theorem for the left Haar

integral there exists a constant c ą 0 such that, for all f P C`c pHq,

ż

H

fphqdh “ c

ż

G

fV pxqdx.

From this equation, (iii) follows by routine arguments from integration theory.(iv) Since G and H are σ-compact, in both groups there is no distinction

between locally null sets and null sets. Let f P L8pHq. Then, for each δ ă }f}8,the set th P H : |fphq| ą δu is not a null set. Thus by what we have shown in thefirst paragraph of the proof of (iii), tx P G : |fV pxq| ą δu is not a null set. Sincethis holds for all δ ă }f}8, it follows that }f}8 ď }fV }8.

For the reverse inequality, let δ ă }fV }8. There is a measurable subset M of Gsuch that mGpMq ą 0 and |fV pxq| ą δ for all x P M . Then M Ď V and |fpxq| ą δfor all x P βpMq. Since M Ď βpMq Ď S, we have mGpβpMqSq ě mGpMq ą 0,which in turn implies mHpβpMqq ą 0. Therefore, the set th P H : fphq ą δu

contains βpMq which has positive measure, and this implies }f}8 ą δ. As before,since this holds for all δ ă }fV }8, we conclude that }f}8 ě }fV }8.

(v) Let f, g P L2pHq. Then fV , gV P L2pGq by (iii). Therefore, the convolutionproducts pf ˚ qgqphq and pfV ˚ |gV qpxq exist for all h P H and x P G, respectively.


For h P H, it follows from (iii) that

pfV ˚ |gV qphq “

ż

G

fV phxqgV pxqdx

“

ż

G

1V pxqfphβpxqqgpβpxqqdx

“

ż

G

1V pxqpLh´1fqpβpxqqgpβpxqqdx

“

ż

G

1V pxqpLh´1fq ¨ gqpβpxqqdx

“

ż

G

ppLh´1fqgqV pyq dy

“1

c2

ż

H

pLh´1fq ¨ gqpyqdy

“1

c2pf ˚ qgqphq.

This completes the proof of the proposition. �

With Proposition 2.6.3 at hand, we are now ready for the main result of thissection.

Theorem 2.6.4. Let G be a locally compact group and H a closed subgroup of G.For every u P ApHq there exists v P ApGq such that v|H “ u and }v}ApGq “ }u}ApHq.If u is positive definite, then v can be chosen to be positive definite.

Proof. (a) To begin with, we consider the case when G is second countable.Let u P ApHq and let f, g P L2pHq be such that u “ f ˚ qg and }u}ApHq “ }f}2}g}2.

Define v : G Ñ C by vpxq “ c2pfV ˚ |gV qpxq, x P G. Then v P ApGq and, byProposition 2.6.3(v), vphq “ pf ˚qgqphq “ uphq for all h P H. Moreover, by assertion(iii) of Proposition 2.6.3,

}v}ApGq ď c2}fV }2}gV }2 “ }f}2}g}2 “ }u}ApHq.

Since }u}ApHq “ }v|H}ApHq ď }v}ApGq, it follows that }v}ApGq “ }u}ApHq.(b) Next, suppose that G is σ-compact. Since G is a normal topological space,

u extends to some uniformly continuous function h on G. The theorem of Kakutaniand Kodaira 1.2.16 then assures that there exists a compact normal subgroup K ofG such that G{K is second countable and h is constant on cosets of G{K. Then uis constant on cosets of H X K in H. Indeed, if x, y P H are such that y´1x P K,then

upxq “ hpxq “ hpypy´1xqq “ hpyq “ upyq.

Since K is compact, HK is a closed subgroup of G. Moreover, since H is σ-compact, the map xpH X Kq Ñ xK is a topological isomorphism from H{H X Konto HK{K. Therefore we can define a function w on HK{K by wpxKq “ upxq,x P H. Then w belongs to ApHK{Kq. Since G{K is second countable, (a) applies tothe closed subgroup HK{K and w and yields the existence of some w1 P ApG{Kq

with w1|HK{K “ w and }w1}ApG{Kq “ }w}ApHK{Kq. Now define v P ApGq byvpxq “ w1pxKq. For h P H, we then have

vphq “ w1phKq “ wphKq “ uphq.


It is also clear that

}v}ApGq “ }w1}ApG{Kq “ }w}ApHK{Kq “ }u}ApHq.

(c) Now let G be an arbitrary locally compact group and let u “ f ˚ qg P ApHq.We prove the existence of a σ-compact open subgroup K of G such that f andg both vanish almost everywhere on HzK. Since CcpHq is dense in L2pHq andCcpHq “ CcpGq|H , there exist sequences pfnqn and pgnqn in CcpGq such that

}fn|H ´ f}2 Ñ 0 and }gn|H ´ g}2 Ñ 0.

Now simply take for K the open subgroup of G generated by the σ-compact setŤ8

n“1psupp fn Y supp gnq.

Then K has the required properties. It follows that u “ 0 on HzK and

u|HXK “ f |HXK ˚ qg|HXK .

Now, since K is σ-compact, by (b) u|HXK admits an extension w P ApKq with}w}ApKq “ }u|HXK}ApHXKq.

Finally, define v to be the trivial extension of w to all of G. Then v|H “ usince both functions agree on H X K and vanish on HzK. Furthermore,

}v}ApGq “ }w}ApKq “ }u|HXK}ApHXKq “ }u}ApHq,

and we are done.Following the construction performed in (a), (b) and (c), it is easily seen that

v is positive definite whenever u is positive definite, that is, if u “ f ˚ qf for some

f P L2pHq with }u}ApHq “ }f}2} qf}2. �

Corollary 2.6.5. Let G be a locally compact group and H a closed subgroupof G. Let IpHq “ tu P ApGq : u|H “ 0u. Then the restriction map u Ñ u|H

from ApGq onto ApHq induces an isometric isomorphism u` IpHq Ñ u|H from thequotient algebra ApGq{IpHq onto ApHq.

Proof. The map u ` IpHq Ñ u|H is an algebra isomorphism of ApGq{IpHq

into ApHq. Proposition 2.4.1(ii) shows that this map is onto and an isometry since

}u|H}ApHq “ inft}v}ApGq : v P ApGq, v ´ u P IpHqu “ }u ` IpHq}

for all u P ApGq. �

Let H be a closed subgroup of G and let

r : ApGq Ñ ApHq, u Ñ rpuq “ u|H

be the restriction map. We conclude this section with briefly studying the adjointmap

r˚ : ApHq˚

“ V NpHq Ñ ApGq˚

“ V NpGq

given by xr˚pSq, uy “ xS, rpuqy for u P ApGq and S P V NpHq. Since r is surjectiveby Proposition 2.4.1(ii), r˚ is injective. Recall that λG and λH denote the regularrepresentation of G and H, respectively.

Proposition 2.6.6. Let H be a closed subgroup of the locally compact groupG. The map r˚ is a w˚-w˚-continuous isomorphism from V NpHq onto V NHpGq

satisfying r˚pλHpxqq “ λGpxq for all x P H.


Proof. It is clear that r˚ is w˚-w˚-continuous. To see that r˚ is a homo-morphism, we first observe that if x P H, then r˚pλHpxqq “ λGpxq. So r˚

preserves products on the set D “ tλHpxq : x P Hu. Since multiplication in avon Neumann algebra is separately continuous in the w˚-topology, it follows thatr˚pST q “ r˚pSqr˚pT q for all S, T P V NpHq.

Observe next that r˚ also preserves involution. Indeed, if T P D then clearlyr˚pT˚q “ pr˚pT qq˚. If T P V NpHq is arbitrary, let pTαqα be a net in the linear spanof D such that Tα Ñ T in the w˚-topology. Then T˚

α Ñ T˚ in the w˚-topologyand hence

r˚pT˚

q “ limα

r˚pT˚

α q “ limα

pr˚pTαqq

˚“ pr˚

pT qq˚.

To see that r˚ is surjective, it suffices to show that X “ r˚pV NpHqq is w˚-closedin V NpGq. Since r˚ is a ˚-homomorphism of the C˚-algebra V NpHq into the C˚-algebra V NpGq, X must be norm-closed. By the open mapping theorem, the unitball X1 of X is contained in r˚pV NpHqδq for some δ ą 0, where

V NpHqδ “ tS P V NpHq : }S} ď δu.

We claim that X1 is w˚-closed. For that, let pTαqα be a net in X1 converging tosome T P V NpGq in the w˚-topology. Then, for each α, there exists Sα P V NpHqδ

such that r˚pSαq “ Tα. After passing to a subnet if necessary, we can assume thatSα Ñ S in the w˚-topology for some S P V NpHqδ. Since r˚ is w˚-w˚-continuous,it follows that Tα “ r˚pSαq Ñ r˚pSq and hence that T “ r˚pSq P X. As V NpGq1 isw˚-closed, T P X1. Thus X1 is w˚-closed, and consequently X must be w˚-closed

by the Krein-qSmulian theorem. This finishes the proof. �

2.7. Existence of bounded approximate identities

Given any nonunital Banach algebra A, it is important to know whether A atleast admits a bounded approximate identity. The theme of this section is to solvethis problem for the Fourier algebra of a locally compact group G. It turns outthat the existence of a bounded approximate identity in ApGq is equivalent to theamenability of G (Theorem 2.7.2).

Of course, the reader will be aware of that there are several different propertiesthat are equivalent to amenability of a locally compact group (Section 1.8). We aregoing to present a very much focused approach to Theorem 2.7.2 mainly using thatamenability is equivalent to that the trivial representation is weakly contained inthe left regular representation.

The following proposition provides the main step towards showing that amena-bility of G implies the existence of a bounded approximate identity.

Proposition 2.7.1. Let G be a locally compact group and let puαqα be a net inP pGq that converges to some u P P pGq uniformly on compact subsets of G. Then

limα

}puα ´ uqv}ApGq “ 0

for every v P ApGq.

Proof. Note first that since uαpeq Ñ upeq, we can assume that the netpuαpeqqα is bounded. Because P pGq X CcpGq spans a dense subspace of ApGq

and

supα

}puα ´ uqw}ApGq ď pupeq ` supα

uαpeqq}w}ApGq

2.7. EXISTENCE OF BOUNDED APPROXIMATE IDENTITIES 73

for every w P ApGq, it is sufficient to show that

limα

}puα ´ uqv}ApGq “ 0

for each v P P pGq X CcpGq. Fix such v and put w “ uv and wα “ uαv. Thenwα, w P P pGq and the supports of wα and w are all contained in the compact setK “ supp v. Since uα Ñ u uniformly on K, we have }wα ´ w}8 Ñ 0. Sincew P P pGqXCcpGq, λGpwq is a positive bounded operator on L2pGq and there existsg P L2pGq such that

w “ g ˚ rg and g ˚ f “ λGpwq1{2f

for all f P CcpGq. Similarly, there exist functions gα P L2pGq such that

wα “ gα ˚ Ăgα and gα ˚ f “ λGpwαq1{2f

for all f P CcpGq.We are going to show that }gα ´ g}2 Ñ 0. For f P CcpGq, we have }Ryf}2 “

Δpyq1{2}f}2 and therefore, using vector-valued integration,

}pwα ´ wq ˚ f}2 “

›

›

›

›

x Ñ

ż

G

rwαpxyq ´ wpxyqsfpy´1q dy

›

›

›

›

2

“

›

›

›

›

x Ñ

ż

G

rwαpyq ´ wpyqsΔpy´1qLy´1fpxq dy

›

›

›

›

2

ď

ż

G

|wαpyq ´ wpyq| ¨ }f}2 dy

“

ż

K

|wαpyq ´ wpyq| ¨ }f}2 dy

ď }wα ´ w}8}f}2|K|.

This implies }λGpwαq ´ λGpwq} Ñ 0 and hence also that c “ supα }λGpwαq} ă 8.Now, employing the continuous functional calculus for C˚-algebras and approxi-mating the function

?t through polynomials uniformly on the interval r0, cs, we

conclude that also

}λGpwαq1{2

´ λGpwq1{2

} Ñ 0.

This in turn yields

limα

}pwα ´ wq ˚ g}2 “ limα

}pλGpwαq1{2

´ λGpwq1{2

qg}2 “ 0

for all g P CcpGq, and hence also

limα

xgα ´ g, f ˚ hy “ limα

xpgα ´ gq ˚ h˚, fy “ 0

for all f, h P CcpGq. Since CcpGq ˚ CcpGq is dense in L2pGq and supα }gα}22 “

supα wαpeq ă 8, it follows that

limα

xgα, fy “ xg, fy

for all f P L2pGq. Since also }gα}22 “ wαpeq Ñ wpeq “ }g}22, we get

limα

}gα ´ g}22 “ 2 }g}

22 ´ 2 lim

αxgα, gy “ 0.


This finally implies that

}uαv ´ uv}ApGq “ }wα ´ w}ApGq

“ }gα ˚ rgα ´ g ˚ rg}ApGq

“ }gα ˚ prgα ´ rgq ` pgα ´ gq ˚ rg}ApGq

ď }gα ´ g}2p}gα}2 ` }g}2q,

which converges to 0, as was to be shown. �

We are now ready for the main result of this section.

Theorem 2.7.2. For a locally compact group G, the following three conditionsare equivalent.

(i) G is amenable.(ii) ApGq has an approximate identity bounded by 1 and consisting of com-

pactly supported positive definite functions.(iii) ApGq has a bounded approximate identity.

Proof. (i) ñ (ii) Since G is amenable, by Theorem 1.8.18 the trivial repre-sentation of G is weakly contained in the left regular representation. Thus, givenany K P KpGq, the collection of all compact subsets of G, and ε ą 0, there ex-ists uK,ε P P pGq associated with λG such that |uK,εpxq ´ 1| ď ε for all x P K.Clearly, since CcpGq is dense in L2pGq, we can in addition assume that uK,ε hascompact support. Now, order the set of all pairs α “ pK, εq, K P K, ε ą 0, byα ě α1 “ pK 1, ε1q if K Ě K 1 and ε ď ε1. Then the net puαqα satisfies the hypothesesof Proposition 2.7.1 with u “ 1G and therefore forms an approximate identity forApGq. Finally, since uαpeq Ñ 1, replacing uα by uαpeq´1uα, we can assume that1 “ uαpeq “ }uα}ApGq. This shows that (ii) holds.

It remains to prove (iii) ñ (i). Suppose that puαqα is an approximate identityfor ApGq with }uα}ApGq ď c ă 8 for all α. We are going to show that }λGpfq} “

}f}1 for every f P C`c pGq. By Proposition 1.8.20 and Theorem 1.8.18 , this property

implies that G is amenable.Let K be any compact subset of G, choose a compact neighbourhood V of e in

G, and put

u “ |V |´1

p1V ˚ 1V ´1Kq P ApGq.

Then, for x P K,

upxq “ |V |´1

ż

G

1V pyq1V ´1Kpy´1xqdy “ 1.

Since }uαu ´ u}ApGq Ñ 0, it follows that uαu Ñ u uniformly on K. Thus, givenε ą 0, there exists an index α such that Repuαpxqq ě 1 ´ ε for all x P K. We nowapply the preceding with K “ supp f , where f P C`

c pGq. Then

Re xuα, fy “

ż

G

fpxqRepuαpxqqdx ě p1 ´ εq}f}1.

On the other hand, we have

|xuα, fy| “ |xλGpfq, uαy| ď c }λGpfq}.

Since ε ą 0 is arbitrary, we conclude that }f}1 ď c }λGpfq} for every f P C`c pGq.

Replacing f with the n-fold convolution product fn in the preceding calculations,


it follows that

}f}n1 “ }fn

}1 ď c }λGpfnq} ď c }λGpfq}

n

and therefore

}f}1 ď }λGpfq} ¨ limnÑ8

c1{n“ }λGpfq} ď }f}1.

This finishes the proof of (iii) ñ (i). �

Theorem 2.7.2 raises the question of whether the Fourier algebra ApGq of anonamenable locally compact group G possesses an approximate identity whichis bounded in some norm weaker than the ApGq-norm or at least an unboundedapproximate identity. This appears to be a very difficult problem, which we aregoing to touch in Chapter 5. However, it should be mentioned that even the weakestpossible variant, namely whether u P uApGq holds for every u P ApGq, appears tobe a widely open problem.

We continue with several interesting applications of Theorem 2.7.2.

Corollary 2.7.3. Let G be an amenable locally compact group and C a com-pact subset of G. Then, given ε ą 0, there exists u P ApGq XCcpGq such that u “ 1on C and }u} ď 1 ` ε.

Proof. Choose 0 ă δ ă 1 such that p1 ` 2δqp1 ´ δq´1 ď 1 ` ε. Since ApGq

is regular, there exists w P ApGq such that w “ 1 on C. Because ApGq has anapproximate identity bounded by 1, by Cohen’s factorization theorem [126] we candecompose w as w “ vw1, where }v} ď 1 and }w1´w} ď δ. Let u1 “ v´vpw´w1q P

ApGq. Then u1 “ 1 on C and }u1} ď 1 ` δ. Finally, choose u2 P ApGq X CcpGq

such that }u2 ´ u1} ď δ and define u P ApGq by the norm-convergent sum

u “ u2 ¨

8ÿ

n“0

pu1 ´ u2qn.

Then u has compact support, u “ 1 on C and

}u} ď}u2}

1 ´ }u1 ´ u2}ď

1 ` 2δ

1 ´ δď 1 ` ε,

as required. �

Corollary 2.7.4. Let H be a proper closed subgroup of a locally compact Gsuch that the ideal IpHq has a bounded approximate identity. Then H is amenable.

Proof. By Theorem 2.7.2, it suffices to show that ApHq has a bounded ap-proximate identity. Choose any x P GzH. The ideal IpxHq also has a boundedapproximate identity, say puαqα. For each α, let vα “ uα|H P ApHq. Then}vα}ApHq ď }uα}ApGq, and since ApHq X CcpHq is dense in ApHq, it is enoughto verify that }vαv ´ v}ApHq Ñ 0 for every v P ApHq X CcpHq. Fix such a v andlet C “ supp v and choose a neighbourhood V of C such that V X xH “ H. SinceApGq is regular, there exists u0 P ApGq such that u0|C “ 1 and supp u0 Ď V .Since the restriction map ApGq Ñ ApHq is surjective, there exists u P ApGq withu|H “ v. Then w “ u0u belongs to IpxHq and w|H “ v. Finally,

limα

}vαv ´ v}ApHq ď limα

}wαw ´ w}ApGq “ 0,

which completes the proof. �


Theorem 2.7.5. For a locally compact group G, G ‰ teu, the following areequivalent.

(i) G is amenable.(ii) The ideal I “ Ipteuq has a bounded approximate identity.(iii) There exists a proper closed subgroup H of G such that IpHq has a

bounded approximate identity.(iv) There exists an amenable proper closed subgroup H of G such that IpHq

has a bounded approximate identity.

Proof. (i) ñ (ii) Let puαqα be an approximate identity for ApGq with }uα} ď 1for all α and let pWβqβ be a neighbourhood basis of e. For each β, there existswβ P P pGq such that supp wβ Ď Wβ and wβpeq “ 1. For each pair pα, βq, let

vα,β “ uα ´ uαpeqwβ P ApGq.

Then vα,β P Ipteuq and

}vα,β} ď }uα} ` |uαpeq| ¨ }wβ} ď 2.

Now let u P Ipteuq and ε ą 0 be given. Since the singleton teu is set of synthesis(see Chapter 6), there exists v P jpteuq such that }v ´ u} ď ε. For β large enough,Wβ X supp v “ H and hence wβv “ 0. It follows that

}vα,βu ´ u} ď }vα,βpu ´ vq} ` }vα,β v ´ v} ` }v ´ u}

ď 3ε ` }vα,βv ´ v} “ 3ε ` }uαv ´ v}

since wβv “ 0. Since puαqα is an approximate identity for ApGq and ε ą 0 isarbitrary, it follows that the net pvα,βqα,β is an approximate identity for Ipteuq.

(ii) ñ (iii) being trivial, assume that (iii) holds. Then the subgroup H mustbe amenable by Corollary 2.7.4, so (iv) holds.

(iv) ñ (i) Since H is amenable, ApHq has a bounded approximate iden-tity. Then ApGq{IpHq, being isometrically isomorphic to ApHq (Corollary 2.7.4),also has a bounded approximate identity. Since both IpHq and ApGq{IpHq havebounded approximate identities, the same is true for ApGq (Proposition 1.1.5).Therefore, G is amenable (Theorem 2.7.2). �

Corollary 2.7.4 and Theorem 2.7.5 suggest the question of which impact on thegroup G the existence of just some closed ideal of ApGq with bounded approximateidentity might have. The remaining part of this section is devoted to clarify thisquestion.

Lemma 2.7.6. Let I be a closed ideal of ApGq with bounded approximate iden-tity. Then the interior hpIq˝ of the hull hpIq is closed in G and the boundary BphpIqq

has measure zero. Moreover, 1GzhpIq˝ P BλpGq “ C˚λ pGq˚.

Proof. Let E “ hpIq and let puαqα be a bounded approximate identity forI. After passing to a subnet if necessary, we can assume that w˚ ´ limα uα “ ufor some u P C˚

λ pGq˚ “ BλpGq. We claim that u “ 1GzE˝ . To see this, let v P I,

f P L1pGq and ε ą 0. Then

xuv, fy “ xu, vfy “ limα

xuα, vfy “ limα

xuαv, fy.

Now choose α such that |xuv, fy ´ xuαv, fy| ď ε and }uαv ´ v} ď ε. Then

|xuv, fy ´ xv, fy| ď |xuv, fy ´ xuαv, fy| ` |xuαv, fy ´ xv, fy|

ď ε ` }f}1}uαv ´ v} ď εp1 ` }f}1q.


Since ε ą 0 is arbitrary, it follows that xuv, fy “ xv, fy for every f P L1pGq, andhence uv “ v for every v P I. This implies that upxq “ 0 on E˝ and upxq “ 1 onGzE, and therefore u “ 1 on GzE˝. Thus u “ 1GzE˝ . In particular, E˝ is closed inG.

It remains to observe that |BpEq| “ 0. For a contradiction, assume that |BpEq| ą

0. Choose a Borel subset V of BpEq such that 0 ă |V | ă 8, and let f “ 1V . Then,since u “ 1 on BpEq and puαqα Ď I,

0 ă |V | “ xu, fy “ limα

xuα, fy “ 0.

This contradiction finishes the proof. �

Lemma 2.7.7. Let I be a non-zero closed ideal in ApGq with bounded approxi-mate identity. Then either G is amenable or hpIq has positive measure.

Proof. Assume that |hpIq| “ 0. To conclude that G is amenable, it sufficesto show that }f}1 “ }λGpfq} for every f P CcpGq with f ě 0, where λG denotesthe left regular representation of G (Section 1.8). Let M be a norm bound for anapproximate identity of I.

Let f P C`c pGq and let C be any compact subset of G such that supp f Ď C,

and let ε ą 0. Since |hpIq| “ 0, there exists an open neighbourhood V of hpIq suchthat |V | ¨ }f}8 ¨ M ď ε. Since the bounded approximate identity of I converges to1 uniformly on compact subsets, which are disjoint from hpIq, there exists u P Isuch that

inf tRepupxqq : x P CzV u ě 1 ´ ε.

Then |xu, fy| ď }u} ¨ }λGpfq} ď M ¨ }λGpfq}. On the other hand,

Repxu, fyq “

ż

C

Repupxqqfpxqdx

ě p1 ´ εq}f}1 `

ż

V

Repupxqqfpxqdx

ě p1 ´ εq}f}1 ´ |V | ¨ }f}8 ¨ M ě p1 ´ εq}f}1 ´ ε.

Since ε ą 0 is arbitrary, it follows that

M ¨ }λGpfq} ě |xu, fy| ě }f}1

for every f P C`c pGq. Thus, for g P C`

c pGq and any n P N,

}g}n1 “ }gn}1 ď M}λGpgnq} ď M ¨ }λGpgq}

n,

and hence }g}1 ď }λGpgq}, as required. �

Corollary 2.7.8. Suppose that G is connected and I is a nonzero closed idealof ApGq with bounded approximate identity. Then G is amenable.

Proof. Assuming that G is not amenable, |hpIq| ą 0 by Lemma 2.7.7. Onthe other hand, by Lemma 2.7.6, |BphpIqq| “ 0 and hpIq˝ is closed in G. Since G isconnected, either hpIq˝ “ H or hpIq “ G. However, since I ‰ t0u, hpIq ‰ G. Thus0 ă |hpIq| “ |hpIq˝ Y BphpIqq| “ |BphpIqq| “ 0, a contradiction. �

Theorem 2.7.9. Suppose that ApGq has a nonzero closed ideal which possessesa bounded approximate identity. Then G has an amenable open subgroup.


Proof. We first observe that G can be assumed to be almost connected. Infact, since I ‰ t0u, there exists x P GzhpIq and, by translating if necessary, wecan assume that x P G0, the connected component of the identity. Fix an opensubgroup H of G such that H{G0 is compact. Then 1H ¨ I can be viewed as aclosed ideal of ApHq, which is nonzero and has a bounded approximate identity. IfG0 has been shown to be amenable, then H is amenable as well because H{G0 iscompact. Thus we can assume that G is almost connected.

Recall that the restriction map r : ApGq Ñ ApG0q, u Ñ u|G0is surjective

and norm decreasing. Therefore, J “ rpIq is a closed ideal of ApG0q and J has abounded approximate identity. Finally, J ‰ t0u since x P G0zhpIq. Corollary 2.7.8now implies that G0 is amenable. �

In Section 6.5, employing operator space theory, we shall explicitly describe allthe closed ideals of ApGq when G is amenable in terms of the closed coset ring ofG.

2.8. The subspaces AπpGq of BpGq

In this section we associate to any unitary representation π of the locally com-pact group G a closed linear subspace AπpGq of BpGq and present several resultsabout these spaces, which will be used to determine BpGq for some specific groupsG (Section 2.9) and also in Chapters 3 and 4.

Definition 2.8.1. Let π be a unitary representation of G. Let

AπpGq “ spantϕξ,η : ξ, η P Hpπqu}¨}

Ď BpGq,

where ϕξ,ηpxq “ xπpxqξ, ηy, x P G. The space AπpGq is often called the Fourierspace associated with the representation π. Moreover, let

V NπpGq “ tπpxq : x P Gu2

“ spantπpxq : x P Guw˚

Ď BpHpπqq.

Note that when π “ λG, then the set ApGq of all coordinate functions of λG isalready a closed linear subspace of BpGq and hence ApGq “ AλG

pGq.If σ is a subrepresentation of π, then AσpGq is a subspace of AπpGq by the very

definition. For a more precise statement, see Lemma 2.8.3 below. We first identifythe dual space of AπpGq.

Lemma 2.8.2. For any representation π of G, the dual space AπpGq˚ of AπpGq

is isometrically isomorphic to V NπpGq.

Proof. Let BpHpπqq˚ denote the norm closure in BpHpπqq˚ of the linear spanof all linear functionals on BpHpπqq of the form

ϕξ,ηpT q “ xTξ, ηy, T P BpHpπqq, ξ, η P Hpπq.

Then, as is well known, BpHpπqq˚ is the unique predual of BpHpπqq. Moreover,let Eπ “ tϕ|V NπpGq : ϕ P BpHpπqq˚u; then E˚

π “ V NπpGq. For each ϕ P Eπ,ϕ ˝ π P AπpGq and

}ϕ|V NπpGq} “ sup

#ˇ

ˇ

ˇ

ˇ

ˇ

nÿ

j“1

λjϕpπpxjqq

ˇ

ˇ

ˇ

ˇ

ˇ

:

›

›

›

›

›

nÿ

j“1

λjπpxjq

›

›

›

›

›

ď 1

+

“ }ϕ ˝ π}

by Lemma 2.1.8. �

2.8. THE SUBSPACES AπpGq OF BpGq 79

Let V be a closed subspace of BpGq “ C˚pGq˚, it follows readily from [270,Theorem 2.7, p. 123] that V is invariant, that is, f ¨ V Y V ¨ f Ď V for everyf P C˚pGq, if and only if V is left and right translation invariant. In this casethere exists a unique central projection p in C˚pGq˚˚ such that V “ BpGq ¨ p[270, Theorem 2.7(iii), p. 123]. Since AπpGq is two-sided translation invariant, itis an invariant subspace of BpGq. We shall see next that conversely every closedinvariant subspace of BpGq is of this form.

Lemma 2.8.3. Let π be a unitary representation of G.

(i) If σ is a subrepresentation of π, then there exists a unique central pro-jection P in

tT P V NπpGq : T ¨ AπpGq Y AπpGq ¨ T Ď AπpGqu Ď BpHpπqq

such that AσpGq “ P ¨ AπpGq.(ii) If V is a closed translation invariant subspace of AπpGq, then V “ AσpGq

for some subrepresentation σ of π.

Proof. (i) follows from [270, Theorem 2.7(iii), p. 123].(ii) Given V , there exists a central projection P in V NπpGq such that V “

P ¨ AπpGq. Let L “ P pHpπqq and define σ by σpxq “ πpxq|L, x P G. Then σ is asubrepresentation of π on L and AσpGq “ V . �

The following theorem gives an explicit description of the functions in AπpGq.

Theorem 2.8.4. Let π be a representation of the locally compact group G.

(i) Let pξnqn and pηnqn be sequences in Hpπq such thatř8

n“1 }ξn}¨}ηn}n ă 8.Then

upxq “

8ÿ

n“1

xπpxqξn, ηny, x P G,

defines an element of AπpGq and }u} ďř8

n“1 }ξn} ¨ }ηn}n.(ii) For each u P AπpGq there exist sequences pξnqn and pηnqn in Hpπq such

that

up¨q “

8ÿ

n“1

xπp¨qξn, ηny and }u} “

8ÿ

n“1

}ξn} ¨ }ηn}n.

Proof. (i) is clear since the series defining upxq is absolutely convergent and}xπp¨qξ, ηy} ď }ξ} ¨ }η} for each ξ, η P Hpπq. (ii) As in Section 2.1, we now use thepolar decomposition of elements in the predual of a von Neumann algebra. Forevery u P AπpGq “ V NπpGq˚, there exist a partial isometry V P V NπpGq and anelement |u| P AπpGq such that u “ V ¨ |u|, }u} “ }|u|} and |u| defines a positivenormal linear functional on V NπpGq. Since the linear functional |u| is positive,there exists a sequence pηnqn in Hpπq such that |u|pxq “

ř8

n“1xπpxqηn, ηny for all

x P G [59, Theorem 1, p.54]. It follows that }|u|} “ |u|peq “ř8

n“1 }ηn}2. Now,

upxq “ pV ¨ |u|qpxq “

8ÿ

n“1

xπpxqV ηn, ηny,

and hence

}|u|} “ }u} ď

8ÿ

n“1

}V ηn} ¨ }ηn} ď

8ÿ

n“1

}ηn} ¨ }ηn} “ }|u|}.

Thus the stated decomposition follows by setting ξn “ V ηn. �


Lemma 2.8.5. Let H and G be locally compact groups, φ : H Ñ G a contin-uous homomorphism and jpfq “ f ˝ φ for every function f on G. Let π be anyrepresentation of G.

(i) Aπ˝φpHq “ jpAπpGqq.(ii) For every v P Aπ˝φpHq there exists u P AπpGq such that v “ jpuq and

}v}BpHq “ }u}BpGq.

Proof. (i) By Theorem 2.8.4, AπpGq consists precisely of those functions u inBpGq for which there exist sequences pξnqn and pηnqn in Hpπq such that

8ÿ

n“1

}ξn} ¨ }ηn} ă 8 and upxq “

8ÿ

n“1

xπpxqξn, ηny

for all x P G. Consequently, jpAπpGqq coincides with the set of all functions of theform vpyq “

ř8

n“1xπpφpyqqξn, ηny, y P H. This shows that jpAπpGqq “ Aπ˝φpHq.(ii) Given v P Aπ˝φpHq, by Theorem 2.8.4 there exist sequences pξnqn and pηnqn

in Hpπq such that vpyq “ř8

n“1xπpφpyqqξn, ηny for all y P H and }v} “ř8

n“1 }ξn} ¨

}ηn}. Thus v “ jpuq for some u P AπpGq with }u} ďř8

n“1 }ξn} ¨ }ηn} “ }jpuq}. Onthe other hand, by Theorem 2.2.1(ii) we always have }jpuq} ď }u}. �

Remark 2.8.6. (1) Let H be the group G equipped with a stronger locallycompact topology than the given one (e.g., the discrete topology), and let φ : H Ñ

G be the identity map. Then the preceding lemma tells us that AπpGq “ Aπ˝φpHq.(2) Let H be a closed subgroup of G and φ : H Ñ G the embedding. Then,

by Lemma 2.8.5, Aπ˝φpHq “ AπpGq|H . In particular, AπpGq|H is closed in BpHq

for every representation π of G. Taking for π the universal representation of G, weconclude that BpGq|H is closed in BpGq.

We now turn to various properties of the assignment π Ñ AπpGq.

Lemma 2.8.7. Let π and ρ be representations of G. Then AπpGqXAρpGq “ t0u

if and only π and ρ are disjoint.

Proof. Suppose that π and ρ are not disjoint. Then there exists a subrepre-sentation σ of π with Hpσq ‰ t0u, which is equivalent to a subrepresentation of ρ,implemented by a unitary map U : Hpσq Ñ φpHpσqq Ď Hpρq. Then

xσpxqξ, ηy “ x Upσpxqξq, Uηy “ xρpxqpUξq, Uηy

for all ξ, η P Hpσq and x P G. This contradicts AπpGq X AρpGq “ t0u.For the converse, assume that π and ρ are disjoint and let σ “ π ‘ ρ. Let Pπ

and Pρ be the central projections in V NσpGq provided by Lemma 2.8.3. Then Pπ

and Pρ are the orthogonal projections from Hpσq onto Hpπq and Hpρq, respectively.Since π and ρ are disjoint, it follows that PπPρ “ 0, and this implies that

AπpGq X AρpGq “ PπAσpGq X PρAσpGq “ PπAσpGq X pI ´ PπqPρAσpGq

Ď PπAσpGq X pI ´ PπqAσpGq “ t0u,

as was to be shown. �

Proposition 2.8.8. Let pπιqιPI be a family of unitary representations of G andπ “ ‘ιPIπι their direct sum. Then AπpGq consists precisely of those functions inBpGq which can be written as u “

ř

ιPI uι, where uι P AπιpGq and

ř

ιPI }uι} ă 8.

2.8. THE SUBSPACES AπpGq OF BpGq 81

Proof. Since each AπιpGq is a subspace of AπpGq, every family puιqιPI sat-

isfying uι P AπιpGq and

ř

ιPI }uι} ă 8, defines an element of AπpGq by settingupxq “

ř

ιPI uιpxq for x P G.Conversely, let u P AπpGq be given. Then, by Theorem 2.8.4, there exist

sequences pξnqn and pηnqn in Hpπq such thatř8

n“1 }ξn} ¨ }ηn} ă 8 and

upxq “

8ÿ

n“1

xπpxqξn, ηny, x P G.

Since Hpπq “ ‘ιPIHpπιq, ξn and ηn can be decomposed as ξn “ř

ιPI ξnιandηn “ř

ιPI ηnι, where ξnι, ηnι P Hpπιq and

}ξn}2

“

ÿ

ιPI

}ξnι}2 and }ηn}

2“

ÿ

ιPI

}ηnι}2.

This implies, for each x P G and n P N,

xπpxqξn, ηny “ÿ

ιPI

xπιpxqξnι, ηnιy.

Now the family of complex numbers pxπιpxqξnι, ηnιyqιPI,nPN is absolutely summable.Indeed, since |xπιpxqξnι, ηnιyq| ď }ξnι} ¨ }ηnι} and

ÿ

ιPI

}ξnι} ¨ }ηnι} ď

˜

ÿ

ιPI

}ξnι}2

¸1{2˜ÿ

ιPI

}ηnι}2

¸1{2

,

it follows that8ÿ

n“1

˜

ÿ

ιPI

}ξnι} ¨ }ηnι}

¸

ď

8ÿ

n“1

}ξn} ¨ }ηn} ă 8.

Thus we can write, for x P G,

upxq “ÿ

ιPI

˜

8ÿ

n“1

xπιpxqξnι, ηnιy

¸

.

Sinceř8

n“1 }ξnι} ¨ }ηnι} ă 8, we can now define uι P AπιpGq by

uιpxq “

8ÿ

n“1

xπιpxqξnι, ηnιy.

Then }uι} ďř8

n“1 }ξnι} ¨ }ηnι} and therefore

ÿ

ιPI

}uι} ď

8ÿ

n“1

˜

ÿ

ιPI

}ξnι} ¨ }ηnι}

¸

ď

8ÿ

n“1

˜

ÿ

ιPI

}ξnι}2

¸1{2˜ÿ

ιPI

}ηnι}2

¸1{2

“

8ÿ

n“1

}ξn} ¨ }ηn}.

This finishes the proof. �

Proposition 2.8.9. Let π and ρ be disjoint representations of G. Then

(i) Aπ‘ρpGq “ AπpGq ‘1 AρpGq.(ii) Vπ‘ρpGq “ VπpGq ‘8 VρpGq.


Proof. (i) We know from Lemma 2.8.7 that AπpGq X AρpGq “ t0u. On theother hand, Aπ‘ρpGq “ AπpGq ` AρpGq by Proposition 2.8.8. It thus remains toshow that if u P Aπ‘ρpGq, u “ u1 ` u2, where u1 P AπpGq and u2 P AρpGq, then}u} ě }u1} ` }u2}.

To that end, let σ “ π ‘ ρ and ξ, η P Hpσq such that upxq “ xσpxqξ, ηy for allx P G and }u} “ }ξ} ¨ }η}. Then, with Pπ and Pρ as in the proof of Lemma 2.8.7,

u1pxq “ xσpxqPπξ, Pπηy and u2pxq “ xσpxqPρξ, Pρηy

for all x P G. Since Pπ ` Pρ “ IHpσq,

}u1} ` }u2} ď }Pπξ} ¨ }Pπη} ` }Pρξ} ¨ }Pρη}

ď`

}Pπξ}2

` }Pπη}2˘1{2 `

}Pρξ}2

` }Pρη}2˘1{2

“ }ξ} ¨ }η} “ }u}.

(ii) follows from (i) and Lemma 2.8.2. since

Aπ‘ρpGq˚

“ pAπpGq ‘1 AρpGqq˚

“ VπpGq ‘8 VρpGq “ Vπ‘ρpGq

by disjointness of π and ρ. �

Corollary 2.8.10. Let V be the closed linear span of all coefficient func-tions of those unitary representations of G which are disjoint from the left regularrepresentation λ of G. Then V “ AσpGq for some representation σ of G andBpGq “ AλG

pGq ‘1 AσpGq.

Proof. Clearly, V is translation invariant. Therefore, by Lemma 2.8.3, V “

AσpGq for some representation σ of G. The second statement follows from Propo-sition 2.8.9. �

Proposition 2.8.11. Let pπιqιPI be a family of pairwise disjoint representationsof G and let σ “ ‘ιPIπι and u P AσpGq. Then there exists a unique decompositionu “

ř

ιPI uι, where uι P AπιpGq. Moreover, }u} “

ř

ιPI }uι}.

Proof. To show the uniqueness, fix λ P I and set ρ “ř

ι‰λ πι. Then πλ andρ are disjoint and hence AσpGq “ Aπλ

pGq ‘ AρpGq by Proposition 2.8.9. Let u “ř

ιPI uι be any decomposition of u as in Proposition 2.8.8, and let v “ř

ιPI,ι‰λ uι.

Then u “ uλ ` v, uλ P AπλpGq, v P AρpGq, and this decomposition is unique. Thus

uλ is uniquely determined for every λ P I.Since }u} ď

ř

ιPI }uι}, it suffices to show that }u} ěř

ιPJ }uι} for every finitesubset J of I. Let π “ ‘ιPJπι and ρ ‘ιPIzJ πι, and let v “

ř

ιPJ uι P AπpGq andw “

ř

ιPIzJ uι P AρpGq. Since π and ρ are disjoint, }u} “ }v} ` }w} by Proposition

2.8.9. Moreover, as J is finite and the π, ι P I, are pairwise disjoint, repeatedapplication of Proposition 2.8.9 yields }v} “

ř

ιPJ }uι} and hence }u} ěř

ιPJ }uι},as required. �

We conclude this section by answering the natural question of when, for tworepresentations π and ρ of G, the spaces AπpGq and AρpGq coincide. To that end,we have to introduce the following notion. Two representations π and ρ of G arecalled quasi-equivalent if there is no subrepresenation of π which is equivalent toa subrepresentation of ρ and no subrepresentation of ρ which is equivalent to asubrepresentation of π. Obviously, equivalence implies quasi-equivalence, and thetwo notions coincide for irreducible representations.

2.9. SOME EXAMPLES 83

Proposition 2.8.12. Let π and ρ be representations of G. Then AπpGq “

AρpGq if and only if π and ρ are quasi-equivalent.

Proof. By [60, Proposition 5.3.1], π and ρ are quasi-equivalent if and only ifthere exists an isomorphism φ from V NπpGq onto V NρpGq such that ρpfq “ φpπpfqq

for every f P L1pGq.Suppose first that AπpGq “ AρpGq. Then AπpGq˚ “ AρpGq˚, and hence there

exists a Banach space isomorphism φ from V NπpGq onto V NρpGq. We have to showthat φ is an algebra isomorphism. For T P V NπpGq, the operator φpT q satisfiesxφpT q, uy “ xT, uy for every u P AπpGq “ AρpGq. Thus

xφpπpfqq, uy “ xπpfq, uy “

ż

G

fpxqupxqdx “ xρpfq, uy

for every f P L1pGq, and hence φpπpfqq “ ρpfq. For S “ πpfq and T “ πpgq,f, g P L1pGq, it follows that

φpST q “ φpπpfqπpgqq “ φpπpf ˚ gqq “ ρpf ˚ gq “ φpSqφpT q.

Now, πpL1pGqq is ultraweakly dense in V NπpGq and the map pS, T q Ñ ST is sep-arately continuous for the ultraweak topologies. Moreover, the map φ remainsto be continuous when V NπpGq and V NρpGq are endowed with their ultraweaktopologies, because these topologies equal the weak topologies σpV NπpGq, AπpGqq

and σpV NρpGq, AρpGqq, respectively. From these facts we conclude that φpST q “

φpSqφpT q for all S, T P V NπpGq. Thus φ is an algebra isomorphism, and conse-quently π and ρ are quasi-equivalent by the criterion mentioned above.

Conversely, assume that π and ρ are quasi-equivalent, and let φ be an algebraisomorphism from V NπpGq onto V NρpGq satisfying φpπpfqq “ ρpfq for every f P

L1pGq. Since the von Neumann algebras V NπpGq and V NρpGq are isomorphic,so are their preduals. Thus there exists an isometry φ˚ : AπpGq Ñ AρpGq whichis defined by the following property: xu, T y “ xφ˚puq, φpT qy for u P AπpGq andT P V NπpGq. For every f P L1pGq, it follows that

ż

G

upxqfpxqdx “ xu, πpfqy “ xφ˚puq, φpπpfqqy “ xφ˚puq, ρpfqy

“

ż

G

φ˚puqpxqfpxqdx.

Since u and φ˚puq are continuous, this equation implies u “ φ˚puq. This showsthat AπpGq “ AρpGq. �

2.9. Some examples

When G is a noncompact locally compact abelian group then, according to

common understanding, the spectrum of BpGq “ Mp pGq is an intractable object.However, there are locally compact groups G, actually certain semidirect productsof abelian groups with compact groups, for which BpGq turns out to be an extensionof ApGq by another Fourier algebra and consequently σpBpGqq can be determined.The situation is as follows.


Let G be a semidirect product G “ N �K, where(1) K is a compact group and N is a locally compact abelian group and both

are second countable, and

(2) the dual space pG of G is countable and decomposes as

pK ˝ q Y tπk : k P Nu,

where q : G Ñ K is the quotient map and each πk is a subrepresentation of the leftregular representation of G.

Proposition 2.9.1. Let G “ N �K as above. Then

BpGq “ ApKq ˝ q ` ApGq.

Proof. Let u P BpGq, so that u “ xπp¨qξ, ηy for some unitary representationπ of G and ξ, η P Hpπq. By condition (2) above, π is completely decomposable.Thus

π “ p‘σPxK

mσpσ ˝ qqq ‘ p‘8k“1nkπkq,

where mσ and nk denote the multiplicity of σ ˝q and πk as subrepresentations of π.

For σ P pK and k P N, let Pσ and Pk denote the orthogonal projections associatedwith mσpσ ˝ qq and nkπk, respectively. By Proposition 2.8.11, there is a uniquedecomposition

upxq “

ÿ

σPxK

xmσpσ ˝ qqpxqPσξ, Pσηy `

8ÿ

k“1

xnkπkpxqPkξ, Pkηy.

By Proposition 2.8.11, this is an �1-direct sum, that is,

}u}BpGq “

ÿ

σPxK

}vσ}BpGq `

8ÿ

k“1

}wk}BpGq,

where vσ “ xmσpσ ˝ qqpxqPσξ, Pσηy and wk “ xnkπkpxqPkξ, Pkηy. Now each vσ liesin BpKq ˝ q and each wk belongs to ApGq. So

u1 “

ÿ

σPxK

vσ P ApKq ˝ q and u2 “

8ÿ

k“1

wk P ApGq,

and u “ u1 ` u2. �

For locally compact groups G as above the spectrum of BpGq and its topologycan be described as follows.

Proposition 2.9.2. Let G “ N �K as above with BpGq “ ApKq ˝ q ` ApGq

and N noncompact. Then G and K embed topologically into σpBpGqq by x Ñ ϕx,where ϕxpvq “ vpxq for v P BpGq, and a Ñ ψa, where ψapv ˝ q ` uq “ vpaq forv P ApKq and u P ApGq, respectively. Moreover, σpBpGqq “ G Y K, G is openin σpBpGqq and K is closed in σpBpGqq, and a net pϕxα

qα, xα “ yαaα, yα P N ,aα P K, converges to ψa for some a P K if and only if aα Ñ a in K and yα Ñ 8

in N .

Proof. Since ApGq is an ideal in BpGq and ApKq ˝ q “ BpGq{ApGq, it is clearfrom Theorem 2.3.7 and from general Gelfand theory that x Ñ ϕx and a Ñ ψa

are both topological embeddings of G and of K into σpBpGqq, respectively, thatσpBpGqq “ G Y K, G is open in σpBpGqq and K is closed in σpBpGqq.

2.9. SOME EXAMPLES 85

Now let pxαqα be a net in G such that ϕxαÑ ψa for some a P K, and let

xα “ yαaα, yα P N , aα P K. For each u P ApGq and v P ApKq we then have

vpaq “ ψapv ˝ qq “ ψapv ˝ q ` uq

“ limα

ϕxαpv ˝ q ` uq

“ limα

pvpaαq ` upyαaαqq.

Taking u “ 0, this gives vpaαq Ñ vpaq for every v P ApKq and hence upyαaαq Ñ 0for each u P ApGq and regularity of ApKq implies that aα Ñ a.

Towards a contradiction, assume that there exist a compact subset C of Nand a subnet pyαβ

qβ of pyαqα such that yαβP C for all β. Then, passing to a

further subnet if necessary, we can assume that yαβÑ y for some y P C. Then

yαβaαβ

Ñ ya and hence

upyaq “ limβ

upyαβaαβ

q “ 0

for all u P ApGq. This contradiction shows that yα Ñ 8 in N .Conversely, if yα Ñ 8 in N and aα Ñ a in K, then

ϕxαpv ˝ q ` uq “ vpaαq ` upxαq Ñ vpaq “ ψapv ˝ q ` uq

for all v P ApKq and u P ApGq, so that ϕxαÑ ψa in σpBpGqq. �

We close this section by presenting a class of examples of locally compact groupswhich satisfy the conditions (1) and (2) set out ahead of Proposition 2.9.1.

Example 2.9.3. Let p be a prime number, Ωp the locally compact field of p-adic numbers and Δp the compact open subring of p-adic integers. Let SLpn,Δpq

be the multiplicative group of nˆn matrices with entries in Δp and determinant ofvaluation 1. This compact group acts on the vector space Ωn

P by matrix multipli-cation. Form the semidirect product G “ Ωn

p �SLpn,Δpq. When n “ 1, this groupwas presented by Fell as a noncompact group which nevertheless has a countabledual space, and it is therefore usually referred to as the Fell group.

Since SLpn,Δpq is compact, the semidirect product G satisfies the hypotheses

that allow to determine pG by applying the Mackey machine outlined in Section 1.4.Since Ωn

p is self-dual, we only have to determine the orbits in Ωnp under the matrix

action of SLpn,Δpq.For x “ px1, . . . , xnq P Ωn

p , let Spxq denote the l8-sphere through x, that is,

Spxq “ ty “ py1, . . . , ynq P Ωnp : max

1ďjďn|yj | “ max

1ďjďn|xj |u.

We claim that the nontrivial orbits in Ωnp are such spheres. Note first that SLpn,Δpq

preserves spheres. Indeed, if A P SLpn,Δpq and Apx1, . . . , xnq “ py1, . . . , ynq, then

max1ďjďn

|yj | “ max1ďjďn

ˇ

ˇ

ˇ

ˇ

ˇ

nÿ

i“1

ajixi

ˇ

ˇ

ˇ

ˇ

ˇ

ď max1ďjďn

|xj |

because |aji| ď 1 for all j, i and the valuation is nonarchimedean, and thenmax1ďjďn |xj | “ max1ďjďn |yj | since SLpn,Δpq is a group. Furthermore, SLpn,Δpq

acts transitively on the spheres. To see this, let px1, . . . , xnq be a nonzero element ofΩn

p and choose w P Ωp with |w| “ max1ďjďn |xj |. Then there exists A P SLpn,Δpq

such that

Apw, 0, . . . , 0q “ px1, . . . , xnq.


In fact, set aj1 “ w´1xj for 1 ď j ď n. Then aj1 P Δp since |w´1xj | ď 1, andsince max1ďjďn |w´1xj | “ 1, it is possible to find aji, 1 ď j ď n, 2 ď i ď n, sothat A “ pajiq1ďjďn,1ďiďn P SLpn,Δpq. The important fact now is that all thesespheres are open in Ωn

p and that they cover Ωnp ztp0, . . . , 0qu.

For each k P Z, choose wk P Ωp with |wk| “ p´k and let χk P xΩnp be the

character corresponding to pwk, 0, . . . , 0q. Let Sk denote the stability group of χk

in SLpn,Δpq and let Gk “ Ωnp � Sk. Then

pG “ {SLpn,Δpq ˝ q Y

´

Ť

kPZtπk,τ : τ P xSku

¯

,

where q : G Ñ SLpn,Δpq is the quotient map and

πk,τ “ indGGkpχk b τ q, k P Z, τ P xSk.

Then each πk,τ is a subrepresentation of the left regular representation of G sincethe restriction of πk,τ to Ωn

p is supported on the open orbit through χk. Thus thegroup G satisfies the hypotheses of Proposition 2.9.1.

2.10. Notes and references

The Fourier and Fourier-Stieltjes algebras of a locally compact group have beenintroduced by Eymard as generalizations of the L1- and measure algebras, L1pGq

and MpGq, respectively, of a locally compact abelian group G and they have beenextensively studied in his seminal paper [73]. Actually, [73] has not only initiated,but also enormously influenced what has since become one of the most popularresearch areas in abstract harmonic analysis. All the material presented in Sections2.1 to 2.5 is taken from [73], and our treatment follows very closely the excellentexposition in [73].

Given a commutative Banach algebra A, the first relevant problem is to deter-mine the Gelfand spectrum of σpAq of A. It is a classical result that, for a locally

compact abelian group G with dual group pG, σpL1p pGqq “ G. This is generalizedby Proposition 2.3.2, which states that for an arbitrary locally compact group G,σpApGqq can be canonically identified with G and which may be considered as oneof the most fundamental results of the subject area. For example, it forms the basisfor Walter’s [280] isomorphism theorems presented in Section 3.4 as well as for thestudy of ideal theory in ApGq (Chapter 6). Equally important is the fact that ApGq

is a regular function algebra (Proposition 2.3.2).In contrast, the Gelfand spectrum of the Fourier-Stieltjes algebra BpGq is much

less understood. It was extensively investigated by J.L. Taylor (see [274] and thereferences therein) for abelian groups and by Walter [280], [281] for general locallycompact groups. It was shown, for instance, that σpBpGqq is a semigroup withmultiplication inherited from W˚pGq, the dual of BpGq, and that then G identifieswith all unitary elements in σpBpGqq. Part of this is treated in detail in Section3.2.

The identification of the Banach space dual of ApGq with the von Neumannalgebra V NpGq, generated by the left regular representation of G on the Hilbertspace L2pGq, plays a central role in the study of ApGq. In particular, the variousresults on the support of operators in V NpGq, established in this section, provedto be invaluable tools subsequently. The fact that ApGq˚ “ V NpGq is also of greatimportance for the operator space structure of ApGq as developed by Ruan [245].It has also motivated the investigation of uniformly continuous and weakly almost

2.10. NOTES AND REFERENCES 87

periodic functionals on ApGq [102], [174], which are analogues of weakly almost

periodic functions on pG when G is abelian [66].Let H be a closed subgroup of the locally compact group G. Then the as-

signment r : u Ñ u|H maps ApGq into ApHq. It was proved by McMullen [212],and independently by Herz [123], that r is actually surjective. More precisely, anyv P ApHq admits an extension u P ApGq with the same norm, }u}ApGq “ }v}ApHq.The proof is fairly technical in that it involves the reduction to second countablegroups and the existence of appropriate Borel cross-sections. Our presentationin Section 2.7 follows [212]. As a consequence one obtains that the adjoint mapr˚ : V NpHq Ñ V NpGq is a w˚-w˚-continuous isomorphism from V NpHq ontoV NHpGq, the w˚-closure in V NpGq of the linear span of the set of all left regularrepresentation operators λGpxq, x P H. This latter fact as well as surjectivity of rare frequently used later in the book, especially in Section 3.3 and Chapter 6.

There are various properties equivalent to amenability of a locally compactgroup G, which have been employed to give different proofs of Leptin’s theorem(Theorem 2.7.2), such as

(1) }f}1 “ }λGpfq} for every f P L1pGq, f ě 0;(2) given any compact subset K of G and δ ą 1, there exists a compact subset

U of G such that |KU | ă δ|U |;(3) the constant one function can be uniformly on compact subsets of G ap-

proximated by functions of the form f ˚ f˚, f P CcpGq.Leptin, who actually was the first to characterize a Banach algebraic property

of ApGq in terms of the group G, used (1) and (2). An alternative proof of Theorem2.7.2 was given by Derighetti [50], based on (3) and on his own result that on theunit sphere of BpGq the compact-open topology coincides with the weak topologyσpBpGq, L1pGqq. The proof presented in Section 2.7 builds on (3) and on Propo-sition 2.7.1, which can be found in [27] and is attributed there to an unpublishedthesis of Nielson. The important question for which locally compact groups G, theFourier algebra ApGq possesses an approximate identity which is bounded in somenorm weaker than the ApGq-norm, has been studied by several authors (see Chap-ter 5). The remaining results of Section 2.7 (Theorem 2.7.5 and 2.7.9), dealing withthe impact on G of the existence of some ideal in ApGq with bounded approximateidentity, are due to Forrest [79,80].

The subspaces AπpGq of BpGq, which we treated in Section 2.8, were intro-duced and studied in [5]. Arsac proved many results beyond those we have, mainlyfollowing the exposition in [5], presented here. For instance, he intensively inves-tigated the assignment π Ñ AπpGq under various aspects, such as forming tensorproducts and direct integrals of representations and inducing representations. Healso clarified the natural question of when AπpGq is a subalgebra of BpGq.

Let 1 ă p, q ă 8 such that 1p `

1q “ 1. For f P LppGq and g P LqpGq, the

convolution product

pf ˚ qgqpxq “

ż

G

fpxyqgpyqdy, x P G,

defines a function in C0pGq such that }f ˚ qg}8 ď }f}p}g}q. Since pf ˚ qgqpxq “ş

Ggpx´1yqfpyqdy, u can be viewed as a coefficient function of the left regular rep-

resentation of G on LqpGq. Define AppGq to be the set of all functions u P C0pGq


for which there exist sequences pfnqn in LppGq and pgnqn in LqpGq such that8ÿ

n“1

}fn}p}gn}q ă 8 and upxq “

8ÿ

n“1

pfn ˚ |gnqpxq

for all x P G. For u P AppGq, let

}u}AppGq “ inf

#

8ÿ

n“1

}fn}p}gn}q

+

,

where pfnqn and pgnqn are sequences with the above two properties. Then, ofcourse, }u}8 ď }u}AppGq.

Theorem 2.10.1. pAppGq, } ¨ }AppGqq is a Banach algebra with respect to point-wise operations, AppGq X CcpGq is dense in AppGq and AppGq is uniformly densein C0pGq.

Theorem 2.10.2. The spectrum of AppGq can be canonically identified with G.More precisely, the map x Ñ γx, where γxpuq “ upxq for u P AppGq, is a homeo-morphism from G onto σpAppGqq. Moreover, AppGq is regular and Tauberian.

The algebras AppGq are usually referred to as the Figa-Talamanca-Herz alge-bras. Note that A2pGq “ ApGq. The preceding two theorems are due to Herz andhave been shown in [122] and [123], respectively.

Several other results presented in this chapter for ApGq, essentially extend tothe algebras AppGq, 1 ă p ă 8. We mention two significant ones. As shown by Herz[123], surjectivity of the restriction map u Ñ u|H remains true for AppGq. However,when p ‰ 2, given ε ą 0, the existence of an extension u P AppGq of v P AppHq

could only be shown to satisfy the norm inequality }u}AppGq ď }v}AppHq ` ε. Inaddition, as also shown in [123], if H is normal and v P AppHq has compactsupport, then given an open subset U of G such that supp v Ď U X H, there existssuch an extension u of v satisfying supp u Ď U . If G is amenable, then AppGq hasan approximate identity of norm bound 1 for every p. Conversely, if AppGq has abounded approximate identity for some p, then G is amenable.

The notions of the Fourier and the Fourier-Stieltjes algebra of a locally compactgroup have been generalized in various different directions, which we now brieflyindicate, confining ourselves to Fourier algebras.

Firstly, let K be a compact subgroup of the locally compact group G. Forrest[83] has introduced the Fourier algebra ApG{Kq of the left coset space G{K. Thisalgebra can simultaneously be viewed as an algebra of functions on G{K and as thesubalgebra of ApGq consisting of all those functions in ApGq which are constant onleft cosets of K. In many respects, ApG{Kq behaves as nicely as does ApGq. Forinstance, as shown in [83], ApG{Kq is regular and semisimple, σpApG{Kqq “ G{Kand ApG{Kq has a bounded approximate identity if and only if G is amenable.The algebras ApG{Kq are precisely the norm closed left translation invariant ˚-subalgebras of ApGq [271].

Secondly, let H be a locally compact hypergroup with left Haar measure. TheFourier space ApHq was defined in analogy to the description of functions in ApGq

in terms of L2-functions (Proposition 2.3.3). In general, ApHq need not be closedunder pointwise multiplication. However, it is an algebra for many important classesof hypergroups, such as double coset hypergroups. See [217] and the referencestherein.


Let G be a topological group, let P pGq denote the collection of all continuouspositive definite functions on G, and let BpGq denote the linear span of P pGq.

By a σ-continuous representation of G into a W˚-algebra M , we shall mean apair pω,Mq such that ω is a homomorphism of G into

Mu :“ tx P M : x˚x “ xx˚“ 1u,

the group of unitaries in M , where 1 is the identity of M , and σ is the weak˚-topology σpM,M˚q defined by the unique predual of M .

Let ΩpGq denote the collection of all σ-continuous representations α “ pω,Mq

of G, such that@

ωpGqDα

“ M . Then BpGq is precisely the collection of all complex-

valued functions φ on G such that φ “ fα for some f P M˚ and some α “ pω,Mq “

pωα,Mαq in ΩpGq, where fαpaq “ xωpaq, fy for all a P G. For each φ P BpGq, define

}φ} :“ }φ}BpGq “ inf

}fα} : fα P M˚, φ “ fα, andα “ pω,Mq P ΩpGq(

.

Also letMΩ :“řÀ

Mωα, the direct summand of the W˚- algebrasMα :“ Mωα

, α P

ΩpGq. Define a σ-continuous homomorphism of G into MΩ by ωΩpaqpαq :“ ωαpaq

for each α “ pωα,Mαq in ΩpGq. Write

W˚pGq “

@

ωΩpGqDσ

.

Theorem 2.10.3. (a) BpGq is a subalgebra of WAP pGq, the space ofcontinuous weakly almost periodic functions on G, containing the con-stant functions. Furthermore, }¨} is a norm on BpGq and pBpGq, }¨}q is acommutative Banach algebra. More specifically, the map ρ : W˚pGq˚ Ñ

BpGq defined by ρpfq :“ f , f P W˚pGq˚, is a linear isometry fromW˚pGq˚ onto BpGq. Furthermore, ρpfq is positive definite if and only iff is positive.

(b) If pω,Mq is any σ-continuous representation of G, then there is a W˚-homomorphism hω from W˚pGq into M such that the diagram

G W˚pGq

M

ωΩ

ωhω

is commutative. Also if f P M˚, then fpxq “ xhωpxq, fy for allx P W˚pGq.

(c) If φ P BpGq and a P G, then the functions laφ, raφ, φ˚, φ are all in BpGq

and

}laφ} “ }φ}, }raφ} “ }φ}, }φ˚} “ }φ}, }φ} “ }φ},

where φ˚pyq “ φpy˚q for each y P M .

Let A be a C˚-algebra and pπ,Hπq be a non-degenerate representation of A.Let πe be the unique extension of π to MpAq, the multiplier algebra of A is thelocally convex topology on MpAq defined by the semi-norms: tPa : a P Au, where

Papmq “ }a ¨ m} ` }m ¨ a}, m P MpAq.


Definition 2.10.4. Let G be a topological group. We call a host algebra ofG a pair pA, ηq, where A is a C˚-algebra and η : G ÞÑ UpMpAqq is a continuoushomomorphism from G into the unitary group of MpAq, such that the mappingη˚ : ReppAq Ñ ReppGq, η˚pπq :“ πe ˝ η is injective, where ReppAq denotes thecollection of ˚- representations ofA, and ReppGq the group of the continuous unitaryrepresentations of G.

Given any C˚- algebra A, let rA denote the universal enveloping von Neumannalgebra A˚˚ of A.

We say that pA, ηq is a full host algebra of a topological group G, if it is a hostalgebra of G and if η˚ is also surjective.

Theorem 2.10.5. Let G be a topological group with a full host algebra A. Then

the von Neumann algebras rA and W˚pGq are isomorphic. In particular, rA˚ isisometrically isomorphic to BpGq. Furthermore, if A1 is any C˚-algebra such that

A˚1 is isometrically isomorphic to the Banach space BpGq, then ĂA1 and rA are either

isomorphic or anti-isomorphic.

Remark 2.10.6. Theorem 2.10.3 was also proved in [173], and Theorem 2.10.5in [181]. See also [182].

CHAPTER 3

Miscellaneous Further Topics

After having introduced in Chapter 2 the Fourier and the Fourier-Stieltjes alge-bras, ApGq and BpGq, of a locally compact group G and studied some elementary,but basic functorial properties as well as the von Neumann algebra V NpGq, the Ba-nach space dual of ApGq, we start in this chapter the more intrinsic and somewhatdeeper investigation of both algebras. The core of the chapter is formed by resultswhich are by now classical, whereas the more recent achievements are postponedto subsequent chapters.

Generalizing the classical description of idempotents in the measure algebraof a locally compact abelian group, Host [129] has identified the integer-valuedfunctions in BpGq. Host’s so-called idempotent theorem, which has found numer-ous applications, is shown in Section 3.1. A natural question is whether either ofthe Banach algebras ApGq and BpGq determines G as a topological group. Thisquestion has been affirmatively answered by Walter [280]. If G1 and G2 are locallycompact groups and BpG1q and BpG2q (respectively, ApG1q and ApG2q) are isomet-rically isomorphic, then G1 and G2 are topologically isomorphic or anti-isomorphic.These results are displayed in Section 3.2, and building on these, Pham [234] hasdetermined the structure of certain homomorphisms between Fourier and Fourier-Stieltjes algebras of general locally compact groups. This is the content of Section3.2.

Takesaki and Tatsuuma [271] have established Galois type correspondencesbetween closed subgroups of G and right translation invariant von Neumann subal-gebras of L8pGq and between closed subgroups of G and von Neumann subalgebrasof V NpGq, which are invariant in some sense. These correspondences are given inSection 3.4. In a similar spirit, in Section 3.5 translation invariant subalgebras ofApGq and of BpGq are studied and related to closed subgroups of G. Let G1 andG2 be locally compact groups. It is an interesting problem of when ApG1 ˆ G2q isalgebraically isomorphic to the projective tensor product of ApG1q and ApG2q. Thisisomorphism problem was treated by Losert [200], and his solution is presented inSection 3.6. Finally, in Section 3.7 various topologies on BpGq are studied, themain emphasis being on comparing the relative topologies on the unit sphere ofBpGq. It is also shown that G has to be compact whenever the w˚-topology andthe norm topology coincide on the unit sphere in BpGq.

3.1. Host’s idempotent theorem

One of the profound discoveries of Cohen [33] was the description of all idempo-

tent measures on a locally compact abelian group. Since, for suchG, BpGq “ Mp pGq,the natural, much more general problem was the characterization of all idempotentsin the Fourier-Stieltjes algebra BpGq of an arbitrary locally compact group G. Thiswas achieved by Host [129]. In this section we are going to present his description

91

92 3. MISCELLANEOUS FURTHER TOPICS

of all integer-valued functions in BpGq, thus in particular of all idempotents inBpGq.

We start with some preliminary facts on operators on Hilbert spaces.

Lemma 3.1.1. Let H be a Hilbert space. Then

(i) If P is an orthogonal projection onto a closed subspace E of H, thenP ě 0, }P } “ 1 unless E “ t0u and E “ tξ P H : }Pξ} “ }ξ}u.

(ii) On the set of orthogonal projections in H, the weak operator topology andthe strong operator topology agree.

(iii) If pTαqα is a net in BpHq such that Tα Ñ T P BpHq in the weak operatortopology and T˚

αTα “ T˚T for all α, then Tα converges to T in the strongoperator topology.

(iv) If pTαqα is a monotone increasing net of self-adjoint operators in H suchthat Tα ď I for all α, then pTαqα converges in the strong operator topologyto some self-adjoint operator T , and T is the least upper bound of pTαqα.In particular, if S P BpHq is such that 0 ď S ď I, then the sequence pSnqn

converges in the strong operator topology to some projection P P BpHq.

Proof. For (i) see [147, p. 109].(ii) We have to show that if pPαqα is a net of orthogonal projections in H and

Pα Ñ P in the weak operator topology, then }Pαξ ´Pξ} Ñ 0 for each ξ P H. Now,

}Pαξ ´ Pξ}2

“ xPαξ, Pαξ ´ Pξy ´ xPξ, Pαξ ´ Pξy

“ xξ, Pαξ ´ PαpPξqy ´ xPξ, Pαξ ´ Pξy

ď |xξ, pPα ´ P qξy| ` |xξ, pP ´ PαqPξy| ´ xPξ, pPα ´ P qξy,

which converges to 0.(iii) If ξ P H, then

}Tαξ ´ Tξ}2

“ xTαξ, Tαξ ´ Tξy ´ xTξ, Tαξ ´ Tξy

“ xT˚αTαξ, ξy ´ xTαξ, T ξy ´ xTξ, Tαξ ´ Tξy

“ xT˚Tξ, ξy ´ xTαξ, T ξy ´ xTξ, Tαξ ´ Tξy,

which converges to xT˚Tξ, ξy ´ }Tξ}2 “ 0.For (iv) see [147, p. 307]. �

For a representation π of G, let πpGq denote the closure of πpGq in BpHpπqq in

the weak operator topology. Then πpGq Ď BpHpπqq1 and πpGq with the weak oper-

ator topology is a compact semitopological semigroup, i.e., for each S P πpGq, the

mappings T Ñ TS and T Ñ ST from πpGq into πpGq are continuous. Furthermore,

π : G Ñ πpGq is a continuous mapping of G onto a dense subset of πpGq.

Lemma 3.1.2. Let π be a unitary representation of G and let ξ, η P Hpπq.Suppose that η is a cyclic vector and that the function u on G defined by upxq “

xπpxqξ, ηy, x P G, is integer-valued. Then

(i) For each S P πpGq, xSξ, ηy is also an integer.

(ii) The set tSξ : S P πpGqu is a discrete subset of Hpπq.

Proof. (i) follows readily by passing to limits.

(ii) Let S, T P πpGq such that }Sξ´Tξ} ă 1{}η}. For each x P G, πpx´1qS and

πpx´1qT belong to πpGq, and hence xSξ, πpxqηy and xTξ, πpxqηy are both integers

3.1. HOST’S IDEMPOTENT THEOREM 93

by (i). However

|xSξ, πpxqηy ´ xTξ, πpxqηy| ď }Sξ ´ Tξ} ¨ }πpxqη} ă 1.

Hence xSξ ´Tξ, πpxqηy “ 0. Since x P G is arbitrary and η is cyclic, it follows thatSξ “ Tξ. So (ii) holds. �

We denote by πpGq`

the set of all positive operators in πpGq. Then πpGq`

is

weakly closed in πpGq.

Lemma 3.1.3. Let π be a unitary representation of G and let ξ, η P Hpπq becyclic vectors. Assume that the function upxq “ xπpxqξ, ηy, x P G, is integer-valued.Then

(i) πpGq`

is a commutative semigroup consisting entirely of projections. In

particular, each S P πpGq`

is a partial isometry.

(ii) E “ tPξ : P P πpGq`

u is a finite subset of Hpπq.

Proof. (i) Let S P πpGq`. By Lemma 3.1.1(iv), the sequence pSnqn converges

to some P P πpGqìn the strong operator topology and P is a projection. We are

going to show that P “ S. Fix x P G. Then Snπpxq P πpGq for all n and Pπpxq P

πpGq. Since the sequence pSnπpxqξqn converges to Pπpxqξ, by Lemma 3.1.2(ii) wecan find n P N such that Snπpxqξ “ Pπpxqξ. We show now that the operatorsSn ´ P and S ´ P have the same kernel. For that, let ω P Hpπq. If pS ´ P qω “ 0,then clearly pSn ´ P qω “ pSn ´ Pnqω “ 0 since S and P commute. Conversely,let pSn ´ P qω “ 0. Then since SP “ P , pSm ´ P qω “ SmńpSn ´ P qω “ 0 for allm ě n. Let m be the smallest even integer such that pSm ´ P qω “ 0. Then

xpSm{2´ P qω, Sm{2ωy “ xpSm

´ P qω, ωy “ 0

and therefore, since PS “ P “ P 2,

}Sm{2pωq}

2“ xSm{2

pωq, Sm{2pωqy “ xP pωq, Sm{2

pωqy

“ xP pωq, PSm{2pωqy “ xPSm{2

pωq, PSm{2pωqy

“ }PSm{2pωq}

2.

So Sm{2pωq is in the range of P by Lemma 3.1.1(i) and hence

pSm{2´ P qpωq “ Sm{2

pωq ´ P pSm{2pωqq “ 0.

This contradicts the minimality of m unless m “ 2. In this case, pS´P qpωq “ 0. Itfollows that Sπpxqξ “ Pπpxqξ. Since this holds for all x P G and since ξ is cyclic,

we conclude that S “ P . This shows that πpGq`

consists entirely of idempotents.

For each S P πpGq`, S˚S and SS˚ both belong to πpGq

ànd are projections.

Hence S is a partial isometry.To see that πpGq

ìs a commutative semigroup, let S, T P πpGq

`. Then, by

what we have seen above, ST and TS are partial isometries in πpGq. Now, forξ P H, in order that }STξ} “ }ξ}, it is necessary and sufficient that Tξ “ ξ “ Sξand so STξ “ ξ. Indeed, if }STξ} “ }ξ}, then }Tξ} ě }STξ} “ }ξ} and so}Tξ} “ }ξ}, whence Tξ “ ξ. Consequently, }ξ} “ }STξ} “ }Sξ} and thereforeSξ “ ξ. Similarly, }TSξ} “ }ξ} if and only if Sξ “ ξ “ Tξ, and so TSξ “ ξ. This

shows that ST “ TS and ST P πpGq`.


(ii) Since πpGq`

consists of projections, the weak operator topology and the

strong operator topology coincide on πpGq`

by Lemma 3.1.1(iii). Hence the set

E “ tPξ : P P πpGq`

u is a norm-compact subset of Hpπq. Then, by Lemma3.1.2(ii), E must be finite. �

The following is the generalization, announced at the outset of this section, ofCohen’s idempotent theorem for locally compact abelian groups to general locallycompact groups.

Theorem 3.1.4. Let G be a locally compact group. The integer-valued func-tions in BpGq are precisely the finite linear combinations with integer coefficientsof translates of characteristic functions of open subgroups of G. In particular, theidempotents in BpGq are exactly the characteristic functions of finite unions oftranslates of open subgroups of G.

To start the proof of the theorem, let u be an integer-valued function in BpGq.We know that there exist a unitary representation π and cyclic vectors ξ, η P Hpπq

such that upxq “ xπpxqξ, ηy for all x P G. Let

A “ tEξ : E P πpGq`

u.

By Lemma 3.1.3, A is a finite subset of Hpπq. For each ω P A, let

Sω “

!

E P πpGq`: Eξ “ ω

)

.

Then Sω is a closed subsemigroup of πpGq `. Indeed, Sω is clearly closed and ifE,F P Sω, then

pFEqξ “ Fω “ F 2ξ “ Fξ “ ω,

so that FE P πpGq`.

For ω P A, let Pω denote the unique minimal element of Sω, and for everyB Ď A, let

QB “

˜

ź

ωPB

Pω

¸

¨

˝

ź

ωPAzB

pI ´ Pωq

˛

‚

and ξB “ QBpξq. It follows from the commutativity of πpGq`

that QB is a pro-jection which is a finite linear combination with integer coefficients of elements ofπpGq

`. Then

ř

BĎAQB “ I. This is clear if A is a singleton. So suppose that A

3.1. HOST’S IDEMPOTENT THEOREM 95

contains at least two elements and fix ω0 P A and let A0 “ Aztω0u. Thenÿ

BĎA

QB “ÿ

BĎA0

QB `ÿ

CĎA0

QCYtω0u

“ÿ

BĎA0

˜

ź

ωPB

Pω

¸

¨

˝

ź

ωPtω0uYpA0zBq

pI ´ Pωq

˛

‚

`

ÿ

CĎA0

¨

˝

ź

ωPCYtω0u

Pω

˛

‚

¨

˝

ź

ωPA0zC

pI ´ Pωq

˛

‚

“ pI ´ Pω0q

ÿ

BĎA0

˜

ź

ωPB

Pω

¸

¨

˝

ź

ωPA0zB

pI ´ Pωq

˛

‚

`Pω0

ÿ

CĎA0

˜

ź

ωPC

Pω

¸

¨

˝

ź

ωPA0zC

pI ´ Pωq

˛

‚

“

ÿ

CĎA0

˜

ź

ωPC

Pω

¸

¨

˝

ź

ωPA0zC

pI ´ Pωq

˛

‚

“

ÿ

CĎA0

QC .

Thus the statementř

BĎA QB “ I follows by induction on the number of elementsin A. In particular,

ř

BĎA ξB “ ξ. Therefore, if for B Ď A, we define uB by

uBpxq “ xπpxqξB, ηy, x P G,

then u “ř

BĎA uB . Each uB is a finite linear combination with integer coefficients

of functions x Ñ xπpxqQξ, ηy, where Q P πpGq`. All these latter functions in BpGq

are integer-valued by Lemma 3.1.2(ii) and hence so is uB . Thus, in order to showthat u is of the form indicated in Theorem 3.1.4, it suffices to verify this for eachuB, B Ď A. To that end, fix B and let v “ uB, P “ PB and ψ “ ξB. ThenPψ “ P 2

Bpξq “ PBpξq “ ψ.The crucial statement in the following lemma is (iii), and this constitutes the

final essential step towards the proof of the theorem.

Lemma 3.1.5. Retain the above notation.

(i) For any Q P πpGq`, either Qψ “ 0 or Q ě P .

(ii) If S P πpGq and Sψ ‰ 0, then S˚S ě P .

(iii) The subset D “ tϕ P πpGqpψq : xϕ, ψy ‰ 0u of Hpπq is finite.

Proof. (i) Note that Q ě PQpξq since Q P SQpξq and PQpξq is the smallestelement of SQpξq. If Qpξq P B, then Q ě PQpξq ě PB “ P , whereas if Qpξq R B,then

Qpψq “ QpPBpξqq “ PBpQpξqq “ 0

since PB ď I ´ PQpξq and pI ´ PQpξqqQpξq “ 0.

(ii) If Sψ ‰ 0, then xS˚Sψ, ψy “ }Sψ}2 ‰ 0 and S˚S is a projection in πpGq`.

Then (i) shows that S˚S ě P .


(iii) We know from Lemma 3.1.2(ii) that πpGqpψq is a discrete subset of Hpπq.Therefore it suffices to show that D is compact. To that end, let pψαqα be a net in

D and, for each α, choose Sα P πpGq such that ψα “ Sαψ. Since πpGq is compact inthe weak operator topology, there exists a subnet pSαβ

qβ which converges to some

S P πpGq in the weak operator topology. Since, by Lemma 3.1.2(i), xSαβψ, ηy “

xψαβ, ηy is a nonzero integer for every β, xSψ, ηy is also a nonzero integer. In

particular, Sαβψ ‰ 0 and hence S˚

αβSαβ

ě P by (ii), and then also S˚S ě P .

Now SP is a weak operator limit of the net pSαβP qβ and all these operators

have absolute value P . It follows that SαβP Ñ SP in the strong operator topology

as well. In particular, }SαβPψ ´ SPψ} Ñ 0. But Pψ “ ψ and Sψ P D, and this

shows that the subnet pSαβψqβ converges in D. So D is compact. �

To finish the proof of Theorem 3.1.4, let G0 “ tx P G : πpxqω “ ωu. Then G0

is a subgroup of G and G0 is open in G since πpGqω is a discrete subset of Hpπq

and the map x Ñ πpxqω from G into H is continuous. Since πpxqω “ πpyqω if andonly if xG0 “ yG0, the function x Ñ vpxq “ xπpxqω, ηy is constant on cosets of G0.But, by Lemma 3.1.5(iii), the set tπpxqω : vpxq ‰ 0u is finite. So v is zero excepton a finite number of cosets of G0. In particular, v is a linear combination withinteger coefficients of translates of the characteristic function of the open subgroupG0 of G.

3.2. Isometric isomorphisms between Fourier-Stieltjes algebras

In this section it is shown that for a locally compact group G, each of thecommutative Banach algebras BpGq and ApGq completely determines the group G.More precisely, two locally compact groups G1 and G2 are topologically isomorphicif BpG1q and BpG2q (respectively, ApG1q and ApG2q) are isometrically isomorphic(Theorem 3.2.5 and Theorem 3.2.6). Recall that for a locally compact abelian group

H, BpHq and ApHq is isometrically isomorphic to Mp pHq and L1p pHq, respectively,

and thatx

xH “ H by the Pontryagin duality theorem. Therefore, the above theorems,which are due to Walter [280], are far reaching generalizations of the correspondingresults of Wendel [288] and of Johnson [140] for abelian groups. As is to beexpected, establishing Theorems 3.2.5 and 3.2.6 requires considerably more effortthan in the abelian case.

Let G be a locally compact group and W˚pGq “ BpGq˚. Then W˚pGq is theenveloping W˚-algebra of C˚pGq. Given T P W˚pGq and u P BpGq, we define abounded complex-valued function Tlpuq on G by

Tlpuqpxq “ xT, lxuy,

where plxuqpyq “ upxyq for x, y P G. Let ωpxq denote the element of W˚pGq suchthat xωpxq, uy “ upxq for all u P BpGq. Moreover, let LTu and RTu in BpGq bedefined by

xLTu, Sy “ xu, TSy and xRTu, Sy “ xu, ST y,

S P W˚pGq. Then, as is readily checked, lxu “ Lωpxqu and hence

Tlupxq “ xT, lxuy “ xT, Lωpxquy

“ xu, ωpxqT y “ xRTu, ωpxqy

“ pRTuqpxq

3.2. ISOMORPHISMS BETWEEN FOURIER-STIELTJES ALGEBRAS 97

for all x P G, u P BpGq and T P W˚pGq. Consequently, Tlu P BpGq and }Tlu} ď

}T } ¨ }u}. Thus, for any S, T P W˚pGq, we can define an element S ˝ T of W˚pGq

through

xS ˝ T, uy “ xS, Tlpuqy

for u P BpGq.

Lemma 3.2.1. Let G be a locally compact group and S, T P W˚pGq.

(i) S ˝ T “ ST , the product of S and T in W˚pGq.(ii) pST qlpuq “ SlpTlpuqq for all u P BpGq.

Proof. (i) Fix T P W˚pGq. The equation S ˝ T “ ST evidently holds whenS “ ωpxq for some x P G. Hence it also holds for all S P E, the linear span ofωpGq. Now, if S is an arbitrary element of W˚pGq, then there is a net pSαqα in Econverging to S in the w˚-topology. Thus, for each u P BpGq,

xS ˝ T, uy “ xS, Tlpuqy “ limα

xSα, Tlpuqy

“ limα

xSα ˝ T, uy “ limα

xSαT, uy

“ xST, uy

and so S ˝ T “ ST .(ii) follows from (i) by using that Tlplxuq “ lxpTluq for all x P G. �

Clearly, ωpxq P σpBpGqq for every x P G. Furthermore, the linear span of the settωpxq : x P Gu is w˚-dense inW˚pGq since otherwise, by the Hahn-Banach theorem,there exist T P W˚pGq and u P BpGq such that xT, uy ‰ 0 and upxq “ xωpxq, uy “ 0for all x P G, which is impossible.

Corollary 3.2.2. For any closed linear subspace E of BpGq “ C˚pGq˚, thefollowing three conditions are equivalent.

(i) T ¨ E Ď E and E ¨ T Ď E for all T P W˚pGq.(ii) T ¨ E Ď E and E ¨ T Ď E for all T P C˚pGq.(iii) E is translation invariant.

Proof. It is easily checked that

ωpxq ¨ u “ lxu and u ¨ ωpxq “ rxu

for all x P G and u P BpGq. Thus (iii) is equivalent to

ωpxq ¨ V Ď V and V ¨ ωpxq Ď V

for all x P G. Since C˚pGq and the linear span of ωpGq are both w˚-dense inW˚pGq, it follows right away from Lemma 3.2.1 that (iii) is also equivalent to (i)and (ii). �

We are now in a position to identify the spectrum of BpGq and the spectrumof its ideal ApGq within W˚pGq “ BpGq˚.

Lemma 3.2.3. Let G be a locally compact group.

(i) An element T of W˚pGq belongs to σpBpGqq if and only if Tl is a nonzeroalgebra homomorphism of BpGq.

(ii) If T, S P σpBpGqq, then TS P σpBpGqq and T˚ P σpBpGqq.(iii) If T P σpBpGqq, but T R ωpGq, then xT, uy “ 0 for all u P ApGq.


(iv) If T P σpBpGqq and T is invertible in W˚pGq, then T “ ωpxq for somex P G. In particular,

ωpGq “ tT P σpBpGqq : T is unitaryu.

Proof. (i) Suppose that T P σpBpGqq. Then

Tlpuvqpxq “ xT, lxpuvqy “ xT, plxuqplxvqy

“ xT, lxuyxT, lxvy

“ TlpuqpxqTlpvqpxq

for all u, v P BpGq and x P G. Moreover, if u P BpGq is such that xT, uy ‰ 0, thenTlpuq ‰ 0. So Tl is a nonzero algebra homomorphism.

Conversely, if T is a nonzero algebra homomorphism, then T ‰ 0 and T ismultiplicative since xT, uy “ Tlpuqpeq for u P BpGq.

(ii) Since T P σpBpGqq and Sl is a homomorphism by (i), by Lemma 3.2.1(i)we have for all u, v P BpGq

xTS, uvy “ xT, Slpuvqy “ xT, SlpuqSlpvqy

“ xT, SlpuqyxT, Slpvqy

“ xTS, uyxTS, vy.

Hence TS is multiplicative, and TS is nonzero since

xTS, 1Gy “ xT, Slp1Gqy “ xT, 1Gy “ 1.

(iii) Assume that xT, u0y ‰ 0 for some u0 P ApGq. Then T |ApGq P σpApGqq andhence there exists x P G such that xT, uy “ upxq for all u P ApGq. Since ApGq is anideal in BpGq, it follows that

xT, vyxT, u0y “ xT, vu0y “ pvu0qpxq “ vpxqxT, u0y

for all v P BpGq and hence T “ ωpxq, which is a contradiction.(iv) Suppose that T is invertible inW˚pGq and, towards a contradiction, assume

that T R ωpGq. Then xT, uy “ 0 for all u P ApGq by (iii). Since ApGq is translationinvariant, it follows that

T P ApGq˝

“ pE ´ ZqW˚pGq

for some central projection Z in W˚pGq which satisfies BpGqZ “ ApGq. ThuspE ´ ZqT “ T , which implies that

E ´ Z “ pE ´ ZqTT´1“ TT´1

“ E

and hence Z “ 0. Thus ApGq “ t0u, which is impossible. This contradiction showsthat T “ ωpxq for some x P G. �

Corollary 3.2.4. Let G be a locally compact group and endow σpBpGqq withthe w˚-topology and the product inherited from W˚pGq (Lemma 3.2.3(ii)). ThenσpBpGqq is a compact semitopological semigroup with separately continuous mul-tiplication and it is invariant under involution. Furthermore, if G1 and G2 arelocally compact groups and σpBpG1qq and σpBpG2qq are topologically isomorphicsemigroups, then G1 and G2 are topologically isomorphic.

Proof. Since BpGq is unital, σpBpGqq is compact, and it is closed under mul-tiplication and involution by Lemma 3.2.3(ii). If φ : σpBpG1qq Ñ σpBpG2qq isan isomorphism, then T P σpBpG1qq is invertible in W˚pG1q if and only if φpT q

3.2. ISOMORPHISMS BETWEEN FOURIER-STIELTJES ALGEBRAS 99

is invertible in W˚pG2q. By Lemma 3.2.3(iv), it follows that φpωpG1qq “ ωpG2q.Finally, if φ is also a homeomorphism then G1 and G2 are topologically isomorphicsince ω : Gi Ñ ωpGiq Ď σpBpGiqq is a topological isomorphism, i “ 1, 2. �

The following theorem is the main result we were aiming at. It describes ex-plicitly the structure of isometric isomorphisms between Fourier-Stieltjes algebras.

Theorem 3.2.5. Let G1 and G2 be locally compact groups. If φ is an isometricalgebra isomorphism from BpG2q onto BpG1q, then there exist a topological iso-morphism or anti-isomorphism α from G2 onto G1 and an element b P G2 suchthat

φpuqpxq “ upb αpxqq

for all u P BpG2q and x P G1. In particular, G1 and G2 are topologically isomor-phic.

Proof. Let e1 and e2 be the identities of G1 and G2, respectively. Thenωpe1q and ωpe2q are the identities of W˚pG1q and W˚pG2q, respectively. Let S “

φ˚pωpe1qq and T “ S˚. Since S is a unitary, so is T . By Lemma 3.2.3(iv), T “ ωpbqfor some b P G2. Then

Tlupyq “ xT, lyuy “ xωpbq, lyuy “ lyupbq “ upybq

for all y P G2 and u P BpG2q, that is, Tl is the right translation operator associatedto b on BpG2q. Consequently, φ ˝ Tl is also an isometric isomorphism from BpG2q

onto BpG1q. Let Λ “ pφ ˝ Tlq˚, then Λ is a linear isometry from W˚pG1q onto

W˚pG2q. Moreover,

xΛpωpe1qq, uy “ xT˚l ˝ φ˚

pωpe1qq, uy

“ xφ˚pωpe1qq, Tluy “ xS, Tluy “ xS ˝ T, uy

“ xωpe2q, uy

by Lemma 3.2.3 and since S is a unitary. So Λpωpe1qq “ ωpe2q.By [145, Theorem 7 and its proof], Λ is a Jordan isomorphism from W˚pG1q

onto W˚pG2q. Let Λ1 denote the restriction of Λ to σpBpG1qq. Then Λ1 is ahomeomorphism from σpBpG1qq onto σpBpG2qq. Also Λ1pT˚q “ Λ1pT q˚ for allT P σpBpG1qq by [145, Lemma 8]. Moreover, if T, S P σpBpG1qq, then eitherΛ1pTSq “ Λ1pT qΛ1pSq or Λ1pTSq “ Λ1pSqΛ1pT q. Indeed, if TS “ ST , this followsfrom [145, Theorem 5]. Otherwise, using [145, Lemma 6], we have

Λ1pT qΛ1

pSq ` Λ1pSqΛ1

pT q “ Λ1pTSq ` Λ1

pST q,

and assuming that Λ1pTSq ‰ Λ1pT qΛ1pSq as well as Λ1pTSq ‰ Λ1pSqΛ1pT q, it followsthat

Λ1pTSq, Λ1

pST q, Λ1pT qΛ1

pSq and Λ1pSqΛ1

pT q

are pairwise distinct elements of σpBpG2qq. However, elements in σpBpG2qq arelinearly independent. So the above equation is impossible. It now follows that Λ1

maps the unitary elements of σpBpG1qq onto the unitary elements of σpBpG2qq.Thus, by Lemma 3.2.3, Λ1 maps ωpG1q onto ωpG2q.

By [145, Theorem 10], we can find central projections Z1 P W˚pG1q and Z2 P

W˚pG2q such that

Λ|W˚pG1qZ1: W˚

pG1qZ1 Ñ W˚pG2qZ2


is a ˚-isomorphism and

Λ|W˚pG1qpI´Z1q : W˚

pG1qpI ´ Z1q Ñ W˚pG2qpI ´ Z2q

is a ˚-anti-isomorphism.For x P G1, let

Hx “ ty P G1 : pxy ´ yxqZ1 “ 0u

and

Kx “ ty P G1 : pxy ´ yxqpI ´ Z1q “ 0u.

Then Hx is a subgroup of G1. In fact, if y1, y2 P Hx, then

xpy1y2qZ1 “ py1xZ1qy2 “ y1pxy2Z1q “ y1py2xZ1q “ py1y2qxZ1,

and hence y1y2 P Hx. Also, e1 P Hx and y P Hx implies y´1 P Hx. Similarly, it isshown that Kx is a subgroup of G1.

Notice next that Hx Y Kx “ G1. This can be seen as follows. If y P G1 then,as we have seen above, either Λpyxq “ ΛpyqΛpxq or Λpyxq “ ΛpxqΛpyq. In the firstcase, recalling that Λ|W˚pG1qpI´Z1q is an anti-isomorphism, we get

Λppyx ´ xyqpI ´ Z1qq “ ΛpyxpI ´ Z1qq ´ ΛpxpI ´ Z1qqΛpypI ´ Z1qq “ 0,

and hence pyx ´ xyqpI ´ Z1q “ 0, whence y P Kx. In the second case, sinceΛ|W˚pG1qZ1

is an isomorphism,

Λppyx ´ xyqZ1q “ ΛpxZ1qΛpyZ1q ´ ΛpxyZ1q “ 0,

which implies that y P Hx. Now, HxYKx “ G1 in turn implies that eitherHx “ G1

or Kx “ G1. Indeed, if there exist y1 P Hx X pG1zKxq and y2 P Kx X pG1zHxq,then y1y2 R Hx Y Kx, which is impossible. Now, let

H “ tx P G1 : Hx “ G1u and K “ tx P G1 : Kx “ G1u.

Then H and K are subgroups of G1. Indeed, if x1, x2 P H, then for any x P G1,

rxpx1x2q ´ px1x2qxsZ1 “ pxx1qZ1x2 ´ x1pxx2qZ1

“ px1xqZ1x2 ´ px1xqx2Z1 “ 0

and hence x1x2 P H. Moreover, e1 P H and x P H implies x´1 P H since, forany y P G1, x

´1yZ1 “ yx´1Z1 if and only if yZ1 “ xyx´1Z1, and this in turn isequivalent to

yxZ1 “ yZ1x “ xyx´1Z1x “ xyZ1.

So H is a subgroup of G1, and similarly it is shown that K is a subgroup. Givenx P G1, as shown above either Hx “ G1 or Kx “ G1. Thus H Y K “ G1 and asbefore it follows that either H “ G1 or K “ G1.

Suppose that H “ G1. We claim that then Λpxyq “ ΛpyqΛpxq for all x, y P

G1. To see this, we can assume that Λpxyq “ ΛpxqΛpyq. Then, as shown above,pxy ´ yxqpI ´Z1q “ 0. On the other hand, pxy ´ yxqZ1 “ 0 since x P G1 “ Hy. Soxy “ yx and hence Λpxyq “ ΛpyqΛpxq. Similarly, ifK “ G1 then Λpxyq “ ΛpxqΛpyq

for all x, y P G1.Thus we have shown that Λ1 is an isomorphism or an anti-isomorphism from

G1 onto G2. Finally, note that if Λ1 is an anti-isomorphism, then x Ñ Λ1px´1q isan isomorphism.

Finally, let b P G2 be as chosen at the beginning of the proof and let Λ be thelinear isometry from W˚pG1q onto W˚pG2q defined above. Now define α : G1 Ñ G2

3.3. HOMOMORPHISMS BETWEEN FOURIER ALGEBRAS 101

by setting, for x P G1, αpxq “ y where y is the unique element of G2 such thatΛpωpxqq “ ωpyq. It then follows that

φpuqpxq “ upbαpxqq

for all u P BpG2q and x P G1. �

Using arguments similar to those in the proof of Theorem 3.2.5 one can showthat Theorem 3.2.5 remains valid if BpGiq is replaced by BλpGiq, i “ 1, 2. Also, thenext theorem can be proved by very much the same arguments as those employedin the proof of Theorem 3.2.5, replacing W˚pGiq by V NpGiq, i “ 1, 2, and noticingthat σpApGqq “ G for any locally compact group G.

Theorem 3.2.6. Let G1 and G2 be locally compact groups. If φ : ApG1q Ñ

ApG2q is an isometric isomorphism, then there exist a topological isomorphism oranti-isomorphism α : G2 Ñ G1 and a P G1 such that φupyq “ upaαpyqq for allu P ApG1q and y P G2. In particular, G1 and G2 are topologically isomorphic.

3.3. Homomorphisms between Fourier and Fourier-Stieltjes algebras

In Theorem 3.2.6 it was shown that if G and H are locally compact groupsand φ : ApGq Ñ ApHq is an isometric isomorphism, then φ is implemented by atopological isomorphism or anti-isomorphism ϕ : H Ñ G and an element a P G.An analogous result holds for the Fourier-Stieltjes algebras (Theorem 3.2.5). Inthis section we determine the structure of continuous homomorphisms from ApGq

into ApHq and from BpGq into BpHq which are either positive (Theorem 3.3.5) orcontractive (Theorem 3.3.7). However, both of these results to some extent rest onTheorem 3.2.6.

Let G and H be locally compact groups, and let φ : ApGq Ñ BpHq be ahomomorphism. Let

U “ th P H : φpuqphq ‰ 0 for some u P ApGqu.

Then U is open in H, and for each h P U , there exists a unique element αphq of Gsuch that φpuqphq “ upαphqq for every u P ApGq. Of course, the map α : U Ñ G iscontinuous. Defining φα : ApGq Ñ �8pHq by

φαpuqphq “

"

upαphqq : if h P U0 : if H P HzU,

we have φ “ φα. Conversely, given any map α : U Ñ G, the preceding formuladefines a homomorphism from ApGq into �8pHq.

By the preceding remarks, this is equivalent to determining the maps α forwhich φα maps ApGq into BpHq. We remind the reader that a map ϕ from a groupU into a group G is called anti-homomorphism if ϕpxyq “ ϕpyqϕpxq for all x, y P U .The easier direction of the desired description of homomorphisms is provided bythe following proposition.

Proposition 3.3.1. Let G and H be locally compact groups and U an open sub-group of H. Suppose that ϕ is a continuous homomorphism or anti-homomorphismfrom U into G and that h P H and g P G. Then the map defined by, for u P ApGq,

φpuqpxq “

"

upgϕphxqq : if x P h´1U0 : if x P Hzh´1U,


is a contractive homomorphism from ApGq into BpHq. Moreover, if g “ eG andh “ eH , then φ is positive.

Proof. Assume first that ϕ is a continuous homomorphism. It is clear thatthen φ is a homomorphism from ApGq into BpHq. To see that φ is contractive,we can assume that g “ eG. Note that Lϕphqpφpuqqpyq “ 0 for y R ϕpUq, whereas

Lϕphqpφpuqqpϕpxqq “ φpuqpϕph´1xqq “ upϕpxqq for x P U . It follows that

}φpuq}BpHq “ }Lϕphq}BpHq “ }Lϕphq|U }BpUq

“ }u ˝ ϕ}BpUq ď }u}BpGq.

Finally, if g “ eG and h “ eH , thennÿ

i,j“1

λiλjφpuqpx´1j xiq “

nÿ

i,j“1

λiλjupϕpxjq´1ϕpxiqq

for x1, . . . , xn P U and λ1, . . . , λn P C, and therefore φpP pUqq Ď P pGq.If ϕ is an anti-homomorphism, the statement of the proposition follows from

the homomorphism case by first applying the inverse map H Ñ H,h Ñ h´1. �It will turn out that the problem of characterizing homomorphisms from ApGq

into BpHq can to some extent reduced to discrete groups.

Proposition 3.3.2. Let G and H be discrete groups and φ : ApGq Ñ BpHq ahomomorphism. Let ϕ : U Ñ G be a map from a subset U of H into G such thatφ “ φϕ as in Proposition 3.3.1. Let F be a subgroup of G containing ϕpUq. Thenthe map ψ defined by

ψpuqpyq “

"

upϕpyqq : if y P U0 : if y P HzU

is a homomorphism from ApF q into BpHq with }ψ} ď }φ}.

Proof. For the main statement of the proposition, we have to show that if u P

ApF q is such that }u}ApF q “ 1, then }ψpuq}BpHq ď }φ}. Since ApF q “ �2pF q˚�2pF qq,by Lemma 2.1.6 we can assume that u “ f ˚ qg, where

f “

mÿ

j“1

αjδaj, g “

mÿ

j“1

βjδbj andmÿ

j“1

|αj |2

“

mÿ

j“1

|βj |2

“ 1,

aj , bj P F, αj , βj P C,m P N. On the other hand, by Lemma 2.1.8,

}ψpuq}ApHq “ sup

#ˇ

ˇ

ˇ

ˇ

ˇ

nÿ

i“1

ciψpuqphiq

ˇ

ˇ

ˇ

ˇ

ˇ

+

,

where the supremum is taken over all h1, . . . , hn P H and c1, . . . , cn P C, n P N,such that }

řni“1 ciωHphiq} ď 1. Nowˇ

ˇ

ˇ

ˇ

ˇ

nÿ

i“1

ciψpuqphiq

ˇ

ˇ

ˇ

ˇ

ˇ

“

ˇ

ˇ

ˇ

ˇ

ˇ

ÿ

hiPU

cipf ˚ qgqpϕphiqq

ˇ

ˇ

ˇ

ˇ

ˇ

“

ˇ

ˇ

ˇ

ˇ

ˇ

ÿ

hiPU

ci

˜

mÿ

j,k“1

αjβkδajb´1k

pϕphiqq

¸ˇ

ˇ

ˇ

ˇ

ˇ

.

Now choose a compact neighbourhood V of the identity in G such that

V V ´1Şta´1

j ak, b´1j bk, a

´1j ϕpxiqbk : 1 ď j, k ď m,xi P Uu “ teGu.


Consider the functions ξ and η in �2pGq defined by

ξ “

mÿ

j“1

αj1ajV and η “

mÿ

j“1

βj1bjV .

Then }ξ}2 “ }η}2 “ |V |1{2, and hence the element u “ ξ ˚ qη of ApGq has normď |V |. Consequently, }φpuq} ď }φ} ¨ |V |. It follows from Lemma 2.1.8 and Corollary2.2.2 that

}φ} ¨ |V | ě

ˇ

ˇ

ˇ

ˇ

ˇ

nÿ

i“1

ciφpuqpxiq

ˇ

ˇ

ˇ

ˇ

ˇ

“

ˇ

ˇ

ˇ

ˇ

ˇ

ÿ

xiPU

ciupϕpxiqq

ˇ

ˇ

ˇ

ˇ

ˇ

“

ˇ

ˇ

ˇ

ˇ

ˇ

ÿ

xiPU

ci

˜

mÿ

j,k“1

αjβk |ajV X ϕpxiqbkV |

¸ˇ

ˇ

ˇ

ˇ

ˇ

“ |V | ¨

ˇ

ˇ

ˇ

ˇ

ˇ

ÿ

xiPU

ci

˜

mÿ

j,k“1

αjβkδajb´1k

pϕpxiqq

¸ˇ

ˇ

ˇ

ˇ

ˇ

.

This proves the required inequality. �

In the proof of Lemma 3.3.4 below we shall use the following fact where, forsimplicity of notation, we set Re a “

12 pa` a˚q for an element a of a C˚-algebra A.

Lemma 3.3.3. Let a, b, c, d be elements of a C˚-algebra A satisfying

Rerα2a ` β2b ` αβc ` αβds ě 0

for all α, β P C. Then a “ b “ c “ d “ 0.

Proof. If f is any positive linear functional on A, then

Rerα2fpaq ` β2fpbq ` αβfpcq ` αβfpdqs ě 0

for all α, β P C. This implies that fpaq “ fpbq “ fpcq “ fpdq “ 0 for all f andhence a “ b “ c “ d “ 0. �

Lemma 3.3.4. The set U is an open subgroup of H and, for each x, y P U , theordered pair tϕpxyq, ϕpyxqu is a permutation of the pair tϕpxqϕpyq, ϕpyqϕpxqu. Inparticular, the set ϕ´1peGq is a closed normal subgroup of U and ϕpxyq “ ϕpyq forevery x P ϕ´1peGq and y P U .

Proof. The adjoint map φ˚ : W˚pHq Ñ V NpGq is a positive linear operatorwith φ˚pIqq “ I and satisfies

φ˚pωHpxqq “

"

λGpϕpxqq : for x P U0 : for x P HzU.

(3.1)

Let x, y P U and α, β P C be arbitrary and consider the element

T “ αωHpxq ` β ωHpyq ` αωHpx´1q ` β ωHpy´1

q

of W˚pHq. A straightforward, but lengthy calculation shows that

φ˚pT 2

q “ 2Retp|α|2

` |β|2qI ` α2φ˚

pωHpx2qq ` β2φ˚

pωHpy2qq

`αβφ˚rωHpxyq ` ωHpyxqs ` αβφ˚

rωHpxy´1q ` ωHpy´1xqsu,


and similarly

φ˚pT q

2“ 2Retp|α|

2` |β|

2qI ` α2λGpϕpxq

2q ` β2λGpϕpyq

2q

`αβ rλGpϕpxqϕpyqq ` λGpϕpyqϕpxqqs

`αβ rλGpϕpxqϕpyq´1

q ` λGpϕpyq´1ϕpxqqsu.

Since T “ T˚, Kadison’s generalized Schwarz inequality [146, Theorem 2] givesφ˚pT 2q ě φ˚pT q2. Setting now

a “ φ˚pωHpx2

qq ´ λGpϕpxq2q, b “ φ˚

pωHpyq2q ´ λGpϕpyq

2q,

c “ φ˚rωHpxyq ` ωHpyxqs ´ rλGpϕpxqϕpyqq ` λGpϕpyqϕpxqqs

and

d “ φ˚rωHpxy´1

q ` ωHpy´1xqs ´ rλGpϕpxqϕpyq´1

q ` λGpϕpyq´1ϕpxqqs,

it follows that Rerα2a ` β2b ` αβc ` αβds ě 0 for all α, β P C and hence c “ 0 byLemma 3.3.3. Thus

φ˚rωHpxyqs ` φ˚

rωHpyxqs “ λGpϕpxqϕpyqq ` λGpϕpyqϕpxqq

for all x, y P U . Since the set of operators λGptq, t P G, is linearly independent, thepreceding equation together with (3.1) implies that xy and yx belong to U . SinceU´1 Ď U , U is a subgroup of H. Moreover, taking into account (3.1) again, itfollows that the ordered pair

tφ˚pωHpxyqq, φ˚

pωHpyxqqu “ tλGpϕpxyqq, λGpϕpyxqqu

is a permutation of the pair tλGpϕpxqϕpyqq, λGpϕpyqϕpxqqu. Since λG is injective,for all x, y P U , we therefore either have ϕpxqϕpyq “ ϕpxyq or ϕpxqϕpyq “ ϕpyxq.This of course implies that ϕ´1peGq is a normal subgroup of U and that ϕpxyq “

ϕpyq for every x P ϕ´1peGq and y P U . �A homomorphism φ : ApGq Ñ BpHq is positive if φpuq is positive definite

whenever u is positive definite. We now consider a nonzero positive homomorphismφ : ApGq Ñ BpHq. Let U and ϕ be defined as in Proposition 3.3.1. For everypositive definite u P ApGq, φpuqpeHq “ }φpuq} since φpuq is positive definite. Asφpuq ‰ 0 for some u, we conclude that eH P U and ϕpeHq “ eG. Also, for eachx P U and every positive definite u P ApGq,

φpuqpx´1q “ φpuqpxq “ upϕpxqq “ upϕpxq

´1q,

which forces x´1 P U and upϕpx´1qq “ upϕpxq´1q, whence ϕpx´1q “ ϕpxq´1.

Theorem 3.3.5. Let G and H be locally compact groups, and let φ : ApGq Ñ

BpHq be a positive homomorphism. Then there exists a continuous group homo-morphism or anti-homomorphism ϕ from an open subgroup U of H into G suchthat, for all u P ApGq,

φpuqpxq “

"

upϕpxqq : if x P U0 : if x P HzU.

Proof. For any topological group X, letXd denote the same groupX with thediscrete topology. Since ϕ´1peGq is a normal subgroup of U by Lemma 3.3.4, we canform the quotient group H0 “ Ud{ϕ´1peGq of Ud. Moreover, let G0 “ ϕpUq Ď Gd,and let ϕ0 : H0 Ñ G0 be the map induced from ϕ. Then ϕ0 is injective and inheritsfrom ϕ the property that the ordered pair tϕ0pxyq, ϕ0pyxqu is a permutation of thepair tϕ0pxqϕ0pyq, ϕ0pyqϕ0pxqu for every x, y P U0.


Now let ρ “ φϕ0: ApG0q Ñ BpH0q be defined by

ρpuqpxq “ upϕ0pxqq

for x P G0. Then ρ is a homomorphism by Proposition 3.3.1 with }ρ} ď }φ}.Since ϕ0 is bijective, ρ maps the set of all finitely supported functions on G0 ontothe set of all finitely supported functions on H0. Consequently, ρ maps ApG0q

injectively onto a dense subalgebra of ApH0q. Now consider ρ as map into ApH0q.Since ρpApG0qq is dense in ApH0q, the adjoint ρ˚ maps V NpH0q injectively onto aw˚-dense subspace of V NpG0q.

Since ρ˚pλH0pxqq “ λG0

pϕ0pxqq for all x P H0, we get

ρ˚pST q ` ρ˚

pTSq “ ρ˚pSqρ˚

pT q ` ρ˚pT qρ˚

pSq

for all S, T P λH0pH0q. Extending by linearity and w˚-continuity, it follows that

ρ˚ : V NpH0q Ñ V NpG0q is a Jordan ˚-homomorphism. By a result in [145],ρ˚ is isometric. It follows that ρ˚ is an isometry from V NpH0q onto V NpG0q,and therefore ρ is an isometric isomorphism from ApG0q onto ApH0q. Then, byTheorem 3.2.6, ϕ0 : H0 Ñ G0 must be a group isomorphism or anti-isomorphism.This of course implies that ϕ : U Ñ G is either a group homomorphism or a groupanti-homomorphism. This finishes the proof. �

Corollary 3.3.6. Suppose that φ : ApGq Ñ ApHq is a positive isomorphism.Then there exists a topological isomorphism or anti-isomorphism ϕ : G Ñ H suchthat φpuq “ u ˝ ϕ for all u P ApGq.

Theorem 3.3.7. Let G and H be locally compact groups, and let φ : ApGq Ñ

BpHq be a contractive homomorphism. Then there exist an open subgroup U of H,a continuous homomorphism or anti-homomorphism ϕ from U into G and elementsg P G and h P H such that, for every u P ApGq,

φpuqpxq “

"

upgϕphxqq : if x P h´1U0 : if x P Hzh´1U.

Proof. As shown at the outset of this section, there exist an open subset Vof H and a continuous map ψ : V Ñ G such that

φpuqpxq “

"

upψpxqq : if x P V0 : if x P HzV.

Fix h P V ´1 and put g “ ψph´1q P G and U “ hV . Define ϕ : U Ñ G by

ϕpxq “ g´1ψph´1xq, x P U.

Then φϕ is a contractive homomorphism from ApGq into BpHq. Indeed, φ is con-tractive and φϕ is a composition of φ with translations by elements of G and H.Moreover, eH P U and ϕpeHq “ g´1ψph´1q “ eG. Therefore, if u P ApGq is positivedefinite, then

φϕpuqpeHq “ upeGq “ }u}ApGq ě }φϕpuq}BpHq.

Thus φϕpuq is also positive definite. The result now follows from Theorem 3.3.5. �

Corollary 3.3.8. Suppose that φ : ApGq Ñ BpHq is a contractive homomor-phism. If φ is also injective, then φ is an isometry.


Proof. By Theorem 3.3.7, after applying translations and the inverse mapx Ñ x´1 of H (if necessary), we can assume that φ “ φϕ, where ϕ is a continuoushomomorphism from an open subgroup U of H into G.

Since ϕ is injective, ϕpUq must be dense in G because otherwise there exists0 ‰ u P ApGq such that u “ 0 on ϕpUq and hence φαpuq “ 0. Let M denote the vonNeumann subalgebra of W˚pHq generated by the operators ωHpyq, y P U . Sinceφ˚pωHpyqq “ λGpyq for y P U and ϕpUq is dense in G, it follows that φ˚ : M Ñ

V NpGq is a surjective ˚-homomorphism. Because }φ˚} “ 1, φ˚ maps the unit ballof W˚pHq onto the unit ball of V NpGq. Thus, for any u P ApGq,

}u}ApGq “ sup t|xu, φ˚pSqy| : S P W˚

pHq, }S} ď 1u

“ sup t|xφpuq, Sy| : S P W˚pHq, }S} ď 1u

“ }φpuq}BpGq

since W˚pHq “ BpHq˚. Hence φ is isometric. �

The following corollary generalizes Theorem 3.2.6. The proof, however, usesTheorem 3.2.6.

Corollary 3.3.9. Suppose that φ : ApGq Ñ ApHq is a contractive homomor-phism such that φpApGqq separates the points of H, and assume that H has at leasttwo elements. Then, for each v P ApHq, there exists u P ApGq such that φpuq “ vand }u} “ }v}.

Proof. Again, by Theorem 3.3.7, we can assume that φ “ φϕ, where ϕ is acontinuous homomorphism from some open subgroup U of H into G. Observe nextthat ϕ is a proper map. Indeed, suppose that there exists a compact subset C ofG such that ϕ´1pCq is not compact. Then, for u P ApGq with u|C “ 1 and anyx P ϕ´1pCq we would have φpuqpxq “ upϕpxqq “ 1, which is impossible since uvanishes at infinity. Now ϕ, being proper and continuous, is a closed map.

Since φpApGqq separates the points of H, ϕ must be injective and HzU canat most have two elements. As H has more than two elements, this implies thatU “ H. Thus we have seen that ϕ is a topological isomorphism from H onto theclosed subgroup ϕpHq of G. Given v P ApHq, we have v ˝ ϕ´1 P ApϕpHqq and}v ˝ ϕ´1} “ }v}, and by Theorem 3.2.6 there exists u P ApGq such that u|ϕpHq “

v ˝ ϕ´1 and }u}ApGq “ }v ˝ ϕ´1}. �

The proof of the preceding corollary shows that φ is the composition of therestriction map ApGq Ñ ApϕpHqq and the isometric isomorphism ApϕpHqq Ñ

ApHq. The following corollary extends Theorem 3.2.6 and [136, Corollary 3.12].

Corollary 3.3.10. Suppose that φ : ApGq Ñ ApHq is a contractive iso-morphism. Then there exist a topological group isomorphism or anti-isomorphismϕ : H Ñ G and an element g P G such that, for all u P ApGq and x P H,

φpuqpxq “ upgϕpxqq.

Theorem 3.3.11. Let G and H be locally compact groups, and let φ : BpGq Ñ

BpHq be an isomorphism which is positive on ApGq. Then there exists a topologicalgroup isomorphism or anti-isomorphism ϕ from H onto G such that φpuq “ u ˝ ϕfor all u P BpGq. In particular, φ and φ´1 are both positive.

3.4. INVARIANT SUBALGEBRAS OF V NpGq AND SUBGROUPS OF G 107

Proof. Recall first that, since Fourier-Stieltjes algebras are semisimple, φ isbounded. Since the restriction φ|ApGq : ApGq Ñ BpHq is positive, by Theorem 3.3.5there exist an open subgroup U of H and a continuous group homomorphism oranti-homomorphism ϕ : U Ñ G such that φpuq “ φϕpuq for all u P ApGq. Applyingthe inversion x Ñ x´1 if necessary, we can assume that ϕ is a homomorphism.Since φ is injective, it follows that ϕpUq is dense in G.

Now consider the adjoint φ˚ : W˚pHq Ñ W˚pGq, which is a bounded linearisomorphism. Then we have φ˚pωHqpHq Ď σpBpGqq, the spectrum of BpGq. Foreach x P U , φ˚pωHpxqq is a multiplicative linear functional on BpGq which mapsu P ApGq to upϕpxqq. By Lemma 3.2.5, this implies that φ˚pωHpxqq “ ωGpyq forsome y P G. It follows that y “ ϕpxq. In addition, the map ωG : G Ñ ωGpGq

is a homeomorphism when ωGpGq Ď σpBpGqq is equipped with the relative w˚-topology, and the analogous statement holds for H. Since φ˚ is a homomorphismfrom W˚pHq onto W˚pGq, where both W˚pHq and W˚pGq carry the w˚-topology,we conclude that ϕ is a homeomorphism from U onto ϕpUq. Thus ϕpUq is a locallycompact subgroup of G. On the other hand, ϕpUq is dense in G, and thereforeϕpUq “ G.

It remains to show that U “ H. Let M denote the w˚-closed subspace ofW˚pHq generated by the set ωHpHq. Then φ˚ maps M onto W˚pGq, and thereforeM “ W˚pHq. This in turn implies that, for every u P BpHq, u|U “ 0 forces u “ 0.Regularity of ApHq implies that U “ H, as required. �

We conclude this section with a corollary which extends [136, Corollary 5.5(ii)]and Theorem 3.2.5.

Corollary 3.3.12. Let G and H be locally compact groups and let φ : BpGq Ñ

BpHq be an isomorphism such that φ|ApGq is contractive. Then there exist a topo-logical group isomorphism or anti-isomorphism ψ from H onto G and an elementa P G such that φpuqpxq “ upaψpxqq for all x P H. In particular, φ is isometric onBpGq and φpApGqq “ ApHq.

Proof. Since φ is an isomorphism and φ|ApGq is contractive, by Theorem 3.3.7there exists a continuous homomorphism or anti-homomorphism ϕ from H onto Gand elements g P G and h P H such that φpuqpxq “ upgϕphxqq for all u P ApGq andx P H. Then let a “ gϕphq and define ψ by ψpxq “ ϕpxq or ψpxq “ ϕphq´1ϕpxqϕphq

depending on whether ϕ is a homomorphism or anti-homomorphism. �

3.4. Invariant subalgebras of V NpGq and subgroups of G

For a locally compact abelian groupG, the Pontryagin duality theorem providesa beautiful and complete one-to-one correspondence between closed subgroups H

of G and closed subgroups HK of the dual group pG such that pH “ pG{HK andzG{H “ HK. In the nonabelian situation one cannot expect such a duality withinthe category of groups. However, Takesaki and Tatsuuma [271] have establishedcorrespondences between closed subgroups of G and certain von Neumann subal-gebras of L8pGq and V NpGq.

Let G be an arbitrary locally compact group. The first theme of this sectionis to provide a Galois type correspondence between closed subgroups of G andright translation invariant von Neumann subalgebras of L8pGq. We start with twopreparatory lemmas.


Lemma 3.4.1. Let E be a right translation invariant w˚-closed linear subspaceof L8pGq. Then E X CpGq is w˚-dense in E.

Proof. Fix f P E. We first show that if u P L1pGq is such that }u}1 “ 1,u ě 0 and ru “ u, then f ˚ u P E. The function u can be regarded as a mean onCbpGq. Hence there exists a net puαqα of means, where each uα is a finite linearcombination uα “

řnα

j“1 cα,jδxα,jof point evaluations, such that uα Ñ u in the

w˚-topology of CbpGq˚. Now, for g P L1pGq,

xf ˚ ru, gy “ xpΔ´1rgq ˚ f, uy “ lim

αxpΔ´1

rgq ˚ f, uαy

“ limα

nαÿ

j“1

cα,jpΔ´1rgq ˚ fqpxα,jq

“ limα

nαÿ

j“1

cα,j

ż

G

fpyxα,jqgpyqdy

“ limα

C

nαÿ

j“1

cα,jRxα,jf, g

G

.

Thus f ˚ ru is the w˚-limit in L8pGq of the net˜

nαÿ

j“1

cα,jRxα,jf

¸

α

Ď E.

Since E is w˚-closed, f ˚ ru P E.Now L1pGq admits an approximate identity consisting of functions u as above.

Since xf ˚ ru, gy “ xf, g ˚ uy for all g P L1pGq, it follows that f ˚ ru Ñ f in thew˚-topology. This completes the proof. �

Lemma 3.4.2. Let A be a right translation invariant W˚-subalgebra of L8pGq.If A ‰ t0u, then A contains the constant functions.

Proof. Let h be the identity of A. Then, for all f P A and x P G,

pRxhqf “ RxphRx´1pfqq “ RxpRx´1pfqq “ f “ hf

and hence Rxh “ h. Choose any nonzero f P A X CpGq (Lemma 3.4.1) and letV “ ty P G : fpyq ‰ 0u. Then h “ 1 locally almost everywhere on V , and sinceRxh “ h and every compact subset of G is contained in a finite union of sets RxV ,x P G, it follows that h “ 1 locally almost everywhere on G. �

The following theorem is the precise formulation of the Galois type correspon-dence announced above.

Theorem 3.4.3. There exists a one-to-one correspondence between closed sub-groups H of G and right translation invariant von Neumann subalgebras A ofL8pGq which is determined by

A “ tf P L8pGq : Lxf “ f for all x P Hu

and

H “ tx P G : Lxf “ f for all f P Au.


Proof. Let H be a closed subgroup of G. Set

ApHq “ tf P L8pGq : Lxf “ f for all x P Hu.

Clearly, ApHq is a right invariant von Neumann subalgebra of L8pGq. Conversely,for a right invariant von Neumann subalgebra A of L8pGq, put

HpAq “ tx P G : Lxf “ f for all f P Au.

Then HpAq is a subgroup of G, and since the map x Ñ Lxf , f P L8pGq, from Ginto L8pGq is continuous for the w˚-topology on L8pGq and A is w˚-closed, HpAq

is closed in G. We claim that

HpApHqq “ H and ApHpAqq “ A.

The inclusions H Ď HpApHqq and A Ď ApHpAqq are both clear from the definitionof ApHq and HpAq. Let x be an arbitrary element of GzH. Choose g P CcpGq withg|H “ 0, 0 ď g ď 1 and gpx´1q “ 1 and put

fpyq “

ż

H

gpt´1yqdt, y P G.

Then Lsf “ f for all s P H, fpeq “ 0 and fpx´1q ‰ 0. Thus f P ApHq, butLxf ‰ f , and hence x R HpApHqq. This shows that HpApHqq “ H.

It remains to prove that ApHpAqq Ď A. By Lemma 3.4.1 it suffices to show thatApHpAqq X CpGq Ď A. For sake of brevity, put B “ ApHpAqq. Let GzH denotethe homogeneous space of right cosets Hx, x P G. Then GzH is a locally compact

Hausdorff space. To any f P B X CpGq we associate the continuous function rf on

GzH defined by rfpHxq “ fpxq.Fix a compact subset K of GzH and let AK denote the algebra of functions

rf |K , where f P A X CpGq. Then AK contains the constant functions (Lemma3.4.2) and is closed under complex conjugation. Moreover, AK separates the pointsof K. Indeed, if Hx1 and Hx2 are two distinct right cosets, then x1x

´12 R H

and hence there exists f P A X CpGq such that Lx1x´12f ‰ f . This implies that

fpx1yq ‰ fpx2yq for some y P G and hence the function Ryf P AXCpGq separatesx1 and x2. The Stone-Weierstrass theorem now yields that AK is uniformly densein CpKq.

Observe next that on bounded spheres in L8pGq the w˚-topology coincideswith the topology defined by the seminorms

f Ñ

ˇ

ˇ

ˇ

ˇ

ż

G

fpxqψpxqdx

ˇ

ˇ

ˇ

ˇ

, ψ P CcpGq, }ψ}1 ď 1.

Therefore, since A is w˚-closed in L8pGq, to show that B X CpGq Ď A, it sufficesto prove that given f P B X CpGq, ε ą 0 and ψ1, . . . , ψn P CcpGq with }ψj}1 ď 1,1 ď j ď n, there exists g P A such that

ˇ

ˇ

ˇ

ˇ

ż

G

rfpxq ´ gpxqsψjpxqdx

ˇ

ˇ

ˇ

ˇ

ď ε

for j “ 1, . . . , n. To that end, let C “ Ynj“1suppψj and K “ HCzH. It follows

from the density of AK in CpKq that there exists h P A X CpGq such that

supxPC

|fpxq ´ hpxq| ď ε.


Let ϕ : C Ñ C be a continuous function such that ϕpzq “ z for all z with |z| ď

}f}8 ` ε and |ϕpzq| ď }f}8 ` ε if |z| ě }f}8 ` ε. Then g “ ϕ ˝ h P A by[256, Proposition 1.18.1] and supxPC |fpxq ´ gpxq| ď ε and therefore

ˇ

ˇ

ˇ

ˇ

ż

G

rfpxq ´ gpxqsψjpxqdx

ˇ

ˇ

ˇ

ˇ

ď ε, 1 ď j ď n.

This completes the proof. �

Let γ : L8pGq Ñ BpL2pGqq be the embedding of L8pGq into BpL2pGqq definedby γpfqg “ fg, g P L2pGq. Then γ is a w˚-w˚-continuous isomorphism ontoγpL8pGqq. In particular, γpL8pGqq is a von Neumann subalgebra of BpL2pGqq

isomorphic to L8pGq.The following proposition is the main step towards establishing a one-to-one

correspondence between right invariant von Neumann subalgebras of γpL8pGqq andvon Neumann subalgebras of V NpGq which are invariant in a sense to be definedlater. The proof of part (ii) builds on the notion of induced representations and atheorem of Blattner and Mackey which provides an isomorphism between certainspaces of intertwining operators. Concerning the Blattner-Mackey theorem, werefer the reader to [15].

Proposition 3.4.4. Let B be a von Neumann subalgebra of γpL8pGqq which isright translation invariant in the sense that Rxf P γ´1pBq for all f P γ´1pBq andx P G. Then there exists a closed subgroup H of G with the following properties.

(i) γ´1pBq “ tf P L8pGq : Lxf “ f for all x P Hu.(ii) V NpGq X B1 “ V NHpGq, where B1 denotes the commutant of B in

BpL2pGqq.

Proof. (i) has been shown in Theorem 3.4.3 with H being defined by

H “ tx P G : Lxf “ f for all f P γ´1pBqu.

We claim that H “ tx P G : λGpxq P B1u. This can be seen as follows. If x P H,h P L2pGq and γpfq P B, then

λGpxqγpfqh “ pLxfqpLxhq “ fpLxhq “ γpfqpλGpxqhq.

Conversely, if λGpxq P B1 and γpfq P B, then for each h P L2pGq,

γpLxfqpLxhq “ Lxpfhq “ λGpxqpγpfqhq “ γpfqpλGpxqhq “ γpfqpLxhq.

Since LxpL2pGqq “ L2pGq, this implies that γpLxfq “ γpfq, and hence Lxf “ f ,so that x P H.

We shall now prove that V NpGq X B1 “ V NHpGq. For any locally compactgroup K, let ρK denote the right regular representation of K and recall that thecommuting algebra of ρKpKq in BpL2pKqq is equal to V NpKq. The unitary rep-

resentation of G obtained by inducing ρK up to G, indGK ρK , can be realized in

the Hilbert space L2pGq, and indGK ρK is then equivalent to ρG. The equivalence isfurnished by the unitary operator U : L2pGq Ñ L2pGq given by

Uξpyq “ ΔGpy´1q1{2ξpy´1

q, ξ P L2pGq, y P G.

For ϕ P C0pG{Hq, let P pϕq P BppL2pGqq be the multiplication operator definedby P pϕqξpyq “ ϕpyHqξpyq. Let P denote the algebra of all operators P pϕq, ϕ P


C0pG{Hq. To show that V NpGq X B1 “ V NHpGq, we now apply the Blattner-Mackey theorem for induced representations [15], which in the current situationasserts the existence of an isomorphism

ρHpHq1

Ñ indGH ρHpGq1X P 1.

Since ρGpxq “ UpindGH ρHqpxqU´1 for all x P G, we thereby obtain an isomorphism

φ : V NpHq Ñ pU indGH ρHpGqU´1q

1X pUPU´1

q1.

By construction of the above isomorphism, φpλHpxqq “ λGpxq for all x P H. Notenext that for ξ P L2pGq, ϕ P C0pG{Hq and y P G,

pUP pϕqU´1ξqpyq “ ΔGpyq´1{2ϕpy´1HqpU´1ξqpy´1

q “ ϕpy´1Hqξpyq.

It follows that the set of all operators UP pϕqU´1, ϕ P C0pG{Hq, equals the set ofall operators γpfq, where f is continuous and constant on right cosets of H and,viewed as a function on GzH, vanishes at infinity. Let C denote the algebra of allthese operators γpfq. Then C is dense in B and this implies that

pUPU´1q

1“ C1

“ B1.

Since φpλHpxqq “ λGpxq for all x P H, we conclude that φpV NpHqq “ V NHpGq

and hence V NHpGq “ V NpGq X B1. �Now recall that for T P V NpGq and u P ApGq, Tu P ApGq is defined by

xTu, Sy“xu, TSy for S P V NpGq, and that Tu satisfies supppTuq Ď psuppT qpsuppuq

whenever u has compact support (Lemma 2.5.7).

Lemma 3.4.5. If u P ApGq X L2pGq, then RTu P L2pGq for every T P V NpGq.

Proof. Let ξ, η P L2pGq be such that upxq “ xλGpxqξ, ηy for all x P G. Then,for any S P V NpGq, xS, uy “ xSpξq, ηy. Now, let T P V NpGq and let pTαqα be anet in V NpGq converging to T in the strong operator topology. Then

}Tαu ´ Tu} “ supt|xTαu ´ Tu, Sy| : S P V NpGq, }S} ď 1u

“ supt|xu, SpTα ´ T qy| : S P V NpGq, }S} ď 1u

“ supt|xSpTα ´ T qξ, ηy| : S P V NpGq, }S} ď 1u

“ supt|xpTα ´ T qξ, S˚ηy| : S P V NpGq, }S} ď 1u

ď }pTα ´ T qξ} ¨ }η},

and hence }Tαu ´ Tu} Ñ 0.Now λGpxqu “ Rxu for all x P G. It follows that Su P L2pGq whenever S is a

finite linear combination of operators λGpxq, x P G. Since these operators are densein V NpGq in the strong operator topology, the net pTαqα above can be chosen sothat Tαu P L2pGq for all α. This implies that Tu P L2pGq for each T P V NpGq. �

Theorem 3.4.6. Let H be a closed subgroup of G and T P V NpGq. ThenT P V NHpGq if and only if suppT Ď H.

Proof. If T P V NHpGq, then T is a w˚-limit of finite linear combinationsof operators λGpxq, x P H. Since suppλGpxq “ txu, it follows from Proposition2.5.6(iv) that supp T Ď H.

Conversely, suppose that suppT Ď H and let V be any open subset of G withV “ HV . For every u P ApGq X CcpGq with supp u Ď V , we then have

supppTuq Ď suppT ¨ supp u Ď HV “ V.


Let PV : L2pGq Ñ L2pGq denote the projection defined by PV h “ 1V h. Since theset of all u P ApGq X CcpGq with supp u Ď V forms a dense linear subspace ofPV pL2pGqq, it follows that T pPV pL2pGqqq Ď PV pL2pGqq, that is, TPV “ PV TPV .Similarly, since suppT˚ “ psuppT q´1 Ď H, we get that T˚PV “ PV T

˚PV as well.Consequently, PV T “ PV TPV . Therefore T and PV commute whenever V is anopen subset of G with V “ HV . Such projections PV generate a von Neumannsubalgebra BH of γpL8pGqq which is right translation invariant in the sense ofProposition 3.4.4 since Rx1V “ 1V x´1 and V x´1 “ HV x´1. By Proposition 3.4.4there exists a closed subgroup K of G such that

γ´1pBHq “ tf P L8

pGq : Lxf “ f for allx P Ku

and V NpGq XB1H “ V NKpGq. Now, if x P G is such that 1V “ Lx1V “ 1xV for all

open subsets V of G with V “ HV , then x P H. It follows that K Ď H and henceV NKpGq Ď V NHpGq. Consequently, T P V NpGq X B1

H Ď V NHpGq, as was to beshown. �

Corollary 3.4.7. Let M be an invariant W˚-subalgebra of V NpGq and letH “ tx P G : λGpxq P Mu. Then M “ V NHpGq.

Proof. Clearly, H is a closed subgroup of G and V NHpGq Ď M. Conversely,if T P M then suppT Ď H. In fact, if λGpxq is a w˚-limit of operators of the formu ¨ T , u P ApGq, then λGpxq P M since M is invariant, and hence x P H by thedefinition of H. Now suppT Ď H implies T P V NHpGq by Theorem 3.4.6. �

A W˚-subalgebra M of V NpGq is said to be invariant if for any T P M andu P ApGq, u ¨ T P M, where xu ¨ T, vy “ xT, uvy for v P ApGq. Clearly, V NHpGq isan invariant W˚-subalgebra for every closed subgroup H of G. However, Corollary3.4.7 tells us that there are no other invariant W˚-subalgebras of V NpGq. Thenext theorem is the result announced prior to Proposition 3.4.4.

The next theorem is the second main result of this section.

Theorem 3.4.8. Let G be a locally compact group. There is a one-to-onecorrespondence between right invariant von Neumann subalgebras B of γpL8pGqq

and invariant von Neumann subalgebras M of V NpGq which is determined by

M “ B1X V NpGq and B “ M1

X γpL8pGqq.

Proof. Let B be a right invariant von Neumann subalgebra of γpL8pGqq andlet H be the closed subgroup of G defined by

H “ tx P G : Lxf “ f for all f P γ´1pBqu.

Then, by Proposition 3.3.4(ii), B1 X V NpGq “ V NHpGq.Let M “ V NHpGq. Then B “ M1 X γpL8pGqq. Indeed, if f P γ´1pBq then for

all x P H and h P L2pGq,

rλGpxqγpfqsh “ λGpxqpfhq “ pLxfqpLxhq “ fpλGpxqhq “ rγpfqλGpxqsh,

so that γpfq P M1. Conversely, if f P L8pGq is such that γpfq P M1, then for eachfixed x P H and all h P L2pGq,

pLx´1fqh “ Lx´1pfλGpxqhq “ Lx´1pγpfqλGpxqhq “ Lx´1pλGpxqfhq “ fh

and therefore Lx´1f “ f locally a.e. on G. Since this holds for all x P H, f P

γ´1pBq. Thus B “ M1 X γpL8pGqq.

3.5. INVARIANT SUBALGEBRAS OF ApGq AND BpGq 113

LetM be an invariant von Neumann subalgebra of V NpGq. By Corollary 3.4.7,M “ V NHpGq, where H “ tx P G : λGpxq P Mu. Let

B “ tγpfq : f P L8pGq such that Lxf “ f for all x P Hu.

Then M “ B1 XV NpGq. To see this, recall that since γ´1pBq is a right translationinvariant w˚-closed subalgebra of L8pGq, by Proposition 3.4.4 there exists a closedsubgroup K of G such that B1 X V NpGq “ V NKpGq, where K is given by

K “ tx P G : Lxf “ f for all f P γ´1pBqu.

Then, by Proposition 3.4.4(i),

γ´1pBq “ tf P L8

pGq : Lxf “ f for all x P Ku.

On the other hand, by the definition of B,

γ´1pBq “ tf P L8

pGq : Lxf “ f for all x P Hu.

This implies that H “ K and hence B1 XV NpGq “ V NHpGq “ M. This completesthe proof of the theorem. �

Remark 3.4.9. It should be pointed out that Takesaki and Tatsuuma [271]defined a notion of invariance for a von Neumann subalgebra M of V NpGq in termsof the normal isomorphism π : V NpGq Ñ V NpGqbV NpGq given by

πpT q “ W´1pT b IqW, T P V NpGq,

where W is the unitary operator on L2pG ˆ Gq defined by

Wfpx, yq “ fpx, xyq, f P L2pG ˆ Gq, x, y P G.

More precisely, they called M invariant if πpMq Ď MbM. They proved that M isinvariant in this sense if and only if there exists a closed subgroup H of G such thatM “ V NHpGq. Corollary 3.4.7 above now shows that this notion of invariance isequivalent to the one we have given.

3.5. Invariant subalgebras of ApGq and BpGq

In this section we are concerned with the structure of translation invariant ˚-subalgebras of BpGq. The first main result (Theorem 3.5.3) says that if such asubalgebra A is w˚-closed and separates the points of G, then A contains ApGq.This yields a description of the w˚-closed, invariant ˚-subalgebras of BpGq whenG is amenable: they turn out to be exactly the subalgebras of BpGq of the formBpG{Nq, where N is a closed normal subgroup of G and BpG{Nq is viewed as asubalgebra of BpGq in the obvious way.

Let P`pGq denote the cone consisting of all u P P pGq such that upxq ě 0 forall x P G, and view P`pGq as a subset of MpGq “ CcpGq˚, the space of all (notnecessarily bounded) Radon measures on G. The following remarkable property ofP`pGq is fundamental for all the results displayed in this section.

Lemma 3.5.1. Let Q Ď P`pGq be a subcone and suppose that Q is closed underpointwise multiplication and that Q separates the identity e from each other elementof G. Then there exists a net puαqα in Q such that uαdx Ñ δe in the w˚-topologyof MpGq.


Proof. It suffices to show that given a compact subset K of G with e P K,an open neighbourhood U of e and ε ą 0, there exists u P Q such that

ż

U

upxqdx “ 1 and

ż

KzU

upxqdx ď ε.

In fact, if f P CcpGq is such that supp f Ď K, then

|xδe, fy ´ xudx, fy| ď

ˇ

ˇ

ˇ

ˇ

ż

U

upxqpfpeq ´ fpxqqdx

ˇ

ˇ

ˇ

ˇ

`

ˇ

ˇ

ˇ

ˇ

ˇ

ż

KzU

fpxqupxqdx

ˇ

ˇ

ˇ

ˇ

ˇ

ď supxPU

|fpeq ´ fpxq| ` }f}8ε.

Let Q1 “ tu P Q : upeq “ 1u. For u P Q1 and 0 ă δ ă 1, let

Uu,δ “ tx P G : upxq ě δu.

Let K, U and ε be given. We claim that there exist u P Q1 and 0 ă δ ă 1 suchthat K X Uu,δ Ď U . Towards a contradiction, assume that

K X Uu,δ X pGzUq ‰ H

for every u P Q1 and 0 ă δ ă 1. Now, the family of sets

tK X Uu,δ X pGzUq : u P Q1, 0 ă δ ă 1u

has the finite intersection property since obviously

Uu,δ X Uv,η Ě Uuv,maxtδ,ηu

for any δ, η ą 0 and u, v P Q1. Since K is compact, it follows that there existsx P GzU with upxq “ 1 for all u P Q1. However, this is a contradiction since Q1

separates e from x. Consequently, there exist v P Q1 and 0 ă δ ă 1 such thatK X Uv,δ Ď U . Then

ż

KzU

vnpxqdx ď δn|KzU |

for all n P N. Fix any η with δ ă η ă 1 and let V “ Uv,η X U . Thenż

U

vnpxqdx ě

ż

V

vnpxqdx ě ηn|V |

and henceˆż

U

vnpxqdx

˙´1 ż

KzU

vnpxqdx ď

ˆ

δ

η

˙n|KzU |

|V |

for all n P N. Since pδ{ηqn Ñ 0, there exists N P N such thatˆż

U

vnpxqdx

˙´1 ż

KzU

vnpxqdx ď ε.

Let u “`ş

UvN pxqdx

˘´1vN P Q. Then

ż

U

upxqdx “ 1 and

ż

KzU

upxqdx ď ε,

as required. �

3.5. INVARIANT SUBALGEBRAS OF ApGq AND BpGq 115

Let M be a von Neumann algebra and A its predual. For x P A, as before let|x| P A denote the absolute value of x and let x˚ P A be defined by xx˚, fy “ xf˚, xy,f P M . Recall that for x P A and f P M the products f ¨ x P A and x ¨ f P Aare defined by xf ¨ x, gy “ xgf, xy and xx ¨ f, gy “ xx, fgy, g P M , respectively (see[60, 12.2.1]).

Lemma 3.5.2. Let X be a closed subspace of A. Suppose that X is M -invariantin the sense that M ¨ X Ď X and X ¨ M Ď X. Then |x| P X and x˚ P X for everyx P X.

Proof. The annihilator XK of X in M is a w˚-closed subspace and an idealsinceX isM -invariant. HenceX is the predual of the von Neumann algebraM{XK.

Let x P X. Then x˚ P X since xx˚, fy “ xf˚, xy “ 0 for f P XK. Let |x| P A and|x|1 P X be the absolute value of x, viewed as a functional on M and as a functionalon M{XK, respectively. The characterization of the absolute value in [60, 12.2.9]shows that |x| “ |x|1 and hence |x| P X. �

A subset A of BpGq is said to be invariant if Lxu,Rxu P A for all u P A andx P G.

Theorem 3.5.3. Let G be a locally compact group and A a w˚-closed invari-ant subalgebra of BpGq. Suppose that A is closed under complex conjugation andseparates the points of G. Then A contains ApGq.

Proof. Since the Dirac measures generate a w˚-dense subalgebra of W˚pGq,the enveloping von Neumann algebra of C˚pGq, and A is w˚-closed and invariant,it follows that A is W˚pGq-invariant. In the following, for u P BpGq we denote by|u| P P pGq the absolute value of u and by u˚ the adjoint of u, viewed as a linearfunctional on W˚pGq. Let

|A| “ t|u| : u P Au, A˚ “ tu˚ : u P Au and Asa “ tu P A : u “ u˚u.

Then, by Lemma 3.5.2, |A| “ A X P pGq and A “ A˚. Hence

Asa “ tu ´ v : u, v P |A|u,

and therefore |A| separates the points of G. Let Re|A| “ tReu : u P |A|u. Since Ais closed under conjugation, Re|A| Ď A X P pGq. Moreover, Re|A| separates e fromevery x P G, x ‰ e. Now, Re|A| is a convex cone and therefore Q “ tu2 : u P Re|A|u

also separates e from every x P G, x ‰ e. As A is an algebra, Q is contained inA and closed under multiplication. By Lemma 3.5.1, there exists a net puαqα in Qsuch that

xuα, fy “

ż

G

fpxquαpxqdx Ñ xδe, fy “ fpeq

for every f P CcpGq.

For ApGq Ď A, it suffices to show that f ˚ rf P A for all f P CcpGq, whererfpxq “ fpx´1q. Fix f P CcpGq. Then f ¨ uα ¨ f˚ P |A| by the W˚pGq-invariance ofA. Since

limα

}f ¨ uα ¨ f˚}BpGq “ lim

αpf ¨ uα ¨ f˚

qpeq “ limα

xuα, f˚

˚ fy “ pf˚˚ fqpeq,

we can assume that pf ¨uα ¨f˚qα is a bounded net in BpGq. Now, for all g P CcpGq,

xf ˚ rf, gy “ pf˚˚ g ˚ fqpeq “ lim

αxuα, f

˚˚ g ˚ fy “ lim

αxf ¨ uα ¨ f˚, gy.


Thus f ˚ rf “ limα f ¨ uα ¨ f˚ in the w˚-topology of BpGq. Since A is w˚-closed and

invariant under complex conjugation, we conclude that f ˚ rf P A, as required. �

Corollary 3.5.4. For a locally compact group G, the following conditions areequivalent.

(i) G is amenable.(ii) The map N Ñ BpG{Nq is a bijection between the set of all closed nor-

mal subgroups of G and the set of all w˚-closed, translation invariant˚-subalgebras A of BpGq with A ‰ t0u.

Proof. (i) ñ (ii) It is clear that for any closed normal subgroup N of G,BpG{Nq is a w˚-closed, translation invariant ˚-subalgebras A of BpGq and thatBpG{N1q “ BpG{N2q implies that N1 “ N2.

Now, let A be any w˚-closed, translation invariant ˚-subalgebra of BpGq andlet N “ tx P G : Lxu “ u for all u P Au. Then N is a closed normal subgroup ofG. Moreover, A viewed as a subalgebra of BpG{Nq separates the points of G{N .Theorem 3.5.3 implies that ApG{Nq Ď A. Since G{N is amenable, ApG{Nq isw˚-dense in BpG{Nq and hence A “ BpG{Nq since A is w˚-closed.

(ii) ñ (i) Let A be the w˚-closure of ApGq in BpGq. By (ii), A “ BpG{Nq forsome closed normal subgroup N of G. But ApGq separates the points of G and soN “ teu and A “ BpGq. However, since A “ BλpGq, this means that λG weaklycontains every representation of G. This in turn implies that G is amenable. �

Let G and H be locally compact groups and let φ : H Ñ G be a continuoushomomorphism. Let j : BpGq Ñ BpHq denote the homomorphism defined byjpuq “ u ˝ φ, u P BpGq.

Corollary 3.5.5. Suppose that φ is injective and H is amenable. ThenjpBpGqq is w˚-dense in BpHq.

Proof. This follows from the preceding corollary since jpBpGqq is an invariant˚-subalgebra of BpHq which separates the points of H. �

In passing we give two immediate applications of Corollary 3.5.5.

Example 3.5.6. (1) Let Gd be the group G with the discrete topology, and letφ : Gd Ñ G be the identity map. Suppose that Gd is amenable. Then Corollary3.5.5 shows that BpGq is w˚-dense in BpGdq.

(2) Let bpGq denote the Bohr compactification of G and let AP pGq be thealgebra of almost periodic functions on G. Recall that G is said to be maxi-mally almost periodic if AP pGq separates the points of G. Let φ : G Ñ bpGq bethe canonical homomorphism. Since jpCpbpGqqq “ AP pGq, by Theorem 2.2.1(ii),jpBpbpGqqq “ BpGq X AP pGq. Consequently, BpGq X AP pGq is w˚-dense in BpGq

provided that G is an amenable maximally almost periodic group.

Next we employ the methods that led to Theorem 3.5.3 to obtain a descriptionof the norm closed, invariant ˚-subalgebras of ApGq.

Theorem 3.5.7. Let G be a locally compact group and let A be a norm-closed,translation invariant ˚-subalgebra of ApGq. If A ‰ t0u, then A “ ApG{Kq for somecompact normal subgroup K of G.

3.6. COMPARISON OF ApG1q pb ApG2q AND ApG1 ˆ G2q 117

Proof. Let K “ tx P G : Lxu “ u for all u P Au. Then K is a closed normalsubgroup of G and A Ď ApG{Kq. Since A ‰ t0u and ApGq Ď C0pGq, K mustbe compact. Moving to G{K, we can assume that K “ teu and then have toshow that A “ ApGq. Observe next that A is V NpGq-invariant. Indeed, we haveu ¨ λpx´1q “ Lxu P A and λpxq ¨ u “ Rxu P A for all u P A and x P G, and the setof all T P V NpGq satisfying A ¨T Ď A and T ¨A Ď A forms a w˚-closed subalgebraof V NpGq. Hence, by Lemma 3.5.2,

A X P pGq “ t|u| : u P Au and A “ tu˚ : u P Au.

As we have seen in the proof of Theorem 3.5.3, for each u P ApGq X P 1pGq, thereexists a net puαqα in A X P 1pGq such that uα Ñ u in the w˚-topology of BpGq “

C˚pGq˚. Since, as we shall see in Theorem 3.7.7, on the unit sphere of BpGq thew˚-topology coincides with the multiplier topology, it follows that

}uαv ´ uv}ApGq Ñ 0

for every v P ApGq. Taking v P A gives uA Ď A for all u P ApGq X P 1pGq, and thisimplies that A is an ideal in ApGq. This in turn yields that A “ ApGq. In fact, ifA is a proper closed ideal of ApGq, then there exists x P G such that upxq “ 0 forall u P A. This is a contradiction since A is translation invariant and A ‰ t0u. �

We finish this section by giving an application of Theorem 3.5.7 to the struc-ture of norm closed, invariant subalgebras of BpGq. Recall that for any unitaryrepresentation π of G, the space AπpGq is the closed linear subspace of BpGq gen-erated by the coefficients of π and that the spaces AπpGq are precisely the closed,invariant subspaces of BpGq (Lemma 2.8.3(ii)). The set of all linear combinationsof coefficients of all unitary representations of G which are disjoint from the leftregular representation is of the form AωpGq for some representation ω, and then wehave the Lebesgue decomposition BpGq “ ApGq ‘1 AωpGq (Proposition 2.8.9(i)).

Corollary 3.5.8. Let G be a locally compact group and let A be a norm closed,translation invariant ˚-subalgebra of BpGq which separates the points of G. Theneither ApGq Ď A or A Ď Aω.

Proof. Recall first that A “ Aπ for some unitary representation π of G. IfA X ApGq “ t0u then π is disjoint from λG (Lemma 2.8.7) and so A Ď Aω. Thussuppose that A X ApGq ‰ t0u. Then, by Theorem 3.5.7, A X ApGq “ ApG{Kq forsome compact normal subgroup K of G. We have to show that K “ teu.

Let x, y P K with x ‰ y. Then there exists u P A such that upxq ‰ upyq.Moreover, there exists v P ApG{Kq such that vpxqvpyq ‰ 0. Then uvpxq ‰ uvpyq,and uv P A X ApGq “ ApG{Kq since ApGq is an ideal in BpGq. This contradictsx, y P K. �

3.6. Comparison of ApG1q pbApG2q and ApG1 ˆ G2q

Let G1 and G2 be locally compact groups. If ui P ApGiq, i “ 1, 2, thenupx1, x2q “ u1px1qu2px2q, xi P Gi, defines an element of ApG1 ˆ G2q with }u} ď

}u1} ¨ }u2}. Thus there exists a canonical contraction φ from the projective tensorproduct ApG1q pbApG2q into ApG1 ˆ G2q. Note that the range of φ is dense inApG1 ˆ G2q. In fact, if ui “ ξi ˚ qηi, ξi, ηi P L2pGiq, i “ 1, 2, then

xT, u1 ˆ u2y “ xT pξ1 b ξ2q, η1 b η2y


for every T P V NpG1 ˆ G2q. Since the linear span of the elementary tensors isdense in L2pG1 ˆ G2q “ L2pG1q b L2pG2q, the claim follows.

If G1 and G2 are both abelian, then ApGiq “ L1pxGiq and ApG1 ˆ G2q “

L1pxG1 ˆ xG2q, and L1pxG1 ˆ xG2q is isometrically isomorphic to L1pxG1q pbL1pxG2q.Consequently, in this case φ is an isometric isomorphism. The theme of this sectionis to treat the problem of when φ is a linear isomorphism for arbitrary locallycompact groups G1 and G2.

For n P N, let �2n be Cn with the Euclidean scalar product and let Mn denotethe algebra of all nˆn-matrices considered as operators on �2n. The predual Mn˚ of

Mn will be identified with �2n pb �2n, where �2n denotes the complex conjugate spaceof �2n and the duality is given by

xξ b η, T y “ xTξ, ηy, ξ, η P �2n, T P Mn.

Then, for m,n P N, there is canonical bilinear bijection

φm,n : Mm bMn Ñ pMm˚ b Mn˚q˚

given by, for ξ1, η1 P �2m, ξ2, η2 P �2n and T1 P Mm, T2 P Mn,

xpξ1 b η1q b pξ2 b η2q, φm,npT1 b T2qy “ xT1ξ1, η1y ¨ xT2ξ2, η2y.

Lemma 3.6.1. }φm,n} “ 1 and }φ´1m,n} “ mintm,nu.

Proof. We identify Mm bMn with Mmn, the matrices acting as operators on�2mn “ �2m b �2n. The above formula implies that

xpξ1 b η1q b pξ2 b η2q, φm,npT qy “ xT pξ1 b ξ2q, η1 b η2y

for every T P Mm bMn, from which we easily conclude that }φm,n} “ 1.Now let N “ mintm,nu and let ε1, . . . , εN be the standard basis of �2N Ď �2m, �2n.

Let ω “řN

i“1 εibεi P �2mb�2n, and define an operator T on �2mb�2n by Tξ “ xξ, ωyω.Then }T } ď }ω}2 and }Tω} “ }ω}3, so that }T } “ }ω}2. On the other hand,}ω}2 “ N .

By definition of φm,n, we have

xpξ1 b η1q b pξ2 b η2q, φm,npT qy “ xxξ1 b ξ2, ωyω, η1 b η2y

“

Nÿ

i“1

xξ1, εiy xξ2, εiy ¨

Nÿ

j“1

xεj b η1y xεj b η2y

for all ξ1, ξ2 P �2m and η1, η2 P �2n. This equation implies

|xpξ1 b η1q b pξ2 b η2q, φm,npT qy| ď }ξ1} ¨ }ξ2} ¨ }η1} ¨ }η2}

“ }ξ1 b η1} ¨ }ξ2 b η2}

and hence }φm,npT q} ď 1 by definition of the projective norm. Since }T } “ N , itfollows that }φ´1

m,n} ě N .

To show that conversely }φ´1m,n} ď N , take any T P Mm bMn. Then there

exist ξ, η P �2m b �2n such that }ξ} “ }η} “ 1 and xTξ, ηy “ }T }. Interpreting ξ and ηas Hilbert-Schmidt operators, by definition of the tensor product of Hilbert spaces,

there exist orthonormal systems tξp1q

i u and tηp1q

i u in �2m and tξp2q

j u and tηp2q

j u in �2n

and λi, μj P C, 1 ď i, j ď N , such thatřN

i“1 |λi|2 “

řNj“1 |μj |2 “ 1 and

ξ “

Nÿ

i“1

λi

´

ξp1q

i b ξp2q

i

¯

and η “

Nÿ

j“1

μj

´

ηp1q

j b ηp2q

j

¯

.

3.6. COMPARISON OF ApG1q pb ApG2q AND ApG1 ˆ G2q 119

Then, using the Cauchy-Schwarz inequality, we get for the projective norm of u,

}u} ďřN

i,j“1 |λiμj | ď N . Since xu, φm,npT qy “ xTξ, ηy “ }T }, we conclude that

}φm,npT q} ě1N }T } and hence }φ´1

m,n} ď N . �

Theorem 3.6.2. Let G1 and G2 be locally compact groups and and let

φ : ApG1q pbApG2q Ñ ApG1 ˆ G2q

be the canonical homomorphism. Then the following three conditions are equivalent.

(i) φ is a surjective (not necessarily isometric) linear isomorphism.(ii) φ is surjective.(ii) At least one of G1 and G2 has an abelian subgroup of finite index.

Proof. (i) ñ (ii) being trivial, suppose that φ is surjective. Then the dualmap φ˚ : V NpG1 ˆ G2q Ñ rApG1q pbApG2qs˚ satisfies }φ˚pT q} ě c}T } for someconstant c ą 0. Let n “ t

1c u and m “ n ` 1.

We are going to show that either V NpG1q or V NpG2q is of type Iďn, i.e., thereare no direct summands of types Ik, k ą n, II or III. Assuming the contrary, itfollows from [270, Chapter V, Proposition 1.35] that there exist pairwise equivalentorthogonal projections P1, . . . , Pm in V NpG1q. Let Uij P V NpG1q, 1 ď i, j ď m,be partial isometries such that U˚

ijUij “ Pj and U˚ij “ Uji (and hence UijU

˚ij “ Pi).

The operators Uij generate a subalgebra of V NpG1q which is isomorphic to Mm

[270, Chapter IV, Proposition 1.8]. In the same way, we obtain a subalgebra ofV NpG2q. The canonical homomorphism

Ψm : Mm bMm Ñ V NpG1q bV NpG2q

is injective and isometric. Indeed, Ψ is injective since Mm bMm is finite dimen-sional, and Ψ is an isometry by uniqueness of the C˚-norm. Now consider thefollowing diagram:

Mm bMm

φm,m ��

Ψm

��

rMm˚pbMm˚s˚

Ωm

��V NpG1q bV NpG2q

φ˚�� rApG1q pbApG2qs˚

.

This diagram commutes. To see this, let ui P ApGiq and Si P Mm, i “ 1, 2. Then

xφ˚pΨmpS1 b S2qq, u1 b u2y “ xΨmpS1 b S2q, u1 ˆ u2y

“ xS1, u1y xS2, u2y,

and, on the other hand, if ui “ ξi b ηi, i “ 1, 2, then

xΩmpφm,mpS1 b S2qq, u1 b u2y “ xφm,mpS1 b S2q,Ωm˚pu1 b u2qy

“ xφm,mpS1 b S2q, pξ1 b η1q b pξ2 b η2qy

“ xS1, u1yxS2, u2y.

Since φ˚ and Ωm are both isometric, it follows that, for any T P Mm pbMm,

}φm,npT q} “ }Ωmpφm,npT qq} “ }φ˚pΨmpT qq} ě c }ΨmpT q} “ c}T }

and hence }φ´1m,n} ď

1c ă m, contradicting Lemma 3.6.1. Thus one of V NpG1q or

V NpG2q has to be of type Iďn. Theorem 1 of [215] (and its proof) now shows thatG1 or G2 has an abelian subgroup of finite index.


(iii) ñ (i) Suppose first that G1, say, is abelian. Then the Fourier transform

furnishes an isometric isomorphism between ApG1q and L1pxG1q. Let L1pxG1, ApG2qq

denote the algebra of ApG2q-valued Bochner integrable function on xG1 with convo-

lution and the usual L1-norm. Then L1pxG1, ApG2qq is isometrically isomorphic to

the projective tensor product L1pxG1q pbApG2q. the isomorphism determined by as-

signing to f bu, f P L1pxG1q, u P ApG2q, the function γ Ñ fpγqu in L1pxG1, ApG2qq.

Now, ApG1 ˆG2q is also isometrically isomorphic to L1pxG1, ApG2qq. In fact, by[270, Chapter IV, Theorem 7.17] and standard facts, we have canonical isometricisomorphisms

L1pxG1, ApG2qq “ rL8

pxG1q bV NpG2qs˚

“ rV NpG1q bV NpG2qs˚

“ V NpG1 ˆ G2q˚ “ ApG1 ˆ G2q.

It is easily verified that the composition

ApG1qpbApG2q Ñ L1pxG1, ApG2qq Ñ ApG1 ˆ G2q

of isometric isomorphisms equals the homomorphism φ. This shows (i) when G1 isabelian.

Now assume that G1 has an abelian subgroup H of finite index and let F be arepresentative system for the right cosets of H in G1. Then every u P ApG1q can

be written as u “ř

xPFČRxu|H , and

}u} ď

ÿ

xPF

} ČRxu|H} ď rG : Hs ¨ }u}.

Thus every w P ApG1q pbApG2q has a representation

w “

8ÿ

j“1

˜

ÿ

xPF

ČRxuj |H b vj

¸

“

ÿ

xPF

˜

8ÿ

j“1

ČRxuj |H b vj

¸

such that }w} “ř8

j“1 }uj} ¨ }vj}, and hence

}w} ď

8ÿ

j“1

›

›

›

›

›

ÿ

xPF

ČRxuj |H

›

›

›

›

›

¨ }vj} ď rG : Hs ¨ }w}.

Of course, we have a similar representation and an analogous norm estimate for ele-ments in ApG1ˆG2q in terms of elements of ApHˆG2q. Since φmaps ApHq pbApG2q

onto ApH ˆ G2q and these spaces have the same codimension in ApG1q pbApG2q

and ApG1 ˆ G2q, respectively, it follows that φ is surjective. �

Corollary 3.6.3. The canonical homomorphism

φ : ApG1q pbApG2q Ñ ApG1 ˆ G2q

is an isometric isomorphism if and only if at least one of G1 and G2 is abelian.

Proof. Suppose first that G1, say, is abelian. In this situation we have alreadyseen in the first part of the proof of the implication (iii) ñ (i) of Theorem 3.6.2that φ is an isometric isomorphism.

Conversely, suppose that φ is an isometric isomorphism and, towards a contra-diction, assume that none of G1 and G2 is abelian. Then (compare the proof ofTheorem 3.6.2) V NpGiq contains a subalgebra isomorphic to M2, i “ 1, 2. Now,

3.7. THE w˚-TOPOLOGY AND OTHER TOPOLOGIES ON BpGq 121

consider the above diagram with m “ 2. Since Ψ2 and Ω2 are both isometric andthe diagram is commutative, the norm of the map

φ˚|Ω2ppM2 pbM2q˚q “ Φ2 ˝ φ´1

2,2 ˝ Ω´12 |Ω2ppM2 pbM2q˚q

equals the norm of φ´12,2. Thus Lemma 3.6.1 implies that }φ˚} ě 2. This contradicts

the hypothesis that }φ˚} “ }φ} “ 1. �

3.7. The w˚-topology and other topologies on BpGq

In this section we study various topologies on the Fourier-Stieltjes algebra BpGq

of a locally compact group G, with emphasis on the question of whether theycoincide on the unit sphere of BpGq.

We define five topologies τuc, τw˚ , τbw˚ , τnw˚ and τM on BpGq by the statementthat a net puαqα in BpGq converges to u P BpGq with respect to

‚ τuc if uα Ñ u uniformly on compact subsets of G;‚ τw˚ if uα Ñ u in the w˚-topology σpBpGq, C˚pGqq;‚ τbw˚ if puαqα is norm bounded and uα Ñ u in τw˚ ;‚ τnw˚ if }uα} Ñ }u} and uα Ñ u in τw˚ ;‚ τM if }puα ´ uqv} Ñ 0 for all v P ApGq.We start the comparison of the topologies with a sequence of lemmas.

Lemma 3.7.1. The following hold.

(i) τM is stronger than τuc.(ii) On norm bounded sets in BpGq, τuc is stronger than τw˚ .

Proof. (i) Let K be any compact subset of G and choose v P ApGq such thatv “ 1 on K. Then, for every x P K,

|uαpxq ´ upxq| ď }puα ´ uqv}8 ď }puα ´ uqv}ApGq.

(ii) Let uα Ñ u in τuc and }uα}, }u} ď C ă 8 for all α. Let f P C˚pGq andε ą 0, and choose g P CcpGq with }g ´ f}C˚pGq ď ε. Then

|xuα, fy ´ xu, fy| ď |xuα ´ u, gy| ` |xuα, f ´ gy| ` |xu, g ´ fy

ď }puα ´ uq|supp g}8}g}8 ` 2C}f ´ g}C˚pGq.

Since }f ´ g}C˚pGq ď ε and uα Ñ u uniformly on compact sets, it follows thatuα Ñ u in τw˚ . �

The following lemma is a special case of [211, Lemma 8.2]. However, the proofgiven here is considerably simpler than the one given in [211].

Lemma 3.7.2. Let puαqα be a net in BpGq such that uα Ñ u P BpGq in thetopology τnw˚ . Let peγqγ be an approximate identity for L1pGq, where every eγ isof the form eγ “ fγ ˚ f˚

γ , fγ P L1pGq, fγ ě 0 and }fγ}1 “ 1. Then, given anyε ą 0, there exist indices α0 and γ0 such that

}eγ0˚ u ´ u} ď ε and }eγ0

˚ uα ´ uα} ď ε

for all α ě α0.

Proof. Let L1pGqe be L1pGq with an identity e adjoined if G is nondiscreteand equal to L1pGq if G is discrete. Then the enveloping C˚-algebra of L1pGqe

equals C˚pGqe. Since }eγ}C˚pGq ď }eγ}1 “ 1, it follows that 0 ď eγ ď e in theC˚-algebra sense and therefore 0 ď pe ´ eγq ˚ pe ´ eγq ď e ´ eγ in C˚pGqe.


Since }uα} Ñ }u}, we may assume that u ‰ 0. For any f P L1pGq and v P BpGq,as eγ “ eγ , we have

|xeγ ˚ v ´ v, fy|2

“ |xv, peγ ´ eq ˚ fy|2

ď }u} ¨ |x|u|, peγ ´ eq ˚ f ˚ f˚˚ peγ ´ eqy|

ď }u} ¨ }f ˚ f˚}C˚pGq|x|u|, pe ´ eγq ˚ pe ´ eγqy|

ď }u} ¨ }f}2C˚pGqx|u|, e ´ eγy.

To deduce this inequality, we have used the fact that if ϕ is a positive linear func-tional on a C˚-algebra A, then ϕpa˚baq ď }b}ϕpa˚aq for all a, b P A. Moreover, thelast inequality holds because |u| is a positive linear functional on C˚pGqe.

Notice next that since |u| is positive and peγqγ is an approximate identity inC˚pGq, x|u|, eγy Ñ x|u|, ey “ }u}. So, given ε ą 0, there exists γ0 such thatx|u|, e ´ eγ0

y ď ε{}u}. Since uα Ñ u in τnw˚ implies that |uα| Ñ |u| [68], takingv “ uα in the above estimate and the supremum over all f P L1pGq with }f}C˚pGq ď

1, it follows that

}eγ0˚ uα ´ uα}BpGq ď }uα}

1{2x|uα|, e ´ eγ0

y1{2

Ñ }u}1{2

x|u|, e ´ eγ0y1{2

ă ε.

Thus there exists α0 such that }uα}1{2x|uα|, e ´ eγ0y1{2 ď ε for all α ě α0. This

completes the proof of the lemma. �

Lemma 3.7.3. Let f P L8pGq and g P L1pGq and suppose that both f and gvanish outside some compact set K in G, and let h “ f ˚g. Then the map u Ñ h˚uis continuous from pBpGq, τbw˚q to pBpGq, τM q.

Proof. Let puαqα be a net in BpGq such that uα Ñ u for some u P BpGq inthe w˚-topology and suppose that C “ sup }uα} ă 8. Let v P ApGq X CcpGq and

let rK be a compact subset of G containing K´1 ¨ supp v. Then, for p P L8pGq andq P L1pGq, we have

xpf ˚ pqv, qy “ xp, f˚˚ pvqqy “ x1Kp, f˚

˚ pvqqy “ xrf ˚ p1Kpqsv, qy.

Since f and Ą1Kp belong to L2pGq, f ˚ Ą1Kp P L2pGq ˚ ČL2pGq “ ApGq and

}pf ˚ pqv}ApGq “ }pf ˚ 1Kpqv}ApGq ď }f}2}Ą1Kp}2}v}ApGq.

Now take p “ g ˚ puα ´ uq. Then

}pf ˚ g ˚ puα ´ uqqv} ď }f}2}rg ˚ puα ´ uq1Ks„

}2}v}ApGq,

which tends to zero since g ˚ puα ´ uq Ñ 0 uniformly on K (compare [59, Lemma13.5.1]). Thus we have seen that }rh ˚ puα ´ uqsv}ApGq Ñ 0 for every v P ApGq X

CcpGq.Now let w P ApGq be arbitrary and let ε ą 0 be given and choose v P ApGq X

CcpGq such that }w ´ v}ApGq ď ε. Notice that, by Lemma 2.1.13, h ˚ uα P BpGq

and}h ˚ uα}BpGq ď }f ˚ g}C˚pGq}uα}BpGq ď C }f}1}g}1

for all α. Then, with c “ pC ` }u}BpGqq}f}1}g}1,

}rpf ˚ gq ˚ puα ´ uqsw}BpGq ď }rpf ˚ gq ˚ puα ´ uqsv}BpGq

`}rpf ˚ gq ˚ puα ´ uqspw ´ vq}BpGq

ď }rpf ˚ gq ˚ puα ´ uqsv}BpGq ` ε c.



The preceding three lemmas now lead to our first theorem.

Theorem 3.7.4. τnw˚ is stronger than τM . In particular, τw˚ and τM coincideon the unit sphere of BpGq.

Proof. Since, by Lemma 3.7.1, τM Ě τuc Ě τw˚ on norm bounded sets inBpGq, we only have to show that if puαqα is a net in BpGq such that, for someu P BpGq, uα Ñ u in the w˚-topology and }uα} Ñ }u}, then uα Ñ u in τM .

Let pUγqγ be a neighbourhood basis of the identity e of G. For each γ, choose anopen, relatively compact neighbourhood Vγ of e such that V ´1

γ “ Vγ and V 2γ Ď Uγ ,

and let fγ “ |Vγ |´11Vγand eγ “ fγ ˚f˚

γ . Then the net peγqγ satisfies the hypothesesof Lemma 3.7.2. Hence, given ε ą 0, there exist α0 and γ0 such that

}eγ0˚ u ´ u}BpGq ď ε and }eγ0

˚ uα ´ uα}BpGq ď ε

for all α ě α0. It follows that, for any v P ApGq and α ě α0,

}puα ´ uqv}ApGq ď }puα ´ eγ0˚ uαqv}ApGq ` }peγ0

˚ u ´ uqv}ApGq

`}reγ0˚ puα ´ uqsv}ApGq

ď 2ε}v}ApGq ` }reγ0˚ puα ´ uqsv}ApGq.

Now, since uα Ñ u in τw˚ and the net puαqα is norm bounded, taking h “ eγ0in

Lemma 3.7.3, we conclude that }reγ0˚ puα ´ uqsv}ApGq ď ε}v}ApGq for all α ě α1

for some α1 ě α0. This completes the proof of the theorem. �

Corollary 3.7.5. τnw˚ is stronger than τuc. In particular, τw˚ and τuc coin-cide on the unit sphere of BpGq.

Proof. This is an immediate consequence of Theorem 3.7.4 and Lemma 3.7.1(i).�

Corollary 3.7.6. Let K be a compact subset of G and

AKpGq “ tu P ApGq : supp u Ď Ku.

Then the norm topology and the w˚-topology coincide on the unit sphere of AKpGq.In particular, if G is compact, then the w˚-topology and the norm topology agreeon the unit sphere of BpGq.

Proof. Let pvαqα Ď AKpGq and v P AKpGq such that }vα} “ 1 “ }v} for all αand v “ w˚-limα vα. Choose u P ApGq with u “ 1 on K. Then vα ´ v “ pvα ´ vquand, by Theorem 3.7.4, pvα ´ vqu Ñ 0. �

We shall see below (Corollary 3.7.8) that the converse of the last assertion inCorollary 3.7.6 also holds. Next we list a number of topologies which coincide onthe unit sphere.

Theorem 3.7.7. Let S denote the unit sphere of BpGq and let puαqα be a netin S and u P S. Then the following are equivalent.

(1) uα Ñ u in the w˚-topology of BpGq.(2) uα Ñ u uniformly on compact subsets of G.(3) For each T P C˚

λ pGq, uα ¨ T Ñ u ¨ T in the norm of C˚λ pGq.

(3’) For each T P V NpGq, uα ¨ T Ñ u ¨ T in the w˚-topology of V NpGq.(4) For each T P C˚

λ pGq and v P ApGq, }T puαvq ´ T puvq}ApGq Ñ 0.(4’) For each T P C˚

λ pGq and v P ApGq, xT, uαvy Ñ xT, uvy.


(5) }uαv ´ uv}ApGq Ñ 0 for every v P ApGq.(5’) For each T P V NpGq and v P ApGq, xuαv, T y Ñ xuv, T y.

Proof. The equivalence of (1) and (2) and of (2) and (3) follows from Theorem3.7.4. Clearly, (5) ñ (3’).

To show that (3’) implies (1), let f P CcpGq and choose v P ApGq with v “ 1on supp f . Then vf “ f and

xuα, λpfqy “

ż

G

uαpxqfpxqdx “

ż

G

uαpxqvpxqfpxqdx

“ xλpuαfq, vy “ xuα ¨ λpfq, vy Ñ xu ¨ λpfq, vy

“ xv ¨ λpfq, uy “ xλpfq, uy.

Thus uα Ñ u in σpBpGq, CcpGqq, which implies (1) by the density of CcpGq inC˚pGq and since the uα and u are norm bounded.

(5) implies (3) since the operators with compact support are dense in C˚λ pGq

and every T P C˚λ pGq can be written as T “ v ¨ T for some v P ApGq with v “ 1 on

suppT and hence uα ¨ T ´ u ¨ T “ puαv ´ uvq ¨ T , which tends to 0 in C˚λ pGq since

the net puαv ´ uvqα is norm bounded.The implications (3) ñ (1), (4’) ñ (1) and (5) ñ (1) are all shown in the same

manner as (3’) ñ (1). Finally, the remaining implications (5) ñ (4), (4) ñ (4’)and (5) ñ (5’) are all evident. �

From Theorem 3.7.7 we deduce the following

Corollary 3.7.8. Let G be amenable and suppose that the w˚-topology andthe norm topology agree on the unit sphere of BpGq. Then G must be compact.

Proof. Since G is amenable, ApGq has a bounded approximate identity puαqα

such that }uα} “ 1 for all α (Theorem 2.8.2). Then }uαv ´ 1Gv}ApGq “ }uαv ´

v}ApGq Ñ 0 for every v P ApGq. Now the implication (5) ñ (1) of Theorem3.7.7 shows that uα Ñ 1G in the w˚-topology of BpGq. Since, by hypothesis,this topology coincides with the norm topology on the unit sphere, it follows that1G P ApGq, and hence G must be compact. �

We are now going to show that the amenability hypothesis in Corollary 3.7.8can be dropped. However, the proof will have to employ a deep result fromrepresentation theory due to S.P. Wang. Actually, we will show a stronger re-sult in that we replace the unit sphere of BpGq by the smaller set P 1

λpGq, whereP 1λpGq “ BλpGq X P 1pGq is the set of all normalized continuous positive definite

functions which are associated to representations of G that are weakly containedin the left regular representation.

Lemma 3.7.9. Let H be an open subgroup of G. If the identity map frompP 1

λpGq, w˚q to pP 1λpGq, } ¨}q is continuous, then the identity map from pP 1

λpHq, w˚q

to pP 1λpHq, } ¨ }q is also continuous.

Proof. For any ϕ P PλppHq, the trivial extension rϕ of ϕ toG belongs to PλpGq.

Indeed, rϕ is associated to the induced representations indGH πϕ and πϕ ă λH implies

indGH πϕ ă indgH λH – λG.


Now let pϕαqα be a net in P 1λpHq converging to ϕ P PλpHq in the w˚-topology.

Then rϕα Ñ rϕ in the w˚-topology of PλpGq and hence

}ϕα ´ ϕ} “ }rϕα ´ rϕ} Ñ 0

by hypothesis. �

Lemma 3.7.10. Let ϕ P expP 1λpGqq and suppose that the identity map from

pexpP 1λpGqq, w˚q to pexpP 1

λpGqq, } ¨ }q is continuous at ϕ. Then πϕ is an isolated

point of pGr.

Proof. Notice first that expP 1λpGqq Ď expP 1pGqq because if ϕ P PλpGq and

ψ P P pGq are such that cϕ´ψ is positive definite for some c ě 0, then ψ P PλpGq. Asmentioned in Chapter 1, if ϕ1, ϕ2 P P 1pGq and the associated GNS-representationsare not equivalent, then }ϕ1 ´ ϕ2} ě 2. By hypothesis, there exists a w˚-opensubset U of expP 1

λpGqq such that

U Ď tψ P expP 1λpGqq : }ψ ´ ϕ} ă 2u.

It follows that πϕ – πψ for all ψ P U . Now the map φ : ψ Ñ πψ from expP 1λpGqq

onto pGr is open (compare [60, Theorem 3.4.10]). Consequently, φpUq “ tπϕu is

open in pGr. �

Theorem 3.7.11. For any locally compact group G, the following conditionsare equivalent.

(i) G is compact.(ii) The w˚-topology and the norm topology coincide on the unit sphere of

BpGq.(iii) The w˚-topology and the norm topology agree on P 1

λpGq.

Proof. The implications (i) ñ (ii) and (ii) ñ (iii) being clear, it only remainsto prove that (iii) implies (i). Towards a contradiction, assume that G is noncom-pact. Then G contains a noncompact, σ-compact, open subgroup H. By Lemma3.7.9, the w˚-topology and the norm topology agree on P 1

λpHq. It now follows from

Lemma 3.7.10 that pHr is discrete. Finally, by [283, Theorem 7.6], a σ-compactlocally compact group with discrete reduced dual is compact. This contradictionproves (iii) ñ (i). �

When studying the w˚-topology on BpGq, the question of when ApGq is w˚-closed in BpGq, arises naturally. It will turn out that for a large class of locallycompact groups G, this forces G to be compact. As for Theorem 3.7.11, we willagain have to utilize some profound result about dual spaces of locally compactgroups, this time one due to Baggett [7].

Lemma 3.7.12. Let K be a compact normal subgroup of G. If ApGq is w˚-closedin BpGq, then ApG{Kq is w˚-closed in BpG{Kq.

Proof. Recall that TK : L1pGq Ñ L1pG{Kq denotes the ˚-homomorphismdefined by TKpfqpxKq “

ş

Kfpxkqdk for almost all x P G, and that TK extends to

a ˚-homomorphism, also denoted TK , from C˚pGq onto C˚pG{Kq. Let T˚K denote

the adjoint map. Then T˚KpBpG{Kqq consists precisely of those functions in BpGq

which are constant on cosets of K. Moreover, since K is compact,

T˚KpApG{Kqq “ ApGq X T˚

KpBpG{Kqq.


Now let puαqα be a net in ApG{Kq converging to some u P BpG{Kq in the w˚-topology. Then, for every f P C˚pGq,

xT˚Kpuαq, fy “ xuα, TKpfqy Ñ xu, TKpfqy “ xT˚

Kpuq, fy.

Since ApGq is w˚-closed in BpGq and T˚Kpuαq P ApGq, it follows that T˚

Kpuq P ApGq

and hence T˚Kpuq P ApGq X T˚

KpBpG{Kqq. Consequently u P ApG{Kq. �

Lemma 3.7.13. If ApGq is w˚-closed in BpGq and H is an open subgroup ofG, then ApHq is w˚-closed in BpHq.

Proof. It suffices to show that the unit ball of ApHq is w˚-closed in BpHq.Thus let puαqα be a net in ApHq and u P BpGq such that

}uα} ď 1, }u} ď 1 and u “ w˚´ lim

αuα.

Let ruα and ru denote the trivial extension of uα and u to G, respectively. Thenruα P ApGq and ru P BpGq with norms ď 1, and for every f P L1pGq,

ż

G

ruαpxqfpxqdx “

ż

H

uαphqfphqdh

Ñ

ż

H

uphqfphqdh “

ż

G

rupxqfpxqdx.

Since pruαqα is a bounded net, it follows that ruα Ñ ru in the w˚-topology of BpGq.Hence, by the hypothesis, ru P ApGq and so u P ApHq. �

Lemma 3.7.14. Let G be an almost connected locally compact group. If ApGq

is w˚-closed in BpGq, then G is compact.

Proof. Since an almost connected locally compact group is a projective limitof Lie groups, there exists a compact normal subgroup K of G such that G{K is aLie group. By Lemma 3.7.12, ApG{Kq is w˚-closed in BpG{Kq. Therefore, we canassume that G is Lie group. Being a compactly generated Lie group, G is secondcountable and hence ApGq is a separable Banach space. By the hypothesis,

ApGq “ BλpGq “ C˚r pGq

˚.

Now a C˚-algebra A with a separable dual Banach space A˚ has a countable dualpA. This appears to be folklore and can easily be seen as follows. Let S denote theset of all pure states of A. Then S is second countable, when equipped with therelative norm topology of A˚. Hence there exist ϕn P S, n P N, such that

S “Ť

nPNtψ P S : }ψ ´ ϕn} ă 2u.

Now, if ϕ, ψ P S are such that }ϕ ´ ψ} ă 2, then the associated irreducible repre-

sentations are equivalent [60, (2.12.1)]. This shows that pA is countable. It follows

that pGr, the reduced dual of G, is countable. Finally, by [7, Theorem 2.5], a sep-arable Lie group with countable reduced dual is compact. This shows that G iscompact. �

Corollary 3.7.15. Let G be any locally compact group and suppose that ApGq

is w˚-closed in BpGq. Then G contains a compact open subgroup.

Proof. Since G{G0 is totally disconnected, there exists an open subgroup Hof G such that H{G0 is compact. By Lemma 3.7.13, ApHq is w˚-closed in BpHq

and hence H is compact by Lemma 3.7.14. �


Lemma 3.7.16. If G is a discrete group and ApGq is w˚-closed in BpGq, thenG is finite.

Proof. Towards a contradiction, assume that G is infinite. Then G has acountable infinite subgroup H. By Lemma 3.7.13, ApHq is w˚-closed in BpHq. As

in the proof of Lemma 3.7.14, we now conclude that pHr is countable. Applying[7, Theorem 2.5] again, it follows that H is finite, contradicting the choice of H. �

Theorem 3.7.17. Suppose that G contains an almost connected open normalsubgroup. Then ApGq is w˚-closed in BpGq if and only if G is compact.

Proof. Let N be an almost connected open normal subgroup of G. ThenApNq is w˚-closed in BpNq (Lemma 3.7.13), and Corollary 3.7.15 implies thatN is compact. Passing to G{N , we have that ApG{Nq is w˚-closed in BpG{Nq.Since G{N is discrete, Lemma 3.7.16 shows that G{N must be finite. Hence G iscompact. �


Host’s idempotent theorem, which in particular identifies the idempotents inBpGq as precisely the characteristic functions of open sets in the coset ring of G, wasproved in [129] and has since become one of the major tools in the investigation ofApGq and BpGq. As samples, we just mention the study of homomorphisms betweenFourier algebras [135] and [136], the description of the closed ideals in ApGq withbounded approximate identities in terms of their hulls [85], and amenability of ApGq

[86]. The idempotent theorem was earlier shown by Cohen [33] for abelian groupsand was announced in the nonabelian case by Lefranc [187], who never publisheda proof. In contrast to Cohen’s proof, which made substantial use of measuretheory on the dual group of G, Host’s proof is considerably simpler and basedonly on elementary facts from operator theory. Recently, Runde [252] obtainedidempotent theorems of a similar flavour for representations on Banach spaces.

Wendel [288] proved that if G and H are two locally compact groups and thegroup algebras L1pGq and L1pHq are isometrically isomorphic, then G and H aretopologically isomorphic. The same conclusion holds if there exists an isometricisomorphism between the measure algebras MpGq and MpHq. In fact, as shown byJohnson [140], such an isomorphism maps L1pGq onto L1pHq. For abelian groupsthese results are very special cases of, but certainly have been a motivation for,Walter’s theorems presented in Section 3.2. Walter’s viewpoint has been to con-sider BpGq and ApGq as dual objects for G and he perfectly succeeded in showingthat both of them completely specify the underlying group G. Subsequently, severalauthors have investigated the problem of which maps between Fourier and Fourier-Stieltjes algebras ensure the groups to be topologically isomorphic. As a sample wemention that, using Walter’s result [280], Arendt and deCanniere [4] have shownthat the pointwise and the positive definite orderings in BpGq and ApGq, respec-tively, determine G up to topological isomorphism. More precisely, the locallycompact groups G and H are topologically isomorphic if (and only if) there existsa bijective linear mapping φ : BpGq Ñ BpHq such that φpBpGq`q “ BpHq` andφpP pGqq “ P pHq, and similarly for Fourier algebras.

For locally compact abelian groups G and H, homomorphisms from ApGq intoBpHq have been completely characterized by Cohen [33] in terms of continuouspiecewise affine maps from sets in the open coset ring of H into G. Cohen’s


achievements have been extremely influential and inspired numerous subsequentinvestigations. As examples of contributions for general locally compact groupswe mention [135], [136], [138]. The results we have presented in Section 3.3 areentirely due to Pham [234]. They provide satisfying descriptions of homomor-phisms which are either positive or contractive. Moreover, in [234] completelybounded and completely contractive homomorphisms and isomorphisms betweenFourier and Fourier-Stieltjes algebras have been studied. We did not touch theseadditional results because we will encounter operator space structures for the firsttime in Chapter 4.

The Galois type correspondence between closed subgroups of a locally compactgroup G and invariant von Neumann subalgebras of L8pGq and V NpGq (Theo-rems 3.4.3 and 3.4.8), which might be considered as an extension of the Pontryaginduality theorem for locally compact abelian groups and the Tannaka-Krein dual-ity theorem for compact groups, is due to Takesaki and Tatsuuma [271]. It is acontinuation of previous investigations of the same authors on duality theorems forlocally compact groups in terms of Hopf-von Neumann group algebras. Our ap-proach, however, follows [271] only to some extent and appears to be considerablysimpler. In particular, it entirely avoids the use of the Hopf-von Neumann algebrastructure of V NpGq. The proof of Theorem 3.4.3 given here is taken from [175].

The material in Section 3.5 is taken from [11], where the structure of two-sided translation invariant w˚-closed ˚-subalgebras of BpGq is studied. The maincharacterization of such subalgebras, when G is amenable, is parallel to that ofw˚-closed right translation invariant ˚-subalgebras of L8pGq, due to Takesaki andTatsuuma [271], as presented in the preceding section. When G is non-amenable,no such structure theorem is known. It should also be pointed out that the problemof characterizing one-sided translation invariant w˚-closed subalgebras of BpGq isopen. Moreover, none of these problems seems to have been treated for the algebrasBppGq, 1 ă p ă 8, formed by all continuous complex-valued functions on G whichare multipliers of AppGq.

Corollary 3.6.3 was previously shown in [221]. Losert [201] mentions that thefollowing generalization of Theorem 3.6.2 for Fourier spaces associated with unitaryrepresentations (compare Section 2.8) could be shown. For i “ 1, 2, let Gi be alocally compact group and πi a unitary representation. Then the spaces Aπ1

pbAπ2

and Aπ1bπ2pG1 ˆ G2q are linearly isomorphic if and only if at leat one of π1 and

π2 is finite dimensional. Then, in addition, Aπ1pbAπ2

and Aπ1bπ2pG1 ˆ G2q are

isometrically isomorphic if and only if one of the representations is 1-dimensional.The material of Section 3.7 up to Theorem 3.7.7 is taken from [105]. Theorem

3.7.11 improves on earlier results due to Derighetti [50] and McKennon [211] andactually verifies a conjecture of [211]. The remaining part of Section 3.7 appearedin [10]. LetH be a closed subgroup of the locally compact group G. In [10] it is alsoshown that the restriction map from BpGq into BpHq is w˚-w˚-continuous (if and)only if H is open in G. This result was in turn generalized in [138], where a sys-tematic study of w˚-continuous homomorphisms between Fourier-Stieltjes algebraswas conducted.

CHAPTER 4

Amenability Properties of ApGq and BpGq

Amenable Banach algebras were introduced by B.E. Johnson. He showed thefundamental result that a locally compact group is amenable if and only if the groupalgebra L1pGq is amenable. We present a proof of the ’only if’ part in Section 4.5.In particular, if G is abelian, then ApGq, being isometrically isomorphic to the L1-

algebra of the dual group pG, is amenable. However, when G is nonabelian, thenApGq need not be weakly amenable, even when G is compact.

In this chapter we will consider the completely bounded cohomology theory ofthe Fourier algebra ApGq and of the Fourier-Stieltjes algebra BpGq. In Section 4.1it is shown that ApGq, equipped with the operator space structure inherited frombeing embedded into V NpGq˚, is a completely contractive Banach algebra. Usingthis we establish in Section 4.2 the fundamental result, due to Ruan [245], that alocally compact group G is amenable precisely when ApGq is operator amenable.Note that the natural operator space structure on L1pGq as predual of L8pGq issuch that all bounded maps from L1pGq into any operator space are automaticallycompletely bounded. Thus the notions of amenability and operator amenabilitycoincide for L1pGq.

Another classical result of B.E. Johnson states that for any locally compactgroup G, L1pGq is always weakly amenable. A simple proof, due to Despic andGhahramani [58], is also included in Section 4.5. In Section 4.3 we are going toprove that ApGq is operator weakly amenable for every locally compact group G. Incontrast, it turns out that ApGq is amenable if and only if G has an abelian subgroupof finite index (Section 4.5). The proof is based on the fact that G has to have anabelian subgroup of finite index if the antidiagonal ΓG “ tpx, x´1q : x P Gu belongsto the closed coset ring of GˆG. The proof of this latter fact depends on completelyboundedness of the flip map ApGq Ñ ApGq, u Ñ qu, where qupxq “ upx´1q foru P ApGq and x P G. It also involves the study of piecewise affine maps (Section4.4) and the structure of ideals in ApGq with bounded approximate identity.

4.1. ApGq as a completely contractive Banach algebra

Let G be a locally compact group. The Fourier algebra ApGq of G is en-dowed with the operator space structure which it inherits from being embeddedinto V NpGq˚ “ ApGq˚˚ (see Appendix A.3 and [69]).

Lemma 4.1.1. Let G and H be locally compact groups. Then the von Neumannalgebras V NpGq bV NpHq and V NpG ˆ Hq are isomorphic.

Proof. We first observe that the two Hilbert spaces L2pGq bL2pHq and L2pGˆ

Hq are isometrically isomorphic via the map φ determined by f b g Ñ f ˆ g,where pf ˆ gqps, tq “ fpsqgptq, f P L2pGq, g P L2pHq, s P G and t P H. ThusBpL2pGˆHqq is isomorphic to BpL2pGq bL2pHqq, which contains V NpGq bV NpGq

129

130 4. AMENABILITY PROPERTIES OF ApGq AND BpGq

as a ˚-subalgebra. Then φ is a unitary equivalence between the representationsλG b λH and λGˆH . Indeed, for x P G and y P H,

λGˆHpx, yqpφpf b gqqps, tq “ φpf b gqpx´1s, y´1tq

“ fpx´1sqgpy´1tq

“ λGpxqfpsqλHpyqgptq

“ φpλGpxqf b λHpyqgqps, tq.

Consequently, φ induces an isomorphism between the von Neumann algebras

V NpGq bV NpHq “ λGpGq2

bλHpHq2

“ pλGpGq b λHpHqq2

and V NpG ˆ Hq “ λGˆHpG ˆ Hq2. �

Lemma 4.1.2. Let G and H be locally compact groups. Then

ApG ˆ Hq – ApGq pbApHq.

Furthermore, the identity

ApG ˆ Hq – ApGq pbApHq

is a (completely) isometric isomorphism.

Proof. By Lemma 4.1.1, V NpG ˆ Hq – V NpGqbV NpHq and, by Proposi-tion A.3.8,

ApGq pbApHq – pV NpGq bV NpHqq˚.

Consequently, ApG ˆ Hq “ V NpG ˆ Hq˚ – ApGqpbApHq. �

In the sequel, let W denote the unitary operator on L2pG ˆ Gq defined by

pWfqps, tq “ fps, stq, f P L2pG ˆ Gq, s, t P G.

Theorem 4.1.3. For any locally compact group G, the Fourier algebra ApGq isa completely contractive Banach algebra.

Proof. With I denoting the identity operator on L2pGq, W satisfies the equa-tion

(4.1) W˚pλGpsq b IqW “ λGpsq b λGpsq

for all s P G. To prove (4.1) it suffices to show that

xW˚pλGpsq b IqWξ1, ξ2y “ xpλGpsq b λGpsqqξ1, ξ2y,

where ξj “ fj b gj , fj , gj P L2pGq, j “ 1, 2. Now, observing that

pλGpsq b Iqξpx, yq “ ξps´1x, yq

4.1. ApGq AS A COMPLETELY CONTRACTIVE BANACH ALGEBRA 131

for every ξ P L2pG ˆ Gq, we get

xW˚pλGpsq b IqWξ1, ξ2y “ pxλGpsq b IqWξ1,Wξ2y

“

ż

G

ż

G

Wξ1ps´1x, yqWξ2px, yq dxdy

“

ż

G

ż

G

ξ1ps´1x, s´1xyq ξ2px, xyq dxdy

“

ż

G

ż

G

f1ps´1xqg1ps´1xyq f2pxqg2pxyq dydx

“

ż

G

ż

G

f1ps´1xq f2pxqg1ps´1yq g2pyq dydx

“ xpλGpsq b λGpsqqξ1, ξ2y,

as required. It follows that the map γ : T Ñ W˚pT b IqW is a normal iso-morphism of V NpGq into the tensor product V NpGq bV NpGq. Moreover, γ is aco-multiplication of V NpGq in the sense that, denoting by i the identity mappingon V NpGq, the diagram

V NpGqγ ��

γ

��

V NpGq bV NpGq

γbi

��V NpGq bV NpGq

ibγ�� V NpGq bV NpGq bV NpGq

is commutative. Indeed, since the operators λGpsq, s P G, generate V NpGq, weonly have to show that

rpγ b iq ˝ γspλGpsqq “ rpi b γq ˝ γspλGpsqq

for all s P G. But this is clear since both sides are equal to the operator λGpsq b

λGpsq b λGpsq.Now γ is a w˚-homomorphism, i.e., a w˚-w˚-continuous homomorphism, from

V NpGq into V NpGq bV NpGq. Hence γ is completely contractive. By Lemma4.1.1,

γpV NpGqq Ď V NpGq bV NpGq – V NpG ˆ Gq.

Thus the dual map γ˚ : V NpG ˆ Gq˚ Ñ V NpGq˚ is also completely contractive[70, Proposition 3.2.2]. Since γ is w˚-w˚-continuous, it follows that

γ˚ “ γ˚|ApGˆGq Ñ ApGq

is completely contractive as well. Identifying ApGq pbApGq with ApG ˆ Gq viaub v Ñ uˆ v, we may now deduce that γ˚pub vq “ uv for all u, v P ApGq. Indeed,for each s P G, we have

xγ˚pu b vq, λGpsqy “ xγ˚ppu b vq, λGpsqy “ xu b v, γpλGpsqqy

“ xu b v, λGpsq b λGpsqy “ xu, λGpsqy ¨ xv, λGpsqy

“ upsqvpsq “ xuv, λGpsqy,

and since the linear span of the set tλGpsq : s P Gu is w˚-dense in V NpGq, itfollows that γ˚pubvq “ uv. Consequently, ApGq is a completely contractive Banachalgebra. �


Remark 4.1.4. It should be noted that two different projective tensor productshave been considered in this paper. The projective tensor product ApGq pbApHq ofFourier algebra ApGq and ApHq considered in Section 3.6 in the “Banach spaceprojective tensor product”, and the projective tensor product pb considered inLemma 4.1.2 should be “operator space projective tensor product”. They are dif-

ferent unless either G or H is an abelian group, i.e., either ApGq “ L1p pGq or

ApGq “ L1p pHq.

4.2. Operator amenability of ApGq

In this section, we will show that the Fourier algebra ApGq is operator amenableexactly when G is an amenable locally compact group. We start by providing asufficient condition for operator amenability of ApGq.

Lemma 4.2.1. Let G be an amenable locally compact group. Suppose that thereexists a bounded net pwαqα in BpG ˆ Gq such that, for every u P ApGq,

(i) }u ¨ wα ´ wα ¨ u} Ñ 0;(ii) }mpwαqu ´ u} Ñ 0.

Then ApGq is operator amenable.

Proof. Since G is amenable, by Theorem 2.8.5 ApGq has a bounded approx-imate identity, puβqβ say. Now consider the elements vα,β “ wαpuβ b uβq of

ApGq pbApGq “ ApG ˆ Gq. Clearly, the net pvα,βqpα,βq is bounded and puβ b uβqβ

is a bounded approximate identity for ApG ˆ Gq. Moreover, pvα,βqpα,βq is an ap-proximate diagonal for ApGq. In fact, for each u P ApGq,

u ¨ vα,β ´ vα,β ¨ u “ ru ¨ wα ´ wα ¨ uspuβ b uβq Ñ 0

by (i), and by (ii),

mpvα,βqu ´ u “ mpwαqru2βu ´ us ` mpwαqu ´ u Ñ 0.

Since ApGq is completely contractive by Theorem 4.1.3, Proposition A.4.2 yieldsthat ApGq is operator amenable. �

For ξ P L2pGq, let wξ : G ˆ G Ñ C be defined by

wξps, tq “ xλGpsqρGptqξ, ξy, s, t P G.

Then wξ P BpG ˆ Gq. Also, let V P BpL2pGqq be defined by

V ξpsq “ Δpsq´1{2ξps´1

q, ξ P L2pGq, s P G.

Then V is self-adjoint and unitary. Furthermore, for all s P G,

ρGpsq “ V ˚λGpsqV.

Let W be as in the preceding section.

Lemma 4.2.2. pV ˚ b IqW˚ “ W pV ˚ b Iq.

Proof. If ξ P L2pG ˆ Gq, we have for almost all ps, tq P G ˆ G,

pV ˚b IqW˚ξps, tq “ Δpsq

´1{2pW˚ξqps´1, tq

“ Δpsq´1{2ξps´1, stq

“ pV ˚b Iqξps, stq

“ W pV ˚b Iqξps, tq,


4.2. OPERATOR AMENABILITY OF ApGq 133

Lemma 4.2.3. Let G be a locally compact group and suppose that pξαqα is a netof unit vectors in L2pGq such that

(i) }W pξα b ηq ´ pξα b ηq} Ñ 0 for all η P L2pGq;(ii) }λGpsqρGpsqξα ´ ξα} Ñ 0 uniformly on compact subsets of G.

Then the net pwξαqα in BpG ˆ Gq satisfies conditions (i) and (ii) in Lemma 4.2.1.

Proof. Let λ “ λG and ρ “ ρG. To establish conditions (i) and (ii) of Lemma4.2.1, it suffices to consider u P ApGq of the form upsq “ xλpsqη, ηy, where η P L2pGq.Then, since ρpsq “ V ˚λpsqV and using Lemma 4.2.2, for all s, t P G,

pwξα ¨ uqps, tq “ xλpsqρptqξα, ξαyxλptqη, ηy

“ xλpsqV ˚λptqV ξα, ξαyxλptqη, ηy

“ xpλpsq b IqpV ˚λptqV b λptqqpξα b ηq, ξα b ηy

“ xpλpsq b IqpV ˚b IqW˚

pλptq b IqW pV b Iqpξα b ηq, ξα b ηy

“ xpλpsq b λpsqqpV ˚λptqV b IqW˚pξα b ηq,W˚

pξα b ηqy.

In addition, we have

pu ¨ wξαqps, tq “ xλpsqη, ηy xλpsqρptqξα, ξαy

“ xλpsqη, ηy xλpsqV ˚λptqV ξα, ξαy

“ xpλpsqV ˚λptqV b λpsqqpξα b ηq, ξα b ηy.

Combining the preceding two equations, we get

pu ¨ wξα ´ wξα ¨ uqps, tq “ xpλpsqV ˚λptqV b λpsqqpξα b ηq, ξα b ηy

´ xpλpsqV ˚λptqV b λpsqqW˚pξα b ηq,W˚

pξα b ηqy

“ xpλpsqV ˚λptq b λpsqqpξα b η ´ W˚pξα b ηq, ξα b ηy

` xpλpsqV ˚λptqV b λpsqqW˚pξα b η ´ W˚

pξα b ηqy.

Consequently,

}u ¨ wξα ´ wξα ¨ u}BpGˆGq ď 2}η} ¨ }ξα b η ´ W˚pξα b ηq},

which tends to 0 by hypothesis (i). Hence condition (i) of Lemma 4.2.1 holds forthe net pwξαqα.

To show that condition (ii) of Lemma 4.2.1 is also satisfied, we may assumethat η P CcpGq. Using the facts that V is unitary and that W commutes withI b V ˚λpsqV “ I b ρpsq, we have

mpwξαqupsq “ wξαps, squpsq

“ xλpsqV ˚λpsqV ξα, ξαyxλpsqη, ηy

“ xpλpsq b λpsqqpI b pV ˚λpsqV qqpη b ξαq, η b ξαy

“ xW˚pλpsq b IqW pI b V ˚λpsqV qpη b ξαq, η b ξαy

“ xW˚pλpsq b IqpI b V ˚λpsqV qW pη b ξαq, η b ξαy

“ xW˚pI b V ˚

qpλpsq b λpsqqpI b V qW pη b ξαq, η b ξαy

“ xW˚pI b V ˚

qW˚pλpsq b IqW pI b V qW pη b ξαq, η b ξαy

“ xpλpsq b IqpI b V qW pI b V qW pη b ξαq,W pI b V qW pη b ξαqy.


Let wα “ W pI b V qWη b ξα and vα “ mpwξαqu ´ u. Then

vαpsq “ pxλpsqV ˚λpsqV ξα, ξαy ´ 1q xλpsqη, ηy

“ xpλpsq b Iqωα, ωαy ´ xpλpsq b Iqpη b ξα, η b ξαqy

“ xpλpsq b Iqpωα ´ pη b ξαqq, ωαy

`xpλpsq b Iqpη b ξαq, ωα ´ pη b ξαy

for all s P G. This implies

}vα} ď 2}η} ¨ }ωα ´ pη b ξαq}.

Let C “ supp η. Since }λpsqρpsqξα ´ ξα} Ñ 0 uniformly on C, it follows that

}ωα ´ pη b ξαq}2

“

ż

G

ż

G

|ωαps, tq ´ pη b ξαqps, tq|2dtds

“

ż

G

ż

G

|ηpsqpξαpsts´1qΔpsq

´1{2´ ξαptq|

2dtds

“

ż

C

|ηpsq| ¨ }λpsqρpsqξα ´ ξα}22ds,

which converges to 0. This implies

}mpwξαqu ´ u} “ }vα} Ñ 0.

So condition (ii) of Lemma 4.2.1 is satisfied. �For any compact subset C of G and ε ą 0, by Proposition 1.8.15 there exists

gC,ε P L1pGq such that gC,ε ě 0, }gC,ε}1 “ 1 and }LsgC,ε ´ gC,ε}1 ď ε for alls P C. Fix such a gC,ε, and for any relatively compact neighbourhood U of e, letfU “ |U |´11U . Then let

fU,C,εptq “

ż

G

gC,εpsqτsfU ptqds,

whereτsfptq “ Δpsqfps´1tsq.

Lemma 4.2.4. The function fU,C,ε defined above has the following properties.

(i) fU,C,ε ě 0 and }fU,C,ε}1 “ 1;(ii) }τsfU,C,ε ´ fU,C,ε}1 ď ε for all s P C.

Proof. The assertions in (i) are clear. To prove (ii), observe first that

τsfU,C,εptq “

ż

G

gC,εpwqfU pw´1ps´1tsqwqdw

“

ż

G

gC,εps´1wqfU pw´1twqdw.

This implies that

}τsfU,C,ε ´ fU,C,ε}1 ď

ż

G

ż

G

|gC,εps´1wq ´ gC,εpwq|fU pw´1twqΔpwqdwdt

“

ż

G

ż

G

|gC,εps´1wq ´ gC,εpwq|fU ptqdwdt

“

ż

G

|gC,εps´1wq ´ gC,εpwq|dw

“ }LsgC,ε ´ gC,ε}1 ď ε

for all s P C. �


Lemma 4.2.5. Let ξU,C,ε “a

fU,C,ε P L2pGq. Then ξU,C,ε ě 0 and }ξU,C,ε}2 “ 1and

(i) }λGpsqρGpsqξU,C,ε ´ ξU,C,ε}22 ď ε for all s P C.

(ii) For each finite subset F of L2pGq, compact subset C of G and ε ą 0,there exists a neighbourhood U0 of e such that

}W pξU,C,ε b ηq ´ pξU,C,ε b ηq}22 ď ε

for all neighbourhoods U of e such that U Ď U0 and all η P F .

Proof. (i) follows from Lemma 4.2.4. Indeed, letting ξ “ ξU,C,ε,

}λGpsqρGpsqξ ´ ξ}22 “

ż

G

ˇ

ˇ

ˇΔpsq

1{2ξps´1tsq ´ ξptqˇ

ˇ

ˇ

2

dt

ď

ż

G

ˇ

ˇΔpsqξps´1tsq2

´ ξptq2ˇ

ˇ dt

“

ż

G

|τsfU,C,εptq ´ fU,C,εptq|dt

“ }τsfU,C,ε ´ fU,C,ε}1 ď ε

for all s P C.(ii) Retaining the notation of (i), for η P L2pGq we have

}W pξ b ηq ´ pξ b ηq}22

“

ż

G

ż

G

|ξpsqηpstq ´ ξpsqηptq|2dsdt

“

ż

G

ż

G

fU,C,εpsq|ηpstq ´ ηptq|2dsdt

“

ż

G

ż

G

ż

G

gC,εpwqfU pw´1swqΔpwq|ηpwsw´1tq ´ ηptq|2dsdtdw.

Since }Lsη ´ η}2 Ñ 0 as s Ñ e, there exists an open neighbourhood U0 of e suchthat

ş

G|ηpstq ´ ηptq|2dt ď ε1{2 for all s P V0 and η P F . Let U0 be an open

neighbourhood of e such that wsw´1 P V0 for all s P U0 and w P supp gC,ε. Then,by the above,

}W pξ b ηq ´ pξ b ηq}22 ď ε

ż

G

ż

G

gC,εpwqfU pw´1swqΔpwqdsdw

“ ε}gC,ε}1}fU }1 “ ε,

whenever U Ď U0. This proves (ii). �

Let A denote the collection of all quadruples α “ pC,F, ε, Uq, where C is acompact subset of G, F a finite subset of L2pGq and ε ą 0, and U is an openneighbourhood of e in G such that the functions ξU,C,ε satisfy (i) and (ii) of Lemma4.2.5 We order A by letting

α “ pC,F, ε, Uq ď α1“ pC 1, F 1, ε1, U 1

q

if and only if C Ď C 1, F Ď F 1, ε1 ď ε and U 1 Ď U .

Corollary 4.2.6. For α P A, let ξα “ ξU,C,ε be as in Lemma 4.2.5. Then

(i) }λGpsqρGpsqξα ´ ξα}2 Ñ 0 uniformly on compact subsets of G.(ii) }W pξα b ηq ´ pξα b ηq}2 Ñ 0 for all η P L2pGq.


Proof. (i) is clear. For (ii), let η P L2pGq, δ ą 0 and a compact subset C0 ofG be given. By Lemma 4.2.5(ii), there exists a neighbourhood U0 of e in G suchthat

}W pξU,C0,δ b ηq ´ pξU,C0,δ b ηq}2 ď δ

for all neighbourhoods U of e such that U Ď U0. Let α0 “ pC0, tηu, δ2, U0q P A.Then, for all α ě α0,

}λGpsqρGpsqξα ´ ξα}2 ď?ε ď δ

for all s P C0 and also

}W pξα b ηq ´ pξα b ηq}2 ď?ε ď δ.

This shows (ii). �

We are now able to present the main result of this section.

Theorem 4.2.7. Let G be a locally compact group. Then ApGq is operatoramenable if and only if G is amenable.

Proof. We only need to show that if G is amenable, then ApGq is operatoramenable. By Lemma 4.2.1, it suffices to find a net pwαqα in BpG ˆ Gq satisfyingthe conditions (i) and (ii) of that lemma. However, if A is as defined prior toCorollary 4.2.5 and for α P A, ξα is as in Corollary 4.2.6, then the net pwαqα inBpG ˆ Gq, defined by

wαps, tq “ xλGpsqρGptqξα, ξαy, s, t P G,

satisfies conditions (i) and (ii) of Lemma 4.2.1. �

A locally compact group G is called an SIN-group (group with small invariantneighbourhoods) if every neighbourhood U of the identity contains a neighbourhoodV of the identity such that x´1V x “ V for all x P G. Clearly, compact groups andlocally compact groups with open centres are SIN-groups. Note that SIN-groupsare unimodular. Before concluding this section, we consider amenable SIN-groups.For such groups G, operator amenability of ApGq can be established by simplerarguments, which we include for readers with a special interest in SIN-groups.The proof, however, uses the fact that closed subgroups of G are sets of synthesis(compare Section 6.1).

Lemma 4.2.8. Let G be an SIN-group and let DG “ tps, sq : s P Gu be thediagonal of GˆG. Let V be a neighbourhood basis of e in G consisting of relativelycompact, conjugation invariant sets. Then there exists a bounded net puV qV PV inBpG ˆ Gq such that uV ps, sq “ 1 for all s P G and uV v Ñ 0 for each v P IpDGq.

Proof. We can define a unitary representation π of G ˆ G on L2pGq by

πps, tqξ “ λGpsqρGptqξ, s, t P G, ξ P L2pGq.

For V P V , define the function uV on G ˆ G by

uV ps, tq “ |V |´1

xπps, tq1V , 1V y.

Then, for all ps, tq P G ˆ G,

uV ps, tq “1

|V |

ż

G

1V ps´1xtq1V pxqdx “|sV t´1 X V |

|V |


and hence, in particular, uV ps, sq “ 1 for all s P G. Clearly, uV is a continuouspositive definite function and therefore }uV } “ uV peq “ 1. If v P jpDGq, then thereexists V P V such that V X supp v “ H. Now jpDGq is dense in IpDGq since thesubgroup DG of G ˆ G is a set of synthesis for ApG ˆ Gq (Theorem 6.1.9). SinceuV v “ 0 for all u P jpDGq and all small enough V , it follows that uV v Ñ 0 forevery v P IpDGq. �

Theorem 4.2.9. If G is an amenable SIN-group, then ApGq is operatoramenable.

Proof. Letm : ApGˆGq Ñ ApGq denote the map defined bympvqpsq “ vps, sq

for v P ApG ˆ Gq and s P G. By Lemma 4.2.1 it suffices to show that there existsa bounded net pwαqα in ApG ˆ Gq such that

(i) }u ¨ wα ´ wα ¨ u} Ñ 0 for every u P ApGq;(ii) }mpwαqu ´ u} Ñ 0 for all u P ApGq.

To show (ii), let pvβqβPB be a bounded approximate identity for ApGq (Theorem2.7.2) and let puV qV PV be the net constructed in Lemma 4.2.8. Let A “ B ˆVB bethe product directed set, and for α “ pβ, pVγqγPBq P A, define wα : G ˆ G Ñ C by

wαps, tq “ uVβps, tqvβpsqvβptq, s, t P G.

Then pwαqα is a bounded net in ApG ˆ Gq. Since mpwαqpsq “ uV ps, sqvβpsq2 “

vβpsq2, the net pmpwαqqαPA forms an approximate identity for ApGq.To see (i), let u P ApGq. Then

u ¨ pvβ ˆ vβq ´ pvβ ˆ vβq ¨ u P IpDGq

for all β. Moreover, for all s, t P G,

pu ¨ wα ´ wα ¨ uqps, tq “ uvβ ps, tqvβpsqvβptqrupsq ´ uptqs

“ uvβ ru ¨ pvβ ˆ vβq ´ pvβ ˆ vβq ¨ usps, tq

and hence

limα

}u ¨ wα ´ wα ¨ u} “ limβ

limvβ

}uvβ ru ¨ pvβ ˆ vβq ´ pvβ ˆ vβq ¨ us} “ 0.

So (i) is satisfied. �

Lemma 4.2.10. Let H be a closed subgroup of G. Then ApGq{IpHq is completelyisometrically isomorphic to ApHq.

Proof. We know already that the map u ` IpHq Ñ u|H is an isometric iso-morphism from ApGq{IpHq onto ApHq (Corollary 2.7.4). Moreover,

IpHqK

“ V NHpGq “ tT P V NpGq : supp T Ď Hu

and V NHpGq is a von Neumann subalgebra of V NpGq which is ˚-isomorphic toV NpHq (Chapter 2). It now follows from [16] that pApGq{IpHqq˚ is completely iso-metrically isomorphic to V NHpGq and hence to V NpHq. Consequently ApGq{IpHq

is completely isometrically isomorphic to ApHq. �

Proposition 4.2.11. Let H be an open subgroup of G. Then 1H ¨ApGq “ tu P

ApGq : u “ 0 on GzHu is completely isometrically isomorphic to ApHq.


Proof. For u P ApHq, let ru be defined by rupxq “ upxq for x P H and ru “ 0otherwise. Then the map φ : u Ñ ru is an isometric isomorphism of ApHq onto1H ¨ ApGq.

Let n P N and ruijs P MnpApHqq with }ruijs}n “ 1. It follows from Lemma4.2.10 that }rĂuij ` IpHqs}n “ 1. Then, given ε ą 0, there exists rvijs P MnpApGqq

such that }rvij `IpHqs} “ }rĂuij `IpHqs} and }rvijs} ď 1`ε. Now, since 1H P BpGq,}1H} “ 1 and hence the cb-norm of the multiplication operator M1H equals 1,

}φnruijs} “ }rruijs}n “ }r1Hvijs}n ď }M1H }cb}rvijs} ď 1 ` ε.

Since ε ą 0 was arbitrary, it follows that }φn} ď 1, and because this holds for alln, we get }φ}cb ď 1.

Finally, the inverse map φ´1 : 1HApGq Ñ ApHq is simply the restriction to1HApGq of the quotient map ApGq Ñ ApGq{IpHq composed with the completeisometry ApGq{IpHq Ñ ApHq (Lemma 4.2.10). Thus }φ´1}cb ď 1 and consequently}φ}cb “ 1. �

Corollary 4.2.12. For a locally compact group G, the following are equivalent.

(i) ApGq has a nonzero closed ideal which is operator amenable.(ii) G has an open amenable subgroup.

Proof. Suppose first that there exists an open amenable subgroup H of Gand consider the closed ideal I “ 1G ¨ ApGq of ApGq. By Proposition 4.2.11, I iscompletely isometrically isomorphic to ApHq. On the other hand, ApHq is operatoramenable since H is amenable (Theorem 4.2.7). Thus (ii) ñ (i).

Conversely, let I be a nonzero closed ideal of ApGq which is operator amenable.Then I has a bounded approximate identity [Theorem A.2.8]. It follows fromTheorem 2.7.9 that G has an amenable open subgroup. �

We now present a class of locally compact groups G for which even BpGq isoperator amenable.

Example 4.2.13. Let G be a semidirect product G “ N � K, where N is alocally compact abelian group andK is a compact group and both are second count-

able. Suppose that pG “ pK ˝ qŤ

tπj : j P Nu, where each πj is a subrepresentationof the regular representation of G. Then

BpGq “ ApKq ˝ q ‘ ApGq

by Proposition 2.6.1. Then BpGq is operator amenable. This follows from The-orem A.4.4 since ApGq and BpGq{ApGq “ ApKq are both operator amenable byRuan’s theorem. Examples of such group are the p-adic motion groups.

4.3. Operator weak amenability of ApGq

If π is a continuous unitary representation of G on the Hilbert space Hpπq,let AπpGq denote the closed linear span in BpGq of the set of all coefficient func-tions xπp¨qξ, ηy, ξ, η P Hpπq. The dual space AπpGq˚ is canonically isomorphicto V NπpGq, the von Neumann subalgebra of BpHpπqq generated by the operatorsπpxq, x P G.

Let π and σ be two disjoint representations of G, that is, there are no subrep-resentations π1 of π and σ1 of σ such that π1 and σ1 are unitarily equivalent. Then,

4.3. OPERATOR WEAK AMENABILITY OF ApGq 139

denoting by ‘p the �p-direct sum,

Aπ‘σpGq “ AπpGq ‘1 AσpGq

andV Nπ‘σpGq “ V NπpGq ‘8 V NσpGq,

respectively.

Lemma 4.3.1. Let G be a noncompact locally compact group. Then the fourrepresentations

1G ˆ 1G, 1G ˆ λG, λG ˆ 1G and λG ˆ λG

of G ˆ G are pairwise disjoint.

Proof. Since G is noncompact, there cannot be a nonzero fixed vector ξ forλG in L2pGq, because any such ξ would define a nonzero constant function in ApGq.Hence none of the representations 1GˆλG, λGˆ1G and λGˆλG can have a nonzerovector in its Hilbert space which is fixed under the action of all elements of G ˆG.Consequently, all three representations are disjoint from 1G ˆ 1G.

Now p1G ˆ λGqps, eqξ “ ξ for all ξ P p1G ˆ λGqH “ C b L2pGq and s P G.But there is no nonzero fixed vector for πpG ˆ teuq, where π is either λG ˆ 1G orλG ˆλG. Hence 1G ˆλG is disjoint from λG ˆ 1G and λG ˆλG. Similarly, λG ˆ 1Gand λG ˆ λG are disjoint. �

Let Ae denote the unitization of a Banach algebra A.

Lemma 4.3.2. Let G be a noncompact locally compact group. Then we have thefollowing completely isometric identifications

ApGqe pbApGq – ApλGˆλGq‘p1GˆλGq

“ span tu, 1G ˆ v : u P ApG ˆ Gq, v P ApGqu

and

ApGqe pbApGqe – ApλGˆλGq‘p1GˆλGq‘pλGˆ1Gq‘p1Gˆ1Gq

“ span tu, 1G ˆ v, w ˆ 1G, 1GˆG : u P ApG ˆ Gq, v, w P ApGqu,

which are implemented by algebra isomorphisms.

Proof. Since ApGq˚ “ V NpGq “ V NλGpGq and ApGq˚

e “ V NλGˆ1GpGq,

pApGqe pbApGqq˚

– V NλGˆ1GpG ˆ Gq bV NpGq

“ pV NpGq ‘8 V N1GpGqq bV NpGq

“ pV NpGq bV NpGqq ‘8 pV N1GpGq bV NpGqq

“ V NpG ˆ Gq ‘8 V N1GˆλGpG ˆ Gq

“ V NpλGˆλGq‘p1GˆλGqpG ˆ Gq

“`

ApλGˆλGq‘p1GˆλGq

˘˚.

It is clear that the resulting identification of the spaces ApGqe pbApGq andApλGˆλGq‘p1GˆλGqpGˆGq is implemented by an algebra isomorphism. The secondequality in the first statement follows from Lemma 4.3.1.

The proof of the second assertion is similar. �

Theorem 4.3.3. Let G be an arbitrary locally compact group. Then ApGq isoperator weakly amenable.


Proof. If G is compact, then ApGq is even operator amenable by Theorem4.2.7 and hence operator weakly amenable. Thus we may assume that G is non-compact. Consider the following direct sum

π “ pλG ˆ λGq ‘ p1G ˆ λGq ‘ pλG ‘ 1Gq ‘ p1G ˆ 1Gq

of pairwise disjoint representations of GˆG (Lemma 4.3.1). Let A “ AπpGˆGq Ď

BpG ˆ Gq. Then, by Lemma 4.3.2

A “ span tu, 1G ˆ v, v ˆ 1G, 1GˆG : u P ApG ˆ Gq, v P ApGqu.

In particular, AπpG ˆ Gq is a subalgebra of BpG ˆ Gq. In the identification ofApGqe pbApGqe in Lemma 4.3.2, the multiplication map m : ApGqe pbApGqe Ñ

ApGqe corresponds to the map R : AπpG ˆ Gq Ñ AλG

À

1GpGq, which restrictsfunctions to the diagonal ΔG “ tps, sq : s P Gu. Let K1 “ kerR and K0 “

K1 X pApGq pbApGqq. Then

K1 “ span tu ´ 1G ˆ Rpuq, u ´ Rpuq ˆ 1G : u P AπpG ˆ Gqu

andK0 “ kerR X ApG ˆ Gq “ tu P ApG ˆ Gq : u|ΔG

“ 0u.

Since the closed subgroup ΔG of G ˆ G is a set of spectral synthesis for ApG ˆ Gq

(Theorem 6.1.9), K0 is the only closed ideal of ApG ˆ Gq with hull equal to ΔG.

In particular, K20 “ K0. As ApG ˆ Gq is an ideal in AπpG ˆ Gq, it follows that

pApGq pbApGqqK1 “ ApG ˆ GqK1 Ď ApG ˆ Gq X K1 “ K0.

On the other hand, using again that ΔG is a set of spectral synthesis,

K0 “ ApG ˆ GqK0 Ď ApG ˆ GqK1 “ pApGq pbApGqqK1.

Combining the preceding two equations with K0 “ K20 , we obtain

K20 “ pApGq pbApGqqK1.

Now, ApGq is completely contractive and ApGq2 “ ApGq because ApGq is a Taube-rian algebra. This shows that the hypotheses of Theorem A.5.4 are satisfied andhence ApGq is operator weakly amenable. �

4.4. The flip map and the antidiagonal

This section serves as a preparation for the next one. We study completelyboundedness of the so-called flip map of the Fourier algebra ApGq and provide acriterion for when the antidiagonal subgroup of G ˆ G belongs to the coset ring.This latter goal is achieved by employing piecewise affine maps.

Let j˚ : BpGq Ñ BpGq be defined by j˚puq “ u, where as before, upxq “ upx´1q,x P G. Then j˚ is a linear isometry. In fact, let upxq “ xπpxqξ, ηy for some unitary

representation π of G and ξ, η P Hpπq with }u} “ }ξ} ¨ }η}. Then upxq “ xπpxqη, ξy

and hence }u} ď }ξ} ¨ }η} and

}u} “ }puq } ď }u} ď }u}.

Hence j˚ : ApGq Ñ ApGq, the flip map, is a linear isometry.

Proposition 4.4.1. Let G be a locally compact group and let j˚ : ApGq Ñ ApGq

be defined by j˚puq “ u, where upxq “ upx´1q, x P G. Then j˚ is an isometry,and j˚ is completely bounded if and only if G contains an abelian subgroup of finiteindex.

4.4. THE FLIP MAP AND THE ANTIDIAGONAL 141

Proof. For any Banach space E, let E denote the complex conjugate spaceof E, i.e., the space E with scalar multiplication λ ¨ x “ λx, x P E, λ P C. Thenthe map u Ñ u is a complete isometry from ApGq onto ApGq. Therefore, if j˚ iscompletely bounded, then so is the map

j˚ : ApGq Ñ ApGq, u Ñ u,

with }j˚}cb “ }j˚}cb. The adjoint of j˚ is the map

j : V NpGq Ñ V NpGq, T Ñ T˚,

which then also is completely bounded by [70, Proposition 3.2.2], and satisfies}j}cb “ }j˚}cb.

We now show that this forces V NpGq to be subhomogeneous, that is, theirreducible representations of V NpGq are finite dimensional and their degrees arebounded. Towards a contradiction, assume that this is not true. Then, for eachn P N, V NpGq contains the full matrix algebra Mn as a ˚-subalgebra [261, Lemma9.3]. For each n, let jn : Mn Ñ Mn stand for taking the transpose of an n ˆ n-matrix. Since entrywise conjugation of matrices is a complete isometry, it followsfrom [70, Proposition 2.2.7] that

n “ }jn}cb “ }j|Mn}cb ď }j}cb “ }j˚}cb “ }j˚}cb

for all n P N, which is a contradiction. So there exists n P N such that everyirreducible representation of V NpGq has dimension n or less.

For k P N, let Pk denote the standard polynomial in k noncommuting variables,that is,

Pkpa1, . . . , akq “

ÿ

p´1qσaσp1qaσp2q . . . aσpkq,

where the sum is taken over all permutations σ of t1, . . . , ku and p´1qσ denotes thesignature of σ. Then, by a result of Amitsur and Levitzky [3], V NpGq satisfies thepolynomial identity P2n “ 0. Since L1pGq is isomorphically contained in V NpGq,L1pGq also satisfies P2n “ 0. Now, Theorem 1 of [215] implies that G has anabelian subgroup of finite index. �

Lemma 4.4.2. A subset C of a group G is a coset if and only if for any r, s, t P

C, rs´1t P C. Moreover, in this case H “ C´1C is a subgroup of G and C “ sHfor every s P C.

Proof. Necessity of the condition is trivial. So suppose the condition is sat-isfied. If s, t P H, then s “ s´1

1 s2 and t “ t´11 t2, where sj , tj P C, j “ 1, 2, and

hencest “ s´1

1 ps2t´11 t2q P C´1C and s´1

“ s´12 s1 P C´1C.

Consequently, H is a subgroup of G. Furthermore, if s P C and t P H, wheret “ t´1

1 t2, t1, t2 P C, then st “ st´11 t2 P C and so sH Ď C. Conversely, C “

ss´1C Ď sH. �

Let G and H be groups and Y Ď H a coset of some subgroup of H. A mapα : Y Ñ G is called affine if

αprs´1tq “ αprqαpsq´1αptq

for all r, s, t P Y . In this case, it follows from Lemma 4.4.2 that αpY q is also a coset.Hence, if s P Y , then the map

β : s´1Y Ñ αpsq´1αpY q, t Ñ αpsq

´1αpstq, t P s´1Y,


is a homomorphism between the subgroups s´1Y of H and αpsq´1αpY q of G. Wenow introduce the less known notion of piecewise affine map.

Definition 4.4.3. Let G and H be groups and Y Ď H. A map α : Y Ñ G iscalled piecewise affine if there are sets Yi, i “ 1, . . . , n in RpHq, the coset ring ofH (see Appendix A.1), with the following properties:

(i) Y “Ťn

i“1 Yi, and the sets Yi are pairwise disjoint;(ii) each Yi is contained in a coset Ki on which there exists an affine map

αi : Ki Ñ G such that αi|Yi“ α|Yi

, 1 ď i ď n.

Lemma 4.4.4. Let Y Ď H and α : Y Ñ G a map. If the graph

Γα “ tps, αpsqq : s P Y u

is a coset in H ˆ G, then Y is a coset in H and α is an affine map.

Proof. Let r, s, t P Y . Then, since Γα is a coset,

pr, αprqqps, αpsqq´1

pt, αptqq “ prs´1t, αprqαpsq´1αptqq P Γα,

which implies that Y is a coset (Lemma 4.4.2). On the other hand, since Γα isa graph, prs´1t, αprs´1tqq P Γα and therefore αprqαpsq´1αptq “ αprs´1tq, whichshows that α is affine. �

Lemma 4.4.5. Let G and H be groups, Y Ď H and α : Y Ñ G a map. IfΓα P RpH ˆ Gq, then α is piecewise affine.

Proof. Since Γα P RpH ˆGq, there exists a finite collection S of subgroups ofH ˆ G such that Γα belongs to the smallest ring of subsets of H ˆ G generated bycosets of elements of S. We may assume that S is closed under forming intersections.We may further assume that if S1, S2 P S and S1 Ď S2, then rS2 : S1s “ 8. NowΓα can be written as a disjoint union Γα “

Ťnj“1 Ej , where each Ej is of the form

Ej “ LjzŤmj

i“1 Mji and Lj and Mji are cosets of subgroups in S and Mji Ď Lj forall j and i.

We claim that each Lj is a graph. Towards a contradiction, assume that for

some j, there are elements ps, t1q and ps, t2q of Lj with t1 ‰ t2. Let t “ t´11 t2 and

let e denote the identity of H. Then, for any element ps, αpsqq of Ej ,

ps, αpsqtq “ ps, αpsqqpe, tq “ ps, αpsqqps, t´11 qps, t2q P LjL

´1j Lj “ Lj

because Lj is a coset. Since Ej Ď Γα is a graph, it follows that ps, αpsqtq P Mji forsome i. Thus

ps, αpsqq “ ps, αpsqtqpe, tq´1P Mjipe, tq

´1.

Hence Ej ĎŤmj

i“1 Mjipe, tq´1 and therefore

Lj ĎŤmj

i“1pMji Y Mjipe, tq´1q.

Hence the subgroup L´1i Li can be covered by finitely many cosets of subgroups

which are of infinite index in L´1i Li, which is impossible (see Lemma A.1.1). Thus

Li is a graph, and we may write

Li “ tps, αipsqq : s P Ki “ qpLiqu,

where q : H ˆ G Ñ H is the projection ph, gq Ñ h. Now, if we let Yi “ qpEiq andNij “ qpMijq, we see that

Yi “ KizŤmi

j“1 Nij

4.4. THE FLIP MAP AND THE ANTIDIAGONAL 143

since q|Lihas inverse s Ñ ps, αipsqq. We also see from Lemma 4.4.4 that αi : Ki Ñ

G is an affine map and αi|Yi“ α|Yi

. This completes the proof. �

Lemma 4.4.6. Let G and H be discrete groups, Y Ď H and α : Y Ñ G apiecewise affine map. Then the map φα : ApGq Ñ BpHq defined by

φαpuqpsq “

"

upαpsqq : if s P Y0 : otherwise,

is a completely bounded homomorphism.

Proof. We shall build up the proof in stages, beginning with homomorphisms,then passing to affine maps and finally to piecewise affine maps.

(i) Suppose first that Y is a subgroup of H and α is a homomorphism. Thenthe map

φYα : ApGq Ñ BpY q, u Ñ φαpuq|Y

is an isometric homomorphism whose adjoint pφYα q˚ : W˚pY q Ñ V NpGq is the

˚-homomorphism such that

pφYα q

˚pωY pyqq “ λGpαpyqq, y P Y.

Here W˚pY q “ BpY q˚ “ C˚pY q˚˚, the enveloping von Neumann algebra of C˚pY q,and for y P Y , ωY pyq is the element of W˚pY q defined by xωY pyq, uy “ upyq for allu P BpY q.

Since Y is a subgroup of H, there is a canonical isometric embedding of BpY q

into BpHq. In fact, for v P BpY q, the trivial extension rv of v to all of H satisfies}rv}BpHq “ }v}BpY q. Now, define mY : BpHq Ñ BpHq by mY puq “ 1Y u P BpHq.Then mY pBpHqq “ trv : v P BpY qu and m˚

Y : BpHq˚ Ñ BpHq˚ satisfies

m˚Y pIq “ ωHpeq and }m˚

Y }cb “ }mY }cb “ }1Y }cb “ 1.

Hence by [70, Corollary 5.1.2], m˚Y is completely positive.

The adjoint of the map v Ñ rv is a ˚-homomorphism

ψ : spantωHpyq : y P Y uw˚

Ñ W˚pY q

such that ψpωHpyqq “ ωY pyq for all y P Y . Hence it follows that φ˚α “

`

φYα

˘˚˝

ψ ˝ m˚Y is completely positive and contractive, and therefore φα is also completely

contractive.(ii) Next we suppose that Y is a coset and α is affine. Fix an element y P Y

and let β : y´1Y Ñ αpyq´1αpY q Ď G be the map defined by

βptq “ αpyq´1αpytq, t P y´1Y.

Then β is a homomorphism between the two subgroups y´1Y of H and αpyq´1αpY q

of G (compare the remark after Lemma 4.4.2). It is easily verified that φα can bewritten as a composition φα “ Ly ˝ φβ ˝ Lαpyq´1 , where Ly : BpHq Ñ BpHq andLαpyq´1 : ApGq Ñ ApGq are the left translation operators

Lyvphq “ vpy´1hq, h P H, and Lαpyq´1uptq “ upαpyqtq, t P G,

respectively. Now the adjoints of these translation operators are multiplicationsgiven by the unitaries λGpαpyq´1q˚ on V NpGq and ωHpyq˚ on W˚pHq, respectively.Since both are complete isometries, it follows that φα is a complete contraction.

(iii) Finally, suppose that α : Y Ñ G is a piecewise affine map. Then Ycan be written as a union Y “

Ťni“1 Yi of pairwise disjoint sets Yi P RpHq with


the property that Yi is contained in a coset Ki and there exists an affine mapαi : Ki Ñ G such that αi|Yi

“ α|Yi, 1 ď i ď n. For each i, let φαi

: ApGq Ñ BpHq

be the algebra homomorphism given by

φαipuqpxq “

"

upαipxqq : if x P Ki

0 : otherwise.

By part (ii), we know that φαiis completely bounded. Now, for every u P ApGq,

φαpuq “

nÿ

i“1

1Yiφαi

puq.

Indeed, if x P GzY , then φαpuqpxq “ 0 and 1Yipxq “ 0 for all i, and if x P Y , then

x P Yj for exactly one j P t1, . . . , nu and

φαjpuqpxq “ upαjpxqq “ upαpxqq,

whereas 1Yipxq “ 0 for i ‰ j. Since Yi P RpHq, 1Yi

P BpHq and the map ωi :BpHq Ñ BpHq given by ωipvq “ 1Yi

v, v P BpHq, is completely bounded sinceBpHq is a completely contractive Banach algebra.

Since the completely bounded maps form a linear space, it follows that φα “řn

i“1 ωi ˝ φαiis completely bounded. This finishes the proof of the lemma. �

Theorem 4.4.7. Let G be a group and let ΓG “ tpx, x´1q : x P Gu. Then ΓG

belongs to the coset ring RpGˆGq if and only if G has an abelian subgroup of finiteindex

Proof. Let G have an abelian subgroup, A say, of finite index and let tx1, . . . ,xnu be a representative system for the left cosets of A in G. For j “ 1, . . . , n, set

Aj “ tpa, xja´1x´1

j q : a P Au.

Then Aj is a subgroup of G ˆ G because A is abelian. Obviously,

ΓG “ px1, x´11 qA1 Y . . . Y pxn, x

´1n qAn,

so that ΓG is a finite union of left cosets of subgroups of G ˆ G, whence ΓG P

RpG ˆ Gq.Conversely, suppose that ΓG P RpG ˆ Gq. Since ΓG is the graph of the map

x Ñ x´1, Lemma 4.4.5 implies that x Ñ x´1 is a piecewise affine map. It thenfollows from Lemma 4.4.6 that the flip map j˚ : ApGq Ñ ApGq, u Ñ u is completelybounded. Finally, Proposition 4.4.1 ensures that G has an abelian subgroup of finiteindex. �

4.5. Amenability and weak amenability of ApGq and of L1pGq

We remind the reader that a Banach algebra A is called amenable if the first co-homology group H1pA,E˚q is trivial for each A-bimodule E, that is, every boundedderivation from A into E˚ is inner, and that A is said to be weakly amenable ifH1pA,A˚q “ t0u (compare Appendix A.2). In this section we study amenabilityand weak amenability of the Fourier ApGq of a locally compact group G. Amenabil-ity turns out to be very restrictive in that for any locally compact group G, ApGq

is amenable (if and) only if G has an abelian subgroup of finite index. Essential in-gredients of the proof are the results on the flip map and the antidiagonal obtainedin the last section. Weak amenability appears to be a less accessible property. A

4.5. AMENABILITY AND WEAK AMENABILITY OF ApGq AND L1pGq 145

sufficient condition is that the connected component of the identity of G be abelian.The question of whether the converse holds remains open.

Lemma 4.5.1. Let pμαqα be a net in MpGq such that, for some μ P MpGq,μα Ñ μ in the w˚-topology σpMpGq, C0pGqq and }μα} Ñ }μ}. Then for each ε ą 0,there exists a compact subset C of G such that

ş

GzCdp|μα| ` |μ|qpxq ă ε eventually.

Proof. Given ε ą 0. Let δ “ ε{4 and choose f P CcpGq such that }f}8 ď 1and |xf, μy ´ }u}| ă δ. There exists α0 such that

|xf, μαy ´ xf, μy| ă δ and |}μα} ´ }μ}| ă δ

for all α ě α0. Denoting by C the support of f , we then have

}μ} ď |xf, μy ` δ| ď |xf, μ ´ μαy| ` |xf, μαy| ` δ

ď 2δ `

ż

C

d|μα|pxq

and henceż

GzC

d|μα|pxq “ }μα} ´

ż

C

d|μα|pxq ď }μ} ` δ ´

ż

GzC

d|μα|pxq ď 3δ

for all α ě α0. Thusż

GzC

dp|μα| ` |μ|qpxq ă 4δ

for all α ě α0, as was to be shown. �

Remark 4.5.2. Let pμαqα be a net in MpGq such that μα Ñ μ in the w˚-topology and |μα|pGq Ñ |μ|pGq. Then |μα| Ñ |μ| in the w˚-topology. To seethis, after passing to a subnet, we may assume that |μα| Ñ ν for some ν P MpGq

since the unit ball of MpGq is w˚-compact. Let f P CcpGq, f ě 0. Then for anyg P CcpGq with |g| ď f , x|μα|, fy ě |xμα, gy| Ñ |xμ, gy|. This implies

xν, fy ě sup t|xμ, gy| : g P CcpGq, |g| ď fu “ x|μ|, fy.

Consequently ν ´ |μ| is positive. Since pν ´ |μ|qpGq “ 0, we conclude that ν “ |μ|.

Lemma 4.5.3. Let pμαqα be a net in MpGq such that, for some μ P MpGq,μα Ñ μ in the w˚-topology and }μα} Ñ }μ}. Then

}pμα ´ μq ˚ f}1 Ñ 0 and }f ˚ pμα ´ μq}1 Ñ 0

for each f P L1pGq.

Proof. We shall prove the first norm convergence, the proof of the second onebeing analogous. Since }μα} Ñ }μ}, after passing to a subnet if necessary, we mayassume that the net p}μα}qα is bounded. Moreover, since }μα ˚ f}1 ď }μα} ¨ }f}1,we may assume that f P CcpGq and then, in addition, that 0 ď f ď 1. For anycompact subset K of G we have

}pμα ´ μq ˚ f}1 ď

ż

K

|pμα ´ μq ˚ f |pxqdx `

ż

GzK

p|μα| ˚ f ` |μ| ˚ fqpxqdx.

Now let gα “ |μα| ˚ f and g “ |μ| ˚ f . Then gα, g P L1pGq and

}gα}1 “ }μα} ¨ }f}1 Ñ }μ} ¨ }f}1 “ }g}1.


By the preceding remark, gα Ñ g and |μα| Ñ |μ| in the w˚-topology. Therefore,given ε ą 0, by Lemma 4.5.1 there exist a compact subset K of G and α0 such that

ż

GzK

p|μα| ˚ f ` |μ| ˚ fqpxqdx ă ε

for all α ě α0. On the other hand, since the map y Ñ Lyf fromG into pC0pGq, }¨}8q

is continuous, the set tLyf : y P Ku is compact in C0pGq and hence there exists α1

such thatş

K|pμα´μq˚f |pxqdx ă ε for all α ě α1. It follows that }pμα´μq˚f}1 ď 2ε

for α ě α0, α1. Since ε ą 0 was arbitrary, this shows that }pμα ´ μq ˚ f}1 Ñ 0. �By the strict topology on MpGq we mean the locally convex topology defined

by the collection of seminorms σf pμq “ }f ˚ μ} ` }μ ˚ f}, f P L1pGq.

Lemma 4.5.4. Let G be a locally compact group, E a pseudo-unital L1pGq-bimodule and D : L1pGq Ñ E˚ a bounded derivation. Then there exists a unique

derivation rD : MpGq Ñ E˚ which extends D and is continuous with respect to the

strict topology on MpGq and the w˚-topology on E˚. In particular, rD is uniquelydetermined by its values on the set tδx : x P Gu.

Proof. We first define a left module action of MpGq on E. Fix u P E and letf P L1pGq and v P E such that u “ f ¨ v. For μ P MpGq, set μ ¨ u “ pμ ˚ fq ¨ v. Thisaction is well defined. Indeed, if g P L1pGq and w P E are such that also u “ g ¨ w,then for any bounded approximate identity phαqα of L1pGq,

pμ ˚ fq ¨ v “ limα

μ ˚ phα ˚ fq ¨ v “ limα

μ ˚ phα ˚ gq ¨ v “ pμ ˚ gq ¨ w.

It is clear that this defines a left module action of MpGq on E. Similarly, onedefines a right module action of MpGq on E. We claim that for each μ P MpGq,w˚-limαpDpμ ˚hαq ´μ ¨Dphαqq exists. To see this, let u P E and choose f P L1pGq

and v P E such that u “ v ¨ f . Then, for any μ P MpGq,

xu, pDpμ ˚ hαq ´ μ ¨ Dphαqy “ xv ¨ f,Dpμ ˚ hαq ´ μ ¨ Dphαqy

“ xv, f ¨ Dpμ ˚ hαq ´ pf ˚ μq ¨ Dphαqy

“ xv,Dpf ˚ μ ˚ hαq ´ Dpfq ¨ pμ ˚ hαqy

´xv,Dpf ˚ μ ˚ hαq ` Dpf ˚ μq ¨ hαy

“ xhα ¨ v,Dpf ˚ μqy ´ xpμ ˚ hαq ¨ v,Dpfqy

Ñ xv,Dpf ˚ μqy ´ xμ ¨ v,Dpfqy.

Therefore, we can define rD : MpGq Ñ E˚ by setting

rDpμq “ w˚´ lim

αpDpμ ˚ hαq ´ μ ¨ Dphαqq.

Then rD extends D since

rDpfq “ w˚´ lim

αpDpf ˚ hαq ´ f ¨ Dphαqq “ w˚

´ limα

Dpfq ¨ hα “ Dpfq

for every f P L1pGq. Furthermore, for μ P MpGq and f P L1pGq, we have

rDpμq ¨ f “ w˚´ lim

αpDpμ ˚ hαq ¨ f ´ μ ¨ Dphαq ¨ fq

“ w˚´ lim

αpDpμ ˚ hα ˚ fq ´ pμ ˚ hαq ¨ Dpfq

´w˚´ lim

αμ ¨ Dphα ˚ μq ` pμ ˚ hαq ¨ Dpfq

“ Dpμ ˚ fq ´ μ ¨ Dpfq.


It remains to show that rD is a derivation. Clearly, using the definition of the stricttopology on MpGq, we have μ ˚ hα Ñ μ in the strict topology for each μ P MpGq.Hence, for μ, ν P MpGq,

rDpμ ˚ νq “ w˚´ lim

αpw˚

´ limβ

Dppμ ˚ hαq ˚ pν ˚ hβqq

“ w˚´ lim

αpw˚

´ limβ

pμ ˚ hαq ¨ Dpν ˚ hβqq ` Dpμ ˚ hαq ¨ pν ˚ hβqq

“ μ ¨ rDpνq ` rDpμq ¨ ν.

The final statement follows from the fact that if μ P MpGq is such that μ ě 0and }μ} “ 1, then there is a net pμαqα in MpGq such that μα Ñ μ in the w˚-topology, where each μα is of the form

řnj“1 λjδxj

, where xj P G, λj ě 0 andřn

j“1 λj “ 1. �

Theorem 4.5.5. Let G be an amenable locally compact group. Then L1pGq isamenable.

Proof. Let E be an L1pGq-bimodule. By Lemma A.2.6, we may assume that

E is pseudo-unital. Let D : L1pGq Ñ E˚ be a derivation and let rD be the extension

of D to MpGq (Lemma 4.5.4). We shall show that rD is inner using Day’s fixedpoint theorem (Theorem 1.8.7). We define an affine action of G on K by

x ¨ ϕ “ δx ¨ ϕ ¨ δx´1 ` rDpδxq ¨ δx´1 , x P G, ϕ P E˚.

This defines a group action of G on E˚ since

pxyq ¨ ϕ “ δxy ¨ ϕ ¨ δpxyq´1 ` rDpδxyq ¨ δpxyq´1

“ δx ¨ pδy ¨ ϕ ¨ δy´1q ¨ δx´1 ` δx ¨ rDpδyq ` p pDpδxq ¨ δyq ¨ δy´1 ˚ δx´1

“ δx ¨ pδy ¨ ϕ ¨ δy´1 ` rDpδyqq ¨ δx´1 ` rDpδxq ¨ δx´1

“ x ¨ py ¨ ϕq

for x, y P G and ϕ P E˚. This action of G on E˚ is affine, that is,

x ¨ pλϕ1 ` p1 ´ λqϕ2q “ λx ¨ ϕ1 ` p1 ´ λqx ¨ ϕ2

for ϕ1, ϕ2 P E˚ and x P G. Using Lemma 4.5.4, it is easy to see that the mapϕ Ñ x ¨ ϕ from E˚ into E˚ is w˚-w˚-continuous for fixed x P G.

Now fix x0 P G and let ϕ0 “ rDpδx0q ¨ δx´1

0. Since }x ¨ ϕ0} ď }ϕ0} ` } rD},

cotx ¨ ϕ0 : x P Gu, the convex hull of the G-orbit of ϕ0, is bounded. Let K denoteits w˚-closure. Then K is a w˚-compact G-invariant subset of E˚. It is also easyto check that the map x Ñ x ¨ϕ0 from G into K is continuous for the w˚-topologyon K. Therefore, by Day’s fixed point theorem for separately continuous actions(Theorem 1.8.7), there exists ϕ P K such that

ϕ “ x ¨ ϕ “ δx ¨ ϕ ¨ δx´1 ` rDpδxq ¨ δx´1

for all x P G. Equivalently, rDpδxq “ ϕ ¨ δx ´ δx ¨ ϕ for all x P G, so that rD isimplemented by ´ϕ. �

Theorem 4.5.6. Let G be any locally compact group. Then L1pGq is weaklyamenable.


Proof. By Lemma 4.5.4 it suffices to show that every bounded derivationfrom MpGq into L8pGq, which is continuous with respect to the strict topology onMpGq and the w˚-topology on L8pGq, is an inner derivation. For x, y P G, we have

δx´1 ¨ Dpδxq “ δx´1 ¨ Dpδxy´1 ˚ δyq(4.2)

“ δx´1 ¨ rδpxy´1q´1 ¨ Dpδxy´1s ¨ δy ` δy´1 ¨ Dpδyq.(4.3)

For ψ P L8pGq, let Repψq and Impψq denote the real and imaginary part of ψ,respectively. Put

S “ tRepδx´1 ¨ Dpδxqq : x P Gu Ď L8R

pGq.

Then S is bounded above by the constant function }D}. Since L8R

pGq is a completevector lattice, ϕ1 “ suppSq exists in L8

RpGq. Moreover, it is not difficult to verify

that

(4.4) suppδx´1 ¨ S ¨ δxq “ δx´1 ¨ suppSq ¨ δx, x P G,

and

(4.5) suppψ ` Sq “ ψ ` suppSq, ψ P L8R

pGq.

Taking supxPG of the real parts in (4.2) and using (4.4) and (4.5), we obtain

ϕ1 “ δy´1 ¨ ϕ1 ¨ δy ` δy´1 ¨ RepDpδyqq,

or equivalentlyRepDpδyqq “ δy ¨ ϕ1 ´ ϕ1 ¨ δy, y P G.

Similarly, considering the imaginary parts and taking supxPG in (4.2), we obtainϕ2 P L8

RpGq such that

ImpDpδyqq “ δy ¨ ϕ2 ´ ϕ2 ¨ δy

for all y P G. Thus, setting ϕ “ ϕ1 ` iϕ2, we get

Dpδyq “ δy ¨ ϕ ´ ϕ ¨ δy, y P G.

Since every positive measure in MpGq is the strict limit of convex combinations ofpoint masses and D is continuous with respect to the strict topology on MpGq andthe w˚-topology on L8pGq, it follows that

Dpμq “ μ ¨ ϕ ´ ϕ ¨ μ

for all μ P MpGq, as required. �

Lemma 4.5.7. Let G be a locally compact group and H a closed subgroup offinite index. Then ApGq is amenable if and only if ApHq is amenable.

Proof. Since ApHq is isomorphic to a quotient of ApGq, amenability of ApGq

implies amenability of ApHq. To see the converse, choose representatives x1 “

e, x2, . . . , xm for the left cosets of H in G. Define a map

φ : ‘mj“1ApHq Ñ ApGq, ‘

mj“1uj Ñ

mÿ

j“1

Lxjuj .

Since the functions Lxjuj are supported on disjoint open sets, φ is a homomorphism,

and it is clearly continuous and surjective. Since ApHq is amenable, so is ‘mj“1ApHq

and hence also ApGq. �

Corollary 4.5.8. Suppose that G has an abelian subgroup of finite index.Then ApGq is amenable.


Proof. This follows from Theorem 4.5.5 and Lemma 4.5.7 observing that if

H a closed abelian subgroup of finite index in G, then ApHq “ L1p pHq and L1p pHq

is amenable (Theorem 4.5.5). �

Theorem 4.5.9. For a locally compact group G, the following conditions areequivalent.

(i) ApGq is amenable.(ii) G has an abelian subgroup of finite index.

Proof. (ii) ñ (i) is Corollary 4.5.8.(i) ñ (ii) Since ApGq is amenable, by [117, Proposition VII.2.15] the kernel of

the homomorphism

Δ : ApGq pbApGq Ñ ApGq, f b g Ñ fg

has a bounded approximate identity, puαqα say. For each α, there are sequencespfα,jqj and pgα,jqj in ApGq such that

8ÿ

j“1

}fα,j}ApGq}gα,j}ApGq ă 8 and uα “

8ÿ

j“1

fα,j b gα,j .

Since the map f Ñ f of ApGq is an isometry, we can define a bounded net pvαqα

in ApGq pbApGq by

vα “

8ÿ

j“1

fα,j b gα,j .

It is immediate that pvαqα is a bounded approximate identity for the kernel of thehomomorphism

Γ : ApGq pbApGq Ñ ApGq, f b g Ñ fg.

For f, g P ApGq, the function px, yq Ñ fpxqgpyq belongs to ApG ˆ Gq, and thisdefines a contractive homomorphism ψ : ApGq pbApGq Ñ ApG ˆ Gq. Let

I “ tu P ApG ˆ Gq : ψpvαqu Ñ uu.

Then I is a closed ideal of ApGˆGq containing ψpker Γq, and pψpvαqqα is a boundedapproximate identity for I. Since G, and hence GˆG, is amenable, Theorem 4.4.7applies and shows that I is of the form

I “ IpEq “ tu P ApG ˆ Gq : u|E “ 0u

for some E P RcpG ˆ Gq. We claim that E “ ΓG. Since ψpvαqpx, x´1q “ 0 forall α and all x P G, we have ΓG Ď E. On the other hand, if px, yq R ΓG then, byregularity of ApG ˆ Gq (Proposition 2.3.2), there exists w P ApG ˆ Gq such thatw|Γ “ 0 and wpx, yq ‰ 0. Then wpx, yqpvαpx, yq ´ 1q Ñ 0, and thus vαpx, yq Ñ 1and px, yq R E. This shows that ΓG “ E P RcpG ˆ Gq. From Theorem 4.4.7, weconclude that G has an abelian subgroup of finite index. �

It has been known for a long time that the measure algebra of a locally compactabelian group is amenable only if the group is discrete. Exploiting this fact andthe preceding theorem, amenability of the Fourier-Stieltjes algebra can now beencompletely characterized in terms of the group structure.


Corollary 4.5.10. The following are equivalent for a locally compact groupG.

(i) The Fourier-Stieltjes algebra BpGq is amenable.(ii) G is compact and has an abelian subgroup of finite index.

Proof. (i) ñ (ii) Since ApGq is a translation invariant ideal of BpGq, it followsthat ApGq is an invariant subspace of BpGq, regarded as a predual of the vonNeumann algebra W˚pGq. Hence there exists a central projection Z in W˚pGq

such that ApGq “ Z ¨ BpGq [270, p.123, Theorem 2.7], where xu ¨ Z, T y “ xu, ZT y

for u P BpGq and T P W˚pGq. Consequently, BpGq “ ApGq ‘ pI ´ Zq ¨ BpGq

and hence ApGq is complemented in BpGq. Since BpGq is amenable, it followsthat ApGq is also amenable [Appendix A.2] and therefore G has an abelian closedsubgroup A of finite index by Theorem 4.5.9. Then A is open in G and hence therestriction map from BpGq to BpAq is surjective. By a standard fact of amenableBanach algebras, BpAq is also amenable. Now, BpAq is isometrically isomorphic

to Mp pAq, the measure algebra of the dual group pA of A. However, amenability of

Mp pAq forces pA to be discrete [25]. So A and hence G must be compact.(ii) ñ (i) If G is compact and has an abelian subgroup of finite index, then

BpGq “ ApGq and ApGq is amenable by Theorem 4.5.9. �

We now turn to weak amenability of Fourier algebras.

Lemma 4.5.11. Let G be a locally compact group and H a closed subgroup ofG.

(i) If ApGq is weakly amenable, then so is ApHq.(ii) If H is open and ApHq is weakly amenable, then so is ApGq.

Proof. Since ApHq is a quotient of ApGq, (i) follows immediately from thebasic hereditary properties of weak amenability for commutative Banach algebras[Appendix A.2].

For (ii), let X be a symmetric Banach ApGq-bimodule and let D : ApGq Ñ Xbe a derivation. Let u P ApGq X CcpGq and choose x1, . . . , xn P G such thatsupp u Ď

Ťnj“1 xjH. For each 1 ď j ď n, let Aj “ 1xjHApGq. Then Aj is a

subalgebra of ApGq, which is isometrically isomorphic to ApHq. Since ApHq isweakly amenable, it follows that D|Aj

“ 0 for all j. In particular, Dp1xjHuq “ 0

for all j and hence Dpuq “ 0 since u “řn

j“1 1xjHu. Since ApGq X CcpGq is dense

in ApGq, we obtain that Dpuq “ 0 for all u P ApGq. �

Lemma 4.5.12. Let G be a locally compact group such that G0, the connectedcomponent of the identity of G, is abelian, and let K be a compact normal subgroupof G such that G{K is a Lie group. Then ApG{Kq is weakly amenable.

Proof. Let q : G Ñ G{K denote the quotient homomorphism. Then the

connected component of the identity of G{K equals qpG0q (see [125]). Thus, sinceG0 is abelian, so is pG{Kq0. In particular, AppG{Kq0q is amenable (Theorem 4.5.9)and hence weakly amenable. Since G{K is a Lie group, pG{Kq0 is open in G{K.It now follows from Lemma 4.5.11(ii) that ApG{Kq is weakly amenable. �

Theorem 4.5.13. Let G be a locally compact group such that G0 is abelian.Then ApGq is weakly amenable.


Proof. We shall first treat the case where G{G0 is compact. Let X be asymmetric Banach ApGq-bimodule and let D : ApGq Ñ X be a derivation. Letu P ApGq and ε ą 0. We are going to show that }Dpuq} ď ε. Since ε ą 0 isarbitrary, this is enough to conclude that Dpuq “ 0.

Since the right translation map x Ñ Rxpuq from G into ApGq is continuous,there is a neighbourhood U of e in G such that

}u ´ Rxpuq} ăε

1 ` }D}

for all x P U . Since an almost connected group is a projective limit of Lie groups,there exists a compact normal subgroup K of G such that K Ď U and G{K is aLie group. Let

pKpuq “

ż

K

Rkpuqdk,

where the integral is a Bochner integral with respect to normalized Haar measureon K. Then

}u ´ pKpuq} ď

ż

K

}u ´ Rkpuq}dk ďε

1 ` }D}.

Now, pKpuq P ApG : Kq, the subalgebra of ApGq consisting of those functions whichare constant on cosets of K. Since ApG : Kq is isometrically isomorphic to ApG{Kq

and ApG{Kq is weakly amenable, we obtain that D|ApG:Kq “ 0. Thus

}Dpuq} “ }Dpuq ´ DppKpuqq} ď }D} ¨ }u ´ pKpuq} ď ε,

as was to be shown.Now drop the hypothesis that G{G0 is compact and note that G contains an

open, almost connected subgroup H. We have shown so far that ApHq is weaklyamenable. By Lemma 4.5.11(ii), this implies that ApGq is weakly amenable. �

Proposition 4.5.14. Let G be a Lie group such that ApGq is weakly amenable.Then every compact subgroup of G has an abelian subgroup of finite index.

Proof. Let K be a compact subgroup of G. Since ApGq is weakly amenable,ApKq and ApK0q are also weakly amenable. But K0 is a compact connected Liegroup and therefore K0 has to be abelian (Theorem 4.5.9). Finally, K0 has finiteindex in K since K is compact and K0 is open in K. �

Corollary 4.5.15. For an almost connected semisimple Lie group G, the fol-lowing conditions are equivalent.

(i) ApGq is amenable.(ii) G is amenable and ApGq is weakly amenable.(iii) G is finite.

Proof. The implications (i) ñ (ii) and (iii) ñ (i) are obvious. For (ii) ñ

(iii), note that an amenable connected semisimple Lie group is compact. So G0 iscompact and hence abelian by Proposition 4.5.14. Since G0 is semisimple, it hasto be trivial, and consequently G is finite. �



The study of amenability and weak amenability and of their operator spacevariants for Banach algebras associated with locally compact groups started withJohnson’s work some forty years ago. Johnson [141] proved that the L1-algebra ofa locally compact group G is amenable precisely when G is amenable. In contrast,L1pGq is weakly amenable for any locally compact group (Theorem 4.5.6). Thecompletely different short proof presented here was found by Despic and Ghahra-mani [58]. Thus, if G is abelian, ApGq is amenable (hence weakly amenable) since

ApGq – L1p pGq. For general locally compact groups, however, the problem of whenthe Fourier algebra is weakly amenable or even amenable, turned out to be moreintricate.

Amenability of ApGq proved to be a very restrictive property. The difficultimplication of Theorem 4.4.7, stating that G must have an abelian subgroup offinite index whenever ApGq is amenable, is due to Forrest and Runde [86]. As toweak amenability, the fact that ApGq is weakly amenable provided that G0, theconnected component of the identity of G, is abelian ([13, Theorem 4.5]), was alsoestablished by Forrest and Runde [86].

Already twenty years earlier, Johnson [143] proved the surprising fact thatApSOp3qq fails to be weakly amenable by constructing a nonzero derivation fromApSOp3qq to V NpSOp3qq. That the same conclusion holds for every nonabeliancompact connected Lie group, was observed by Plymen [238]. See also [89].

Actually, there is strong evidence that this conjecture is true. Recently, Choiand Ghandehari [29], exploiting representation theory, have explicitly constructednonzero derivations from ApGq to V NpGq for such important examples as the ax`b-group and the 3-dimensional reduced Heisenberg group. Moreover, these authorssucceeded to show that a connected and simply connected Lie group G, for whichApGq is weakly amenable, has to be solvable.

The main result of this chapter, Theorem 4.2.7, is due to Ruan [245] wherehe introduced the operator space structure of the Fourier algebra ApGq. This hasbeen an important breakthrough in the study of the subject during the last twentyyears. The short and elegant proof for the special case of an SIN-group (Theorem4.2.9) was found by Spronk [262] using the deep fact, established by Takesaki andTatsuuma [272] that closed subgroups of a locally compact group G are sets ofspectral synthesis for ApGq (see Theorem 6.1.9).

Theorem 4.2.7 is the operator space analogue of Johnson’s result [141] thatthe L1-group algebra of a locally amenable group is amenable (compare Theorem4.5.5).

In contrast, the Fourier algebra ApGq is always weakly operator amenable (The-orem 4.3.3). This result was proved by Spronk [262]. We conclude these notes bymentioning two further open problems.

(1) Is it true that BpGq is weakly amenable if and only if G is compact and G0

is abelian (Question 2.10 in [264])?(2) When is BpGq operator amenable? (Question 3.8 in [264])?

CHAPTER 5

Multiplier Algebras of Fourier Algebras

An important object associated to any (nonunital) commutative Banach alge-bra A is the multiplier algebra MpAq of A, that is, the algebra of all bounded linearmaps T : A Ñ A satisfying the equation T pabq “ aT pbq for all a, b P A. WhenA is faithful, then the map a Ñ Ta, where Tapbq “ ab for b P A, is a continuousembedding of A into MpAq.

Let G be a locally compact group. Then MpApGqq consists of all boundedcontinuous functions u on G such that uApGq Ď ApGq, and since ApGq is anideal in BpGq, BpGq embeds continuously into MpApGqq. If G is abelian, thenas shown by Wendel [288], MpL1pGqq “ MpGq, and hence MpApGqq “ BpGq. Itis not difficult to see that this holds true, more generally, when G is amenable.One of the profound achievements in abstract harmonic analysis has been that theconverse holds, that is MpApGqq “ BpGq forces G to be amenable. This was shownby Nebbia [219] for discrete groups G and by Losert [201] for nondiscrete G. Wepresent these results in Sections 5.2 and 5.3, respectively.

A challenging problem arising in this context is to find elements of MpApGqqz

BpGq if G is nonamenable. It turned out that there is a certain subalgebra ofMpApGqq, the algebra McbpApGqq of all so-called completely bounded multipliers,which is much more accessible than the whole algebra MpApGqq. Sections 5.4 and5.5 are devoted to exploring McbpApGqq and its close relationship with uniformlybounded, nonunitary representations of the group G.

We have seen in Section 2.7 that ApGq possesses a bounded approximate iden-tity if and only if G is amenable. In view of the importance of the existence ofan approximate identity, the question for which nonamenable groups G the Fourieralgebra has an approximate identity bounded in a norm weaker than the ApGq-norm, has become a major issue in the past. For the class of those connectedreal Lie groups, the Levi factor of which has finite centre, there is now a completecharacterization of the groups G for which ApGq admits a completely bounded ap-proximate identity (compare the references in Section 5.8). In Section 5.6 we showthat if Γ is a lattice in a second countable locally compact group G, then ApGq

has a cb-norm bounded approximate identity if (and only if) ApΓq does so. This isapplied in Section 5.7 with G “ SLp2,Rq and Γ “ F2, the free group on two gen-erators, to deduce the existence of such an approximate identity for ApSLp2,Rqq

after establishing the existence for ApF2q.

5.1. Multipliers of ApGq

To start with, we remind the reader that if A is a faithful commutative Banachalgebra, then a mapping T : A Ñ A is a multiplier of A if T pabq “ aT pbq for alla, b P A. If T P MpAq, the algebra of all multipliers of A, then there is a unique

153

154 5. MULTIPLIER ALGEBRAS OF FOURIER ALGEBRAS

bounded continuous function pT on the spectrum σpAq of A such that

zT paqpϕq “ pT pϕqpapϕq

for all a P A and ϕ P σpAq. Thus pT pA Ď pA and conversely, if A is semisimple,

then every bounded continuous function f on ΔpAq with the property that f pA Ď pAdefines a multiplier Tf of A by letting, for a P A, Tf paq be the unique element of A

satisfying {Tf paqpϕq “ fpϕqpapϕq for all ϕ P σpAq.Turning to the Fourier algebra ApGq of a locally compact group G and identi-

fying σpApGqq with G, so that the Gelfand homomorphism of ApGq is simply theidentity mapping, we can then define a multiplier of ApGq to be a bounded contin-uous function f on G with the property that fApGq Ď ApGq. The multiplier normof f is given by

}f}MpApGqq “ supt}fu}ApGq : u P ApGq, }u}ApGq ď 1u.

With this norm, MpApGqq is a semisimple commutative Banach algebra andMpApGqq contains the Fourier-Stieltjes algebra BpGq because ApGq is an idealin BpGq. However, the norm on BpGq need not coincide with the multiplier norm.

Before attacking the problem of when MpApGqq “ BpGq, we collect a numberof basic results about MpApGqq which will be used in either this or later sections.

Lemma 5.1.1. Let H be a closed subgroup of a locally compact group G. Ifu P MpApGqq, then u|H P MpApHqq and

}u|H}MpApHqq ď }u}MpApGqq.

Proof. Given v P ApHq, there exists w P ApGq with w|H “ v. So pu|Hqv “

puwq|H and uw P ApGq since u P MpApGqq. This shows that pu|HqApHq Ď ApHq,that is, u|H P MpApHqq. Since

}pu|Hqv}ApHq ď inft}uw}ApGq : w P ApGq, w|H “ vu

ď }u}MpApGqq ¨ inft}w}ApGq : w P ApGq, w|H “ vu

“ }u}MpApGqq}v}ApHq,

it follows that }u|H}MpApHqq ď }u}MpApGqq. �

Proposition 5.1.2. Let f be a continuous function on a locally compact groupG. Then the following four conditions are equivalent.

(i) f is a multiplier of ApGq.(ii) There exists a (unique) σ-weakly continuous operator Mf on V NpGq such

that Mf pλGpxqq “ fpxqλGpxq for all x P G.(iii) f is bounded on G and there exists a constant c ą 0 such that }λGpfgq} ď

c }λGpgq} for all g P L1pGq.(iv) fv P BλpGq for every v P BλpGq.

Proof. (i) ñ (ii) Let mf : ApGq Ñ ApGq be the mapping u Ñ fu. Theadjoint operator Mf of mf is σ-weakly continuous and by the duality betweenV NpGq and ApGq, xλGpxq, uy “ upxq and therefore

xMf pλGpxqq, uy “ xλGpxq, fuy “ fpxqupxq “ fpxqxλGpxq, uy

for all u P ApGq and x P G. Thus Mf pλGpxqq “ fpxqλGpxq for all x P G. Since thelinear span of the set tλGpxq : x P Gu is σ-weakly dense in V NpGq, Mf is the onlyσ-weakly continuous operator on V NpGq satisfying this formula.

5.1. MULTIPLIERS OF ApGq 155

(ii) ñ (iii) Clearly, for all x P G,

|fpxq| “ }fpxqλGpxq}V NpGq “ }Mf pλGpxqq}V NpGq.

Since a σ-weakly continuous operator on V NpGq is norm bounded, this shows thatf is bounded on G. Thus, if g P L1pGq then fg P L1pGq and hence

xMf pλGpgqq, uy “

ż

G

gpxqxMf pλGpxqq, uydx

“

ż

G

gpxqfpxqxλGpxq, uydx

“ xλGpfgq, uy

for all u P ApGq. It follows that }λGpfgq} ď }Mf } ¨ }λGpgq}, so that (iii) holds.(iii) ñ (iv) Let v P BλpGq and g P L1pGq. Since fg P L1pGq, the duality

between C˚λ pGq and BλpGq yieldsˇ

ˇ

ˇ

ˇ

ż

G

fpxqgpxqvpxqdx

ˇ

ˇ

ˇ

ˇ

“ |xλGpfgq, vy| ď }λGpfgq} ¨ }v}BλpGq

ď c }v}BλpGq}λGpgq}.

Thus λGpgq Ñş

Gfpxqgpxqvpxqdx is well defined on λGpL1pGqq and extends to a

bounded linear functional on C˚λ pGq. Hence there exists w P BλpGq with

ż

G

gpxqfpxqvpxqdx “

ż

G

gpxqwpxqdx, g P L1pGq.

This implies that fv “ w P BλpGq.(iv) ñ (i) Suppose that fv P BλpGq for all v P BλpGq. It follows form the

closed graph theorem that the map

mf : BλpGq Ñ BλpGq, v Ñ fv,

is bounded. For v P BλpGq X CcpGq, we have

fv P BλpGq X CcpGq “ ApGq X CcpGq.

Since ApGq X CcpGq is dense in ApGq and the ApGq-norm and the BλpGq-normcoincide on ApGq, we conclude that fv P ApGq for every v P ApGq. So f is amultiplier of ApGq. �

Remark 5.1.3. Let f P MpApGqq. Then, according to Proposition 5.1.2, wecan associate with f operators mf ,mf ,Mf , and Mf on ApGq, BλpGq, V NpGq, andC˚

λ pGq, respectively. Specifically, mf puq “ fu for u P ApGq, mf is the extension

of mf to BλpGq given by the same formula, Mf is the adjoint of mf , and Mf isthe restriction of Mf to C˚

λ pGq. It follows from the proof of Proposition 5.1.2 that

pMf q˚ “ mf . Thus

}mf } ď }mf } “ }Mf } ď }Mf } “ }mf },

and so all four norms are equal to }f}MpApGqq.

Lemma 5.1.4. The unit ball of MpApGqq is σpL8pGq, L1pGqq-closed in L8pGq.

Proof. Let puαqα be a net in the unit ball of MpApGqq which converges tosome u P L8pGq in the σpL8pGq, L1pGqq-topology. Let f P L1pGq. Then

}λGpuαfq} “ }MuαpλGpfqq} ď }λGpfq},


for all α and, for all ξ, η P L2pGq,

limα

xλGpuαfqξ, ηy “ limα

ż

G

uαpxqfpxqpξ ˚ rηqpx´1q dx

“

ż

G

upxqfpxqpξ ˚ rηqpx´1q dx

“ xλGpufqξ, ηy.

Hence we also have }λGpufq} ď }λGpfq}. The proof of Proposition 5.1.2, (iii) ñ

(iv), shows that for any v P BλpGq, uv is equal to a function in BλpGq locallyalmost everywhere. In particular, u is locally almost everywhere equal to a con-tinuous function w on G. Thus, by (iv) ñ (i) of Proposition 5.1.2, w P MpApGqq.Moreover, }w}MpApGqq “ }mw} ď 1. This proves that the unit ball of MpApGqq is

σpL8pGq, L1pGqq-closed. �

Recall that BpGq is the dual space of C˚pGq and that, as we shall see later,MpApGqq “ BpGq for amenable groups G. We next observe that MpApGqq is adual Banach space for any locally compact group G.

Proposition 5.1.5. Let G be an arbitrary locally compact group. If E is thecompletion of L1pGq with respect to the norm

}f}E “ sup

"ˇ

ˇ

ˇ

ˇ

ż

G

fpxqupxqdx

ˇ

ˇ

ˇ

ˇ

: u P MpApGqq, }u}MpApGqq ď 1

*

,

then E˚ “ MpApGqq. More precisely, every bounded linear functional φ on E is ofthe form

φpfq “

ż

G

fpxqupxqdx, f P L1pGq,

for some u P MpApGqq, and then }φ} “ }u}MpApGqq.

Proof. It is clear that for every u P MpApGqq, the functional

f Ñ

ż

G


extends to a bounded linear functional φu on E and that }φu} ď }u}MpApGqq.

Conversely, let φ P E˚ with }φ} “ 1. Since the restriction of φ to L1pGq is abounded linear functional on L1pGq, there exists v P L8pGq such that

φpfq “

ż

G

fpxqvpxqdx, f P L1pGq.

Since |ş

Gfpxqvpxqdx| ď 1 for all f P L1pGq with }f}E ď 1, it follows from the Hahn-

Banach theorem and the definition of } ¨ }E that v is contained in the σpL8, L1q-closure of the unit ball of MpApGqq. Therefore, by Lemma 5.1.4, v is locally almosteverywhere equal to some u P MpApGqq, and }u}MpApGqq ď 1. This finishes theproof. �

We now proceed to show that if G is an amenable locally compact group,then MpApGqq “ BpGq and the multiplier norm and the BpGq-norm coincide. Forv P BpGq, let mvpuq “ vu for u P ApGq.

Lemma 5.1.6. If v P BpGq, then }mv}MpApGqq ď }v}BpGq and equality holds ifG is amenable.


Proof. Notice first that mv P MpApGqq since ApGq is an ideal in BpGq and

}mv}MpApGqq “ supt}vu}ApGq : u P ApGq1u ď }v}BpGq.

Now assume that G is amenable. Since BpGq “ C˚pGq˚ and CcpGq is dense inC˚pGq, given ε ą 0 there exists f P CcpGq with }f}C˚ “ 1 and |xv, fy| ě }v}BpGq´ε.As G is amenable, there exists u P ApGq such that u “ 1 on supp f and }u}ApGq ď

1 ` ε. It follows that

}v}BpGq ´ ε ď |xv, fy| “ |xv, ufy| “ |xuv, fy| ď }uv}ApGq,

and since }u}ApGq ď 1 ` ε, this implies that

}mv}MpApGqq ě}v}BpGq ´ ε

1 ` ε.

Since ε ą 0 is arbitrary, we conclude that }mv}MpApGqq ě }v}BpGq. �

Lemma 5.1.7. Let G be an amenable locally compact group and let v be amultiplier of ApGq. Then v P BpGq.

Proof. By the closed graph theorem, there exists c ą 0 such that }vu}ApGq ď

c }u}ApGq for all u P ApGq. Let s1, . . . , sn P G and ε ą 0. Since G is amenable,there exists u P ApGq such that upsjq “ 1 for j “ 1, . . . , n and }u}ApGq ď 1` ε. Forany c1, . . . , cn P C this implies

ˇ

ˇ

ˇ

ˇ

ˇ

nÿ

j“1

cjvpsjq

ˇ

ˇ

ˇ

ˇ

ˇ

“

ˇ

ˇ

ˇ

ˇ

ˇ

nÿ

j“1

cjpvuqpsjq

ˇ

ˇ

ˇ

ˇ

ˇ

“

ˇ

ˇ

ˇ

ˇ

ˇ

C

nÿ

j“1

cjλpsjq, vu

Gˇ

ˇ

ˇ

ˇ

ˇ

ď }vu}ApGq

›

›

›

›

›

λ

˜

nÿ

j“1

cjδsj

¸›

›

›

›

›

V NpGq

ď cp1 ` εq

›

›

›

›

›

λ

˜

nÿ

j“1

cjδsj

¸›

›

›

›

›

V NpGq

.

Since ε ą 0 is arbitrary,ˇ

ˇ

ˇ

ˇ

ˇ

nÿ

j“1

cjvpsjq

ˇ

ˇ

ˇ

ˇ

ˇ

ď c

›

›

›

›

›

λ

˜

nÿ

j“1

cjδsj

¸›

›

›

›

›

V NpGq

.

Let i denote the identity map Gd Ñ G, where Gd denotes the group G with thediscrete topology. Since the finite linear combinations of Dirac functions are densein C˚pGdq, this inequality shows that v P Bλ˝ipGdq. Since v is continuous, it followsthat v P BλpGq. �

The following theorem is an immediate consequence of Lemma 5.1.6 and Lemma5.1.7

Theorem 5.1.8. Let G be an amenable locally compact group. Then

MpApGqq “ BpGq and }v}BpGq “ }v}MpApGqq

for every v P BpGq.

Proposition 5.1.9. Let u P BpGq be such that Mu is an isometry of ApGq.Then u “ αχ for some character χ of G and α P C with |α| “ 1.


Proof. Observe first that upxq ‰ 0 for every x P G. Indeed, for any v P ApGq

and x P G, we have

}pLxuqv} “ }LxpuLx´1vq} “ }uLx´1v} “ }v},

so that Lxu is also an isometric multiplier. Therefore, we only have to verify thatupeq ‰ 0. For a contradiction, assume that upeq “ 0 and let v P P 1pGq X ApGq.Then, given any 0 ă ε ă 1, by Corollary 2.5.12 there exists w P P 1pGq XApGq suchthat }uvw} ď ε. This contradicts 1 “ }vw} “ }uvw}.

We claim next that Mu : ApGq Ñ ApGq is surjective. Since Mu is an isometryand a multiplier, I “ MupApGqq is a closed ideal of ApGq. Since upxq ‰ 0 for allx P G and ApGqpxq ‰ t0u, the hull of I is empty and hence I “ ApGq. Now, asMu is surjective, Mu´1 is also a multiplier and then }u´1} “ }Mu´1} “ 1 since}Mu} “ 1. Now }u}8 ď 1 and }u´1}8 ď 1 imply that |upxq| “ 1 for every x P G.

Finally, there exist a unitary representation π of G and ξ, η P Hpπq such thatξ is cyclic, }ξ} “ }η} “ 1 and upxq “ xπpxqξ, ηy for all x P G. It follows that

1 “ |upxq| “ |xπpxqξ, ηy| ď }ξ} ¨ }η} “ }u} “ 1.

Thus πpxqξ P Cη for all x P G, whence Hpπq is 1-dimensional. Thus ξ “ αη forsome α P T and then upxq “ αχpxq for some character χ of G. �

The converse to Theorem 5.1.8 also holds, that is, the condition that everymultiplier of the Fourier algebra of a locally compact group G is given by an elementof BpGq, forces G to be amenable. This result, which was established by Nebbia[219] for discrete groups and by Losert [201] for nondiscrete groups, is no doubtone of the highlights in the theory of multipliers of Fourier algebras and its proofrequires fairly sophisticated arguments. Currently, there seems to be no approachcovering both cases simultaneously. Even though some of Nebbia’s tools are validfor nondiscrete groups as well, the essential steps require discreteness of G. On theother hand, Losert’s treatment supposes G to be nondiscrete. As a consequence ofthis situation, the two cases are presented separately, the discrete one in the nextsection and the nondiscrete one in Section 5.3. In both cases we follow closely theoriginal articles.

Slightly more general, the final result will be the following theorem.

Theorem 5.1.10. For a locally compact group G, the following statements areequivalent.

(i) G is amenable.(ii) MpApGqq “ BpGq.(iii) ApGq is closed in MpApGqq.

Remark 5.1.11. (1) Condition (iii) means that there exists a constant c ą 0such that, for all u P ApGq,

supt}uv}ApGq : v P ApGq1u ě c }u}ApGq.

By the open mapping theorem, this in turn is equivalent to ApGq being closed inMpApGqq.

(2) If G is amenable, ApGq has a bounded approximate identity of norm bound1, so that in (1) the constant c can be chosen to be equal to 1.

(3) To prove Theorem 5.1.10, it only remains to show the implication (iii) ñ

(i). In fact, (i) ñ (ii) holds by Theorem 5.1.8 and (ii) ñ (iii) follows since condition(ii) implies that the norm on BpGq is equivalent to the multiplier norm.


We close this section with a fairly simple result which might support the ex-pectation that Theorem 5.1.10 is true.

Proposition 5.1.12. Let G be a nonamenable locally compact group and sup-pose that ApGq admits a sequential approximate identity . Then MpApGqq properlycontains BpGq.

Proof. Let punqn be a sequence in ApGq such that }unv ´ v} Ñ 0 for everyv P ApGq. Then, since G is not amenable, it follows from Theorem 2.7.2 thatsupnPN }un} “ 8. On the other hand, as unv Ñ v for every v P ApGq, the sequenceof multipliers Mun

of ApGq is pointwise bounded. The uniform boundedness prin-ciple now asserts that the sequence of multiplier norms }Mun

}, n P N, is bounded.This shows that on BpGq Ď MpApGqq the BpGq-norm and the multiplier norm arenot equivalent. Thus BpGq is not closed in MpApGqq and, in particular, not equalto MpApGqq. �

There are several nonamenable locally compact groups, for instance SLp2,Rq,SLp2,Zq and free groups (see Section 5.7), the Fourier algebras of which possess asequential (multiplier bounded) approximate identity.

Theorem 5.1.13. For any locally compact group G, we have UCp pGq “ ApGq ¨

V NpGq if and only if G is amenable.

Proof. Assume first that G is amenable and view V NpGq as an ApGq-Banachmodule. Since ApGq has a bounded approximate identity (Theorem 2.7.2), it followsfrom the Cohen’s factorization theorem (see [126, p.268]) that ApGq ¨ V NpGq is

closed in V NpGq and hence equal to UCp pGq.To attack the much more complicated reverse implication, let Φ denote the

map from the projective tensor product ApGq pbV NpGq into UCp pGq defined byΦpu b T q “ u ¨ T , u P ApGq, T P V NpGq. By hypothesis, Φ is surjective and hence

the algebras ApGq pbV NpGq and UCp pGq are topologically isomorphic. Thus the

adjoint map Φ˚ : UCp pGq˚ Ñ rApGq pbV NpGqs˚ is bounded below, that is, there

exists a constant c ą 0 such that }Φ˚pϕq} ě c}ϕ} for all ϕ P UCp pGq˚.Now, view ApGq as a subspace of V NpGq˚ and, for u P ApGq, let rpuq denote

the restriction of the associated functional to the subspace UCp pGq. Then, for eachT P V NpGq and v P ApGq,

xΦ˚prpuqq, v b T y “ xrpuq,Φpv b T qy “ xu, v ¨ T y.

We claim that r is an isometry. To see this, note that C˚λ pGq is ultraweakly (or

weak˚) dense in V NpGq, and hence the unit ball C˚λ pGq1 is ultraweakly dense in

the unit ball V NpGq1. Since C˚λ pGq Ď UCp pGq, it follows that }rpuq} “ }u} for all

u P ApGq. So

}Φ˚˝ rpuq} ě c }rpuq} “ c}u}

and therefore Φ˚ ˝ rpApGqq is closed in rApGq pbV NpGqs˚.Recall next that rApGq pbV NpGqs˚ is canonically isometrically isomorphic to

BpApGq, V NpGq˚q, the isomorphism Ψ being given by, for ϕ P rApGq pbV NpGqs˚,

xΨpϕqpuq, T y “ xϕ, u b T y, u P ApGq, T P V NpGq.

Let Ω “ Ψ ˝ Φ˚ ˝ r : ApGq Ñ BpApGq, V NpGq˚q. Then ΩpApGqq is closed inBpApGq, V NpGq˚q by what we have seen above. Moreover, for u, v P ApGq and


T P V NpGq,

xΩpvqpuq, T y “ xΦ˚prpuqq, u b T y “ xvu, T y.

Thus Ω equals the multiplication operator on ApGq defined by v. It follows thatApGq is closed in MpApGqq, and consequently G must be amenable by Theorem5.1.10. �

5.2. MpApGqq “ BpGq implies amenability of G: The discrete case

If A and B are commutative Banach algebras and B is a C-module for somesubalgebra C of A, then one can introduce the notion of a C-multiplier from Ainto B. However, we refrain from doing so and instead restrict to two special caseswhich play a role in the sequel. We start with an arbitrary locally compact groupG. Later, G will be assumed to be discrete.

Definition 5.2.1. A bounded linear operator φ from C0pGq into V NpGq iscalled a multiplier of C0pGq into V NpGq if φpufq “ u ¨ φpfq for all u P ApGq

and f P C0pGq. Recall that the module action of ApGq on V NpGq is defined byxu ¨T, vy “ xT, uvy for u, v P ApGq and T P V NpGq. Let MpC0pGq, V NpGqq denotethe linear space of all such multipliers. It is clear that equipped with the operatornorm, MpC0pGq, V NpGqq is a Banach space.

Definition 5.2.2. A bounded linear operator Γ from ApGq into the spaceMpGq of bounded regular Borel measures on G is called a multiplier of ApGq intoMpGq if Γpuvq “ uΓpvq holds for all u, v P ApGq. Endowed with the operator norm,the space MpApGq,MpGqq is a Banach space. We denote by MpApGq, L1pGqq theclosed subspace of MpApGq,MpGqq which consists of all Γ such that Γpuq P L1pGq

for all u P ApGq.

The following simple lemma shows that the spaces of multipliers defined aboveare isomorphic.

Lemma 5.2.3. The spaces MpApGq,MpGqq and MpC0pGq, V NpGqq are isomet-rically isomorphic, the isomorphism Φ being given by

xf,Γpuqy “ xΦpΓqpfq, uy

for u P ApGq, f P C0pGq and Γ P MpApGq,MpGqq.

Proof. For Γ P MpApGq,MpGqq, f P C0pGq and u P ApGq, we have

|xΦpΓqpfq, uy| “ |xf,Γpuqy| ď }Γ} ¨ }u} ¨ }f}.

Thus ΦpΓqpfq P V NpGq, and ΦpΓq is a bounded linear operator from C0pGq intoV NpGq. Moreover, for v P ApGq,

xΦpΓqpvfq, uy “ xvf,Γpuqy “ xf, vΓpuqy

“ xΦpΓqpfq, uvy “ xv ¨ ΦpΓqpfq, uy.

This shows that ΦpΓq P MpC0pGq, V NpGqq. Given T P MpC0pGq, V NpGqq, foreach u, the assignment f Ñ xT pfq, uy defines an element Γpuq of MpGq and

}Γpuq} “ supt|xf,Γpuqy| : f P C0pGq, }f}8 ď 1u

“ supt|xT pfq, uy| : f P C0pGq, }f}8 ď 1u

ď }u} ¨ supt}T pfq} : f P C0pGq, }f}8 ď 1u.

5.2. MpApGqq “ BpGq IMPLIES AMENABILITY OF G: DISCRETE CASE 161

Furthermore,

xf,Γpuvqy “ xT pfq, uvy “ xu ¨ T pfq, vy “ xf,Γpuvqy

“ xT pufq, vy “ xΓpvq, ufy “ xuΓpvq, fy.

Thus Γ is an element of MpApGq,MpGqq satisfying ΦpΓq “ T and }Γ} ď }T }.Combining these facts shows that Φ is an isometric isomorphism. �

If μ P MpGq, then Γμ : ApGq Ñ MpGq, u Ñ μu defines an element ofMpApGq,MpGqq and, using that }u}8 ď }u}ApGq for u P ApGq, it is easily seen that}Γμ} ď }μ}. Conversely, every Γ P MpApGqq,MpGqq is given by a Radon measurewhich, however, in general is unbounded. Indeed, let T “ ΦpΓq P MpC0pGq, V NpGqq

and u P ApGq X CcpGq, and choose v P ApGq such that v “ 1 on supp u. Then

|xT, uy| “ |xT, uvy| “ |xu ¨ T, vy| “ |xu,Γpvqy|

ď }u}8}v}ApGq}Γ}.

We shall see in Theorem 5.2.5 below that every Γ P MpApGq,MpGqq is given by abounded Radon measure if and only if G is amenable.

Let E be the subspace of C0pGq consisting of all functions f of the form fpxq “ř8

j“1 ujpxqgjpxq, where uj P ApGq and gj P C0pGq are such thatř8

j“1 }uj}ApGq}gj}8

ă 8. We define a norm on E by setting

}f}E “ inf

#

8ÿ

j“1

}uj}ApGq}gj}8 : fpxq “

8ÿ

j“1

ujpxqgjpxq for allx P G

+

,

where the infimum is taken over all such representations of f . It is easy to verifythat, with this norm, E is a Banach space.

Lemma 5.2.4. The space MpC0pGq, V NpGqq is isometrically isomorphic to thedual space of E, the duality being given by the formula

xφT , fy “

8ÿ

j“1

xT pgjq, ujy

for T P MpC0pGq, V NpGqq and f “ř8

j“1 ujgj.

Proof. Let f “ř8

j“1 ujgj P E, so thatř8

j“1 }uj}ApGq}gj}8 ă 8. We first

show that if f “ 0 and T P MpC0pGq, V NpGqq, thenř8

j“1xT pgjq, ujy “ 0. For this,

let ε ą 0 be given and choose N P N such thatř8

j“N`1 }uj}ApGq}gj}8 ď ε. For any

compact and symmetric subset C of G with |C| ą 0, let vC “ |C|´1p1C2 ˚ 1Cq P

ApGq. Then 0 ď vC ď 1 and vC |C “ 1. Therefore, the functions vC form anapproximate identity for C0pGq. Thus there exists v P ApGq such that

Nÿ

j“1

}gj ´ vgj}8}uj}ApGq ď ε.

We next observe thatř8

j“1xT pvgjq, ujy “ 0. Indeed, since

xT pvgjq, ujy “ xvT pgjq, ujy “ xT pgjq, vujy “ xujT pgjq, vy “ xT pujgjq, vy,

we get, by continuity of T ,8ÿ

j“1

xT pvgjq, ujy “

8ÿ

j“1

xT pujgjq, vy “ xT, fy “ 0.


It now follows thatˇ

ˇ

ˇ

ˇ

ˇ

8ÿ

j“1

xT pgjq, ujy

ˇ

ˇ

ˇ

ˇ

ˇ

“

ˇ

ˇ

ˇ

ˇ

ˇ

8ÿ

j“1

xT pgj ´ vgjq, ujy

ˇ

ˇ

ˇ

ˇ

ˇ

ď }T } ¨

Nÿ

j“1

}gj ´ vgj}8}uj}ApGq

`}T }p1 ` }v}8q ¨

8ÿ

j“N`1

}gj}8}uj}ApGq

ď ε ¨ }T }p2 ` }v}8q.

Since ε ą 0 was arbitrary, it follows thatř8

j“1xT pgjq, ujy “ 0.

This shows that the valueř8

j“1xT pgjq, ujy does not depend on the representa-

tion of f P E. To each T P MpC0pGq, V NpGqq, we can therefore associate a linearfunctional FT on E by setting

xFT , fy “

8ÿ

j“1

xT pgjq, ujy, f “

8ÿ

j“1

ujgj P E.

It is clear that FT is bounded and }FT } ď }T } since

|xFT , fy| ď }T } ¨

8ÿ

j“1

}uj}ApGq}gj}8

for any representation f “ř8

j“1 ujgj of f . To see the converse }FT } ě }T }, we

simply choose g P C0pGq with }g}8 ď 1 and }T pgq} ě }T } ´ ε and then v P ApGq

with }v}ApGq ď 1 and |xT pgq, vy| ě }T pgq} ´ ε. Then

|xFT , vgy| “ |xT pgq, vy| ě }T pgq} ´ ε ě }T } ´ 2ε.

Thus T Ñ FT is an embedding of MpC0pGq, V NpGqq into E˚.Finally, every F P E˚ arises in this manner. In fact, since

|xF, ugy| ď }F } ¨ }ug}E ď }F } ¨ }u}ApGq}g}8.

for each g P C0pGq and u P ApGq, the assignment u Ñ xF, ugy defines an elementT pgq P ApGq˚ “ V NpGq. Moreover,

xT pvgq, uy “ xF, uvgy “ xT pgq, uvy “ xv ¨ T pgq, uy

for all u, v P ApGq. Consequently, T P MpC0pGq, V NpGqq and F “ FT . �

Theorem 5.2.5. For any locally compact group G, the following four conditionsare equivalent.

(i) G is amenable.(ii) For every f P C0pGq there exist u P ApGq and g P C0pGq such that

f “ ug.(iii) MpApGq,MpGqq “ MpGq.(iv) MpApGq, L1pGqq “ L1pGq.

Proof. (i) ñ (ii) Since ApGq has a bounded approximate identity and C0pGq

is a Banach ApGq-module, ApGqC0pGq is closed in C0pGq by the of Cohen-Hewittfactorization theorem [126, p.268]. On the other hand, ApGqCcpGq “ CcpGq since


for any compact subset C of G there exists v P ApGq such that v “ 1 on C. SoApGqC0pGq is dense in C0pGq, and hence (ii) follows.

(ii) ñ (iii) If (ii) holds, then the space E defined above coincides with C0pGq

and Lemma 5.2.3 shows that MpApGq,MpGqq “ MpGq using that MpApGq,MpGqq

is isometrically isomorphic to MpC0pGq, V NpGqq (Lemma 5.2.3).(iii) ñ (iv) If T is an ApGq-multiplier from ApGq into L1pGq, then by (iii) there

exists μ P MpGq such that T puq “ uμ and uμ P L1pGq for all u P ApGq. However,this clearly forces μ to be absolutely continuous with respect to Haar measure.

(iv) ñ (i) By Proposition 1.8.20, it suffices to prove that }λpfq} “ }f}1 forevery nonnegative function f in L1pGq. To that end, observe first that, for anyf P L1pGq, u P ApGq and ξ P L2pGq,

}λpufqξ}2 “ }pufq ˚ ξ}2 ď } |uf | ˚ |ξ| }2

ď } }u}8|f | ˚ |ξ| }2

ď }u}8}λp|f |q} ¨ }ξ}2.

Thus }u ¨ λpfq} “ }λpufq} ď }u}8}λp|f |q} and hence }u ¨ λpfq} ď }u}8}λpfq} forany nonnegative f P L1pGq and all u P ApGq.

Now recall that the two multiplier algebras MpApGq,MpGqq and MpC0pGq,V NpGqq are isometrically isomorphic and that an ismomorphism is given by themultiplier Tf : C0pGq Ñ V NpGq, where for g P C0pGq, Tf pgq is given by xTf pgq, uy “

xMf puq, gy, that is, xuf, gy “ xTf pgq, uy “ xu ¨ λpfq, gy. Then Tf puq “ u ¨ λpfq foru P ApGq since

xTf puq, vy “ xMf pvq, uy “ xfv, uy “ xf, uvy

“ xλpfq, uvy “ xu ¨ λpfq, vy.

Since ApGq is uniformly dense in C0pGq, it follows that

}Mf } “ sup t}Tf puq} : u P C0pGq, }u}8 ď 1u

“ sup t}u ¨ λpfq} : u P ApGq, }u}8 ď 1u

and hence, whenever f ě 0, }Mf } ď }λpfq}. On the other hand, since MpApGq,L1pGqq “ L1pGq, by the closed graph theorem there exists a constant c ą 0 suchthat }f}1 ď c }Mf } for all f P L1pGq and hence }f}1 ď c }λpfq} whenever f ě 0.Now, since f ě 0,

}f ˚ f˚}1 “

ż

G

ż

G

fpyqf˚py´1xqdydx

“

ż

G

ż

G

fpyqfpx´1yqΔpx´1yqdxdy

“

ż

G

ż

G

fpyq

ˆż

G

fpxyqΔpyqdx

˙

dy “ }f}21.

Thus, for any nonnegative f in L1pGq,

}f}1 “ }f ˚ f˚}1{21 ď c1{2λpf ˚ f˚

q1{2

“ c1{2}λpfq} ď c1{2

}f}1.

By induction we obtain }f}1 ď c1{2n}λpfq} for every n P N, and hence

}f}1 ď limnÑ8

c1{2n}λpfq} “ }λpfq}.

So we have seen that }f}1 “ }λpfq} for every nonnegative f P L1pGq, and thisimplies that G is amenable, as pointed out above. �


From now on we only consider discrete groups G. Then MpGq “ l1pGq andMpApGq, l1pGqq can be identified with the space of all functions f on G such thatfu P l1pGq for all u P ApGq. Indeed, let T P MpApGq, l1pGqq be given and forevery x P G, choose ux P ApGq with uxpxq ‰ 0. Define f : G Ñ C by fpxq “

T puxqpxquxpxq, x P G. Then, for any v P ApGq and x P G,

uxpxqfpxqvpxq “ T puxqpxqvpxq “ T puxvqpxq “ uxpxqT pvqpxq

and hence T pvqpxq “ fpxqvpxq since uxpxq ‰ 0.Define L to be the completion of the space of finitely supported functions on

G with the norm }f}L “ rpf ˚ f˚q2peqs1{4, where the power is meant as convolutionpower. The following lemma (actually, a somewhat more general version of it) wasshown in [235].

Lemma 5.2.6. Let G be a discrete group and let f be a function on G withfinite support F . Then there exists a function u on G, which is of absolute valueone on F , such that

}uf}L ď 2?2 }f}2.

Proof. Let F “ tx1, . . . , xnu and f “řn

j“1 cjδxj, and suppose first that

cj P R for all j. Let r1, . . . , rn be different Rademacher functions on the intervalr0, 1s and for each t P r0, 1s, define a function ut on G by

ut “

nÿ

j“1

rjptqδxj.

Then, by definition of the norm } ¨ }L and of ut,

ż 1

0

}utf}4Ldt “

ż 1

0

rputfq ˚ putfq˚

s2peqdt

“

ż 1

0

˜

nÿ

j1,...,j4“1

4ź

i“1

cjirjiptqδepxj1x´1j2

xj3x´1j4

q

¸

dt

“

nÿ

j1,...,j4“1

˜

4ź

i“1

cji

¸

δepxj1x´1j2

xj3x´1j4

q

ż 1

0

4ź

i“1

rjiptq dt.

Let Q denote the set of all q “ pq1, . . . , q4q in t1, . . . , nu4 such that

ż 1

0

˜

4ź

i“1

rjiptq

¸

dt ‰ 0.

Since the products of Rademacher functions form an orthonormal system, foreach q P Q there exists a permutation σq of q “ tq1, . . . , q4u such that σqpqq “


pp1, p2, p1, p2q P Q. Thus, for every q P Q, we haveś4

i“1 cqi ě 0. This implies that

ż 1

0

}utf}4Ldt “

ÿ

qPQ

˜

4ź

i“1

cqi

¸

δepxq1x´1q2 xq3x

´1q4 q ď

ÿ

qPQ

˜

4ź

i“1

cqi

¸

“

ÿ

qPQ

˜

4ź

i“1

cqi

¸

¨

ż 1

0

4ź

i“1

rqiptqdt

“

nÿ

j1,...,j4“1

˜

4ź

i“1

cji

¸

¨

ż 1

0

4ź

i“1

rjiptqdt

ď

ż 1

0

ˇ

ˇ

ˇ

ˇ

ˇ

nÿ

j1,...,j4“1

˜

4ź

i“1

cji

¸˜

4ź

i“1

rjiptq

¸ˇ

ˇ

ˇ

ˇ

ˇ

dt

ď

ż 1

0

ˇ

ˇ

ˇ

ˇ

ˇ

nÿ

j“1

cjrjptq

ˇ

ˇ

ˇ

ˇ

ˇ

4

dt.

By a well-known result about Rademacher functions [295, Theorem (8.4)], it followsthat

ż 1

0

}utf}4Ldt ď 4

˜

nÿ

j“1

c2j

¸2

“ 4 }f}42.

Now allow f to be complex-valued, say f “ f1`if2, where f1 and f2 are real-valued.Then the preceding inequality yields

ż 1

0

}utf}4Ldt ď

ż 1

0

p}utf1}L ` }utf2}Lq4dt

ď

ż 1

0

p2 ¨ maxt}utf1}L, }utf2}Lq4 dt

ď 24 ¨ 4 ¨ maxt}f1}42, }f2}

42u

ď 26}f}42.

This implies that there exists at least one t0 P r0, 1s such that

}ut0f}L ď 2?2 }f}2.

Since ut0 is of absolute value one on F , the proof is complete. �

Lemma 5.2.7. Let G be a discrete group. Then Mp�8pGq, BpGqq “ �2pGq.

Proof. Note first that, since G is discrete, �2pGqBpGq Ď ApGq and hence�2pGq Ď Mp�8pGq, BpGqq. To prove the reverse inclusion, let v P Mp�8pGq, BpGqq

and for any finite subset F of G, let vF “ 1F v P ApGq. Then, denoting by | ¨ | themultiplier norm on Mp�8pGq, BpGqq,

}vF }ApGq “ }1F v}BpGq ď |v| ¨ }1F }8 “ |v|.

By Lemma 5.2.6, there exists a function u of absolute value one on F such that

}uvF }L ď 2?2 }vF }2.

Now, for any finitely supported function f on G,

}f}2 ď }f}1{3ApGq

}f}2{3L .


It follows that for some function u of absolute value one,

}vF }2 “ }uvF }2 ď }uvF }1{3ApGq

}uvF }2{3L

ď p2?2q

2{3}uvF }

2{3ApGq

}vF }2{32

and therefore

}vF }2 ď 8 }uvF }ApGq “ 8 }p1Fuqv}BpGq ď 8 |v|.

Since F is an arbitrary finite subset of G, we conclude that v P �2pGq. �

Theorem 5.2.8. Let G be a discrete group. Then MpApGqq “ BpGq if andonly if G is amenable.

Proof. We only have to show that if MpApGqq “ BpGq, then G is amenable.Towards a contradiction, suppose that G fails to be amenable. Then, by Theorem5.2.5, (iv) ñ (i), there exists f P MpApGq, l1pGqq such that f R l1pGq. Let g “

|f |1{2; then g R l2pGq, but g P MpApGq, l2pGqq. Indeed, for each v P ApGq, we haveÿ

xPG

|gpxqvpxq|2

ď }v}8 ¨

ÿ

xPG

|fpxqvpxq| ă 8.

If now hg P BpGq for all h P l8pGq, then g P l2pGq by Lemma 5.2.7 So there existsh P l8pGq such that hg R BpGq. However,

phgqApGq “ hpgApGqq Ď h l2pGq Ď l2pGq Ď ApGq.

This shows that hg P MpApGqq even though hg R BpGq. Consequently, G has tobe amenable. �

Lemma 5.2.9. Let G be a discrete group, and suppose that ApGq is closed inMpApGqq. Then �2pGq is closed in MpApGq, �2pGqq.

Proof. Since ApGq is closed in MpApGqq, the original norm and the multipliernorm of ApGq are equivalent. Thus there exists a constant c ą 0 such that, for everyu P ApGq,

supt}uv}ApGq : v P ApGq, }v}ApGq ď 1u ě c}u}ApGq.

It follows that �2pGq is closed in MpApGq, �2pGqq once we have shown the existenceof a constant C ą 0 such that }uv}2 ě C}u}2 for all u P �2pGq and all v P ApGq

wirh }v}ApGq ď 1. By Lemma 5.2.7, �2pGq “ Mp�8pGq, BpGqq. Thus there exists

a constant d ą 0 such that }uw}ApGq ě 2d}u}2 for all u P �2pGq and all w P �8pGq

with }w}8 ď 1.Now, fix u P �2pGq and choose w P �8pGq with }w}8 ď 1 and }uw}ApGq ě d}u}2.

Note that c and d do not depend on u and w. For any v P ApGq with }v}ApGq ď 1,it follows that

}uvw}ApGq ě c}uw}ApGq ě cd}u}2.

Now, for f P �2pGq, f “ f ˚ δe and hence }f}ApGq ď }f}2. Therefore

}uvw}ApGq ď }uvw}2 ď }uv}2,

and hence }uv}2 ě cd}u}2, as required. �

For 1 ď p ă 8, let MpApGq, LppGqq denote the space of all multipliers fromApGq into LppGq, that is, the space all bounded linear operators T : ApGq Ñ LppGq

satisfying T puvq “ uT pvq for all u, v P ApGq. Equipped with the operator norm,MpApGq, LppGqq is a Banach space. Since }u}8 ď }u}ApGq for every u P ApGq,

5.3. MpApGqq “ BpGq IMPLIES AMENABILITY OF G: NONDISCRETE CASE 167

the map f Ñ Mf , where Mf puq “ fu, from LppGq into MpApGq, LppGqq is normdecreasing.

Lemma 5.2.10. Let 1 ă p ă 8 and suppose that on LppGq Ď MpApGq, LppGqq

the } ¨ }p-norm and the multiplier norm are equivalent. Then the same is true forL1pGq Ď MpApGq, L1pGqq, and hence L1pGq is closed in MpApGq, L1pGqq.

Proof. By hypothesis, there exists a constant c ą 0 such that

}g}p ď c ¨ supt}gv}p : v P ApGq, }v}ApGq ď 1u.

Let f P L1pGq and set g “ |f |1{p P LppGq. Choose v P ApGq with }gv}p ě12c}g}p

and }v}ApGq ď 1. Then

}f}1 “ }g}pp ď p2cqp}gv}

pp “ p2cqp}fvp}1

ď p2cqp supt}fu}1 : u P ApGq, }u}ApGq ď 1u.

This proves the first statement of the lemma. In particular, it follows that themultiplier norm on L1pGq Ď MpApGq, L1pGqq is a complete norm, and this in turnimplies that L1pGq is closed in MpApGq, L1pGqq. �

5.3. MpApGqq “ BpGq implies amenability of G: The nondiscrete case

This section is devoted to prove Theorem 5.1.10 for nondiscrete locally compactgroups. The proof is fairly intricate and technical and requires several new ideas.The exposition is based on [201] and some handwritten notes provided by Losert.

Lemma 5.3.1. Let T P C˚λ pGq and let K be a compact subset of G, and consider

L2pKq as a closed subspace of L2pGq. Then the operator

T : L2pKq Ñ L2

pGq, g Ñ T pgq,

is compact.

Proof. Of course, we can assume that K has positive measure |K|. Since thesubspace of compact operators is closed in BpL2pKq, L2pGqq and CcpGq is dense inC˚

λ pGq, we can moreover assume that T “ λGpfq, where f P CcpGq. If g P L2pKq

with }g}2 ď |K|´1{2, then }g}1 ď }g}2 ¨ |K|1{2 ď 1 and hence the function

λGpfqg “ f ˚ g “

ż

K´1

gpy´1qf ˚ δy´1 dy P L2

pGq

belongs to the closed absolutely convex hull of the set A “ tf ˚ δa : a P Ku. Now,since the map x Ñ f ˚ δx from G into L2pGq is continuous, A is compact and so isits absolutely convex hull. �

An interesting simple consequence of the proof of Lemma 5.3.1 is the well-known fact that C˚

λ pGq is unital only if G is discrete. Indeed, if an operator λGpfq

is invertible in C˚λ pGq, then the set

tλGpfqg : g P L2pKq, }g}2 ď |K|

´1{2u

can be relatively compact only if L2pKq is finite dimensional.In the following two lemmas, f denotes a fixed continuous function on R such

that 0 ď f ď 1, fptq “ 0 for t ď 1{4 and fptq “ 1 for t ě 1{2. Then, for anyoperator T on L2pGq, the self-adjoint operator fpT q on L2pGq is defined.


Lemma 5.3.2. Let T P V NpGq such that ´I ď T ď I, u P ApGq X P 1pGq andδ ą 0 such that xT, uy ą 1 ´ δ2{2. Then the following hold.

(i) If x P G is such that xλGpx´1qTλGpxq, uy ą 1 ´ δ2{2, then

xfpT qλGpxqfpT q, Lxuy ą 1 ´ 2δ.

(ii) Let T1, . . . , Tn P V NpGq, n P N, be pairwise commuting operators suchthat 0 ď Tj ď I for all j and x

řnj“1 T

2j , uy ă pδ2{4nq2. Then the operator

S “

˜

nź

j“1

pI ´ Tjq

¸

T

˜

nź

j“1

pI ´ Tjq

¸

satisfies xS, uy ą 1 ´ δ2.

Proof. (i) Since u P ApGq X P 1pGq, there exists ξ P L2pGq with }ξ}2 “ 1 andupyq “ xλGpyqξ, ξy for all y P G. Since 2p1 ´ tq ě 1 for t ď 1{2, by definition of fwe have p1´ fptqq2 ď 2p1´ tq for t ď 1. As ´I ď T ď I, this and the monotony ofthe functional calculus imply that

pI ´ fpT qq2

“ p1 ´ fq2pT q ď 2pI ´ T q.

Since fpT q is self-adjoint, this in turn implies

}λGpxqξ ´ fpT qλGpxqξ}22 “ xpI ´ fpT qq

2λGpxqξ, λGpxqξy

ď 2xpI ´ T qλGpxqξ, λGpxqξy

“ 2p1 ´ xλGpx´1qTλGpxq, uyq ă δ2.

Because xT, uy ą 1´ δ2{2, the same inequality holds with e instead of x. It followsthat

xfpT qλGpxqfpT q, Lxuy “ xfpT qλGpxqfpT qξ, λGpxqξy

“ xλGpxqfpT qξ, fpT qλGpxqξy

“ xλGpxqξ, λGpxqξy ` xξ, λGpx´1qfpT qλGpxqξ ´ ξy

` xfpT qξ ´ ξ, λGpx´1qfpT qλGpxqξy

ě 1 ´ }λGpx´1qfpT qλGpxqξ ´ ξ}2

´ }fpT qξ ´ ξ}2 ¨ }fpT qλGpxqξ}2

ą 1 ´ 2δ.

(ii) Let R “śn

j“1pI ´ Tjq. Since the operators Tj are pairwise commuting,

I ´ R “

nÿ

j“1

˜

Tj ¨

j´1ź

i“1

pI ´ Tiq

¸

,

as is easily verified by induction on n. Since 0 ď Tj ď I, 0 ďśj´1

i“1 pI ´ Tiq ď I forj “ 1, . . . , n, and hence

}Rξ ´ ξ}2 “

›

›

›

›

›

˜

nÿ

j“1

Tj

j´1ź

i“1

pI ´ Tiq

¸

ξ

›

›

›

›

›

2

ď

nÿ

j“1

›

›

›

›

›

j´1ź

i“1

pI ´ Tiq

›

›

›

›

›

¨ }Tjξ}

ď

nÿ

j“1

}Tjξ}2.


Now by hypothesis, for each j,

}Tjξ}22 “ xT 2

j ξ, ξy “ xT 2j , uy ă pδ2{4nq

2

and therefore

}Rξ ´ ξ}2 ă δ2{4.

This in turn implies

xS, uy “ xSξ, ξy “ xRTR, uy “ xTRξ,Rξy

“ xTξ, ξy ` xTRξ,Rξ ´ ξy ` xTRξ ´ Tξ, ξy

ě xTξ, ξy ´ }T } ¨ }Rξ}2 ¨ }Rξ ´ ξ}2 ´ }T } ¨ }Rξ ´ ξ}2 ¨ }ξ}2

ě xTξ, ξy ´δ2

4´

δ2

4“ xT, uy ´

δ2

2

ą 1 ´δ2

2´

δ2

2“ 1 ´ δ2

since }ξ}2 ď 1, }T } ď 1, }R} ď 1 and xT, uy ą 1 ´δ2

2 . �

As a heuristic motivation for the next theorem and its proof we mention thefollowing fact for a nondiscrete locally abelian group G and u1, . . . , un P ApGq. Let

uj “ pfj with fj P L1p pGq, so that }uj} “ }fj}1. Then, since pG is noncompact, one

can find characters χ1, . . . , χn P pG such that the functions δχ1˚ f1, . . . , δχn

˚ fn in

L1p pGq have pairwise almost disjoint supports.

Proposition 5.3.3. Suppose that G is nondiscrete and let uj P ApGq XP 1pGq

and xj P G, j “ 1, . . . , n. Then, given 0 ă ε ă 1, there exist wj P P 1pGq,j “ 1, . . . , n, such that

›

›

›

›

›

nÿ

j“1

μjLxjpujwjq

›

›

›

›

›

ApGq

ě p1 ´ εqnÿ

j“1

|μj |

for any choice of μ1, . . . , μn P C.

Proof. Suppose that there exist operators Si P C˚λ pGq and positive definite

functions wi P P 1pGq, i “ 1, . . . , n, with the following properties:(1) xSi, Lxi

puiwiqy ě 1 ´ ε;(2) the subspaces SipL

2pGqq are pairwise orthogonal, and so are the subspacesS˚i pL2pGqq.

Let Ni denote the null space of Si, 1 ď i ď n. It follows from (2) and standardfunctional analysis arguments that the orthogonal complements NK

i are pairwiseorthogonal. Since

}Si} “ supt}Siξ} : ξ P NKi , }ξi} ď 1u,

(2) implies that }řn

i“1 αiSi} “ max1ďiďn |αi| ¨ }Si} for any α1, . . . , αn P C. Conse-quently, for any choice of μ1, . . . , μn P C,

ˇ

ˇ

ˇ

ˇ

ˇ

C

nÿ

i“1

αiSi,nÿ

j“1

μjLxjpujwjq

Gˇ

ˇ

ˇ

ˇ

ˇ

ď

›

›

›

›

›

nÿ

j“1

μjLxjpujwjq

›

›

›

›

›

,

whenever |αi| ď 1 for all i. Now let αi “ |μi|{μi and vj “ Lxjpujwjq, 1 ď i, j ď n,

choose βij P C with |βij | “ 1 and |xSi, vjy| “ βijxSi, vjy. Then, observing that


}vj} “ }ujwj} “ pujwjqpeq “ 1, we haveC

nÿ

i“1

αiSi,nÿ

j“1

μjvj

G

“

nÿ

j“1

μj

˜

xαjSj , vjy `

ÿ

i‰j

xαiSi, vjy

¸

ě

nÿ

j“1

|μj |

˜

xSj , vjy ´

ÿ

i‰j

|xSi, vjy|

¸

ě

nÿ

j“1

|μj |

˜

2xSj , vjy ´

nÿ

i“1

|xSi, vjy|

¸

“

nÿ

j“1

|μj |

˜

2xSj , vjy ´

nÿ

i“1

βjixSi, vjy

¸

ě

nÿ

j“1

|μj |p2p1 ´ εq ´ 1q

“ p1 ´ 2εqnÿ

j“1

|μj |.

This shows that›

›

›

›

›

nÿ

j“1

μjLxjpujwjq

›

›

›

›

›

ApGq

ě p1 ´ 2εqnÿ

j“1

|μj |,

the desired inequality.The existence of operators Si and functions wi satisfying properties (1) and

(2) above is now shown by induction on n, using Lemmas 5.3.1 and 5.3.2 and thefunction f fixed before Lemma 5.3.2.

Let n “ 1 and let pgαqα be an approximate identity of L1pGq such that, foreach α, gα ě 0, }gα}1 “ 1 and g˚

α “ gα. Then λpgαq Ñ I in C˚λ pGq and }λpgαq} ď 1

and λpgαq˚ “ λpgαq for every α. For sufficiently large α, the operator T “ λpgαq

satisfies ´I ď T ď I, xT, u1y ě 1 ´ ε2{8 and

xλpx´11 qTλpx1q, u1y “ xT, Lx1

Rx1u1y ě 1 ´ ε2{8.

Now let R1 “ fpT q, R11 “ fp2T q, S1 “ R1λpx1qR1 and w1 “ 1G. Then applying

Lemma 5.3.2(i) with δ “ ε{2, it follows that xS1, Lx1pu1w1qy ě 1´ ε. Moreover, we

have 0 ď R1, R11 ď I and R1R

11 “ R1 “ R1

1R1, since 1 ě f ě 0 and fptqfp2tq “ fptqfor all t P R.

To carry out the inductive step, suppose that Ri, Pi P C˚λ pGq and wi P P 1pGq,

1 ď i ď n, have been constructed such that 0 ď Ri, Pi ď I, RiPi “ Ri, PiPj “ 0 fori ‰ j and the operators Si “ RiλGpxiqRi satisfy (1). Let Qi “ fp2Piq, 1 ď i ď n.Then

fpPiqQi “ fpPiq “ QifpPiq and QiQj “ 0, i ‰ j,

since fptqfp2tq “ fptq for all t. Now define operators T1, T2 P C˚λ pGq by

T1 “

nÿ

i“1

Q2i and T2 “ λpxn`1qT1λpx´1

n`1q.

Since the module action of ApGq on V NpGq is continuous, we have un`1¨Tk P C˚λ pGq

for k “ 1, 2.


By Lemma 5.3.1, for every non-empty compact subset K of G, the operators

h Ñ un`1 ¨ T1phq and h Ñ un`1 ¨ T2phq

from L2pKq into L2pGq are compact. Since G is nondiscrete, L2pKq is infinitedimensional, and hence there exists h P L2pGq such that }h}2 “ 1 and

xun`1 ¨ Tjphq, hy ă

ˆ

ε2

32n

˙2

, j “ 1, 2.

Let wn`1 “ xλp¨qh, hy P P 1pGq. Then

xTj , un`1wn`1y ă

ˆ

ε2

32n

˙2

, j “ 1, 2.

As in the case n “ 1 above, we can see that there exists T P C˚λ pGq such that

´I ď T ď I, xT, un`1wn`1y ă 1 ´ ε2{16 and

xλpx´1n`1qTλpxn`1q, un`1wn`1y “ xT, Lxn`1

Rxn`1pun`1wn`1qy ą 1 ´

ε2

16.

Define operators T 1, Rn`1 and R1n`1 by

T 1“

nź

i“1

pI ´ R1iqT

nź

i“1

pI ´ R1iq, Rn`1 “ fpT 1

q and R1n`1 “ fp2T 1

q.

Since pI ´ R1iqR

1i “ R1

i ´ R1i “ 0, we have T 1R1

i “ 0 and hence

Rn`1R1n`1 “ R1

n`1Rn`1 “ Rn`1 and R1iR

1n`1 “ R1

n`1R1i “ 0, 1 ď i ď n.

Taking δ “ ε{?8, Lemma 5.3.2(ii) implies

xT 1, un`1wn`1y ą 1 ´ε2

8and xλpx´1

n`1qT 1λpxn`1q, un`1wn`1y ą 1 ´ε2

8.

Finally, setting Sn`1 “ Rn`1λpxn`1qRn`1 and δ “ ε{2, Lemma 5.3.2(i) gives

xSn`1, Lxn`1pun`1wn`1qy ą 1 ´ ε.

This completes the inductive step and hence the proof of the lemma. �

Corollary 5.3.4. Let G be nondiscrete and suppose that there exists a con-stant c ą 0 such that

supt}uv}ApGq : v P ApGq, }v}ApGq ď 1u ě c }u}ApGq

for all u P ApGq. Then, given u P ApGq X P 1pGq and x1, . . . , xn P G, there existsv P ApGq X P 1pGq such that

nÿ

j“1

}v ¨ Lxju}ApGq ě

cn

4.

Proof. By Proposition 5.3.3, taking μj “ 1 and uj “ u for all j and ε “ 1{2,there exist wj P P 1pGq, 1 ď j ď n, such that

›

›

›

›

›

nÿ

j“1

Lxjpuwjq

›

›

›

›

›

ěn

2.


By hypothesis, there exists v P ApGq with }v} ď 1 and›

›

›

›

›

v ¨

nÿ

j“1

Lxjpuwjq

›

›

›

›

›

ě c ¨

›

›

›

›

›

nÿ

j“1

Lxjpuwjq

›

›

›

›

›

ěcn

2.

Write v “ v1 ` iv2, where v1 and v2 are hermitean. Then }vi} ď 1, i “ 1, 2, and›

›

›

›

›

vi ¨

nÿ

j“1

Lxjpuwjq

›

›

›

›

›

ěcn

4

for at least one i. Thus we can assume that v is hermitean and satisfies }v ¨řn

j“1 Lxjpuwjq} ě cn{4. Now, let v “ v` ´v´, v`, v´ P P pGq. Then }v`}`}v´} “

}v} ď 1, and this implies that›

›

›

›

›

v`¨

nÿ

j“1

Lxjpuwjq

›

›

›

›

›

ěcn

4}v`

}

or›

›

›

›

›

v´¨

nÿ

j“1

Lxjpuwjq

›

›

›

›

›

ěcn

4}v´

}.

It follows that at least one of the two elements }v`}´1v` and }v´}´1v´ of ApGq X

P 1pGq has the desired property. �

Lemma 5.3.5. For real-valued functions f, g P L2pGq we have

ż

G

|fpxq2

´ gpxq2|dx ď

˜

ˆż

G

pfpxq2

` gpxq2qdx

˙2

´ 4

ˆż

G

fpxqgpxqdx

˙2¸1{2

.

Proof. In fact, since f and g are real-valued,ż

G

|fpxq2

´ gpxq2|dx ď }f ´ g}2}f ` g}2

“`

}f}22 ´ 2xf, gy ` }g}

22

˘1{2 `}f}

22 ` 2xf, gy ` }g}

22

˘1{2

“`

}f}42 ` 2}f}

22}g}

22 ` }g}

42 ´ 4xf, gy

2˘1{2

,

which equals the right hand side of the stated inequality. �

In the sequel, LUCpGq denotes the space of all left uniformly continuous func-tions on G. Recall that LUCpGq consists of all those bounded continuous func-tions f on G for which the map x Ñ Lxf from G into L8pGq is continuous.Every T P V NpGq defines an operator, also denoted T , on the tensor productL2pGq b L2pGq as follows. If T “ w˚ ´ limnÑ8

řni“1 αniλpxniq, xni P G, αni P C,

then, for ξ, η P L2pGq,

T pξ b ηq “ limnÑ8

nÿ

i“1

αniλpxniqξ b λpxniqη.

Let m be a mean on LUCpGq, which is given by an integral as in the next lemma.Then m cannot be (left) invariant unless G is compact. The following lemmaprovides a bound for the lack of invariance.


Lemma 5.3.6. Let 0 ă ε ă 1 and let U, V,W be relatively compact neighbour-hoods of e in G with U´1W Ď V and |V | ă p1 ` εq|W |. Let ξ, η P L2pGq andT P V NpGq such that }ξ}2 “ }η}2 “ 1 and }T } ď 1 and suppT Ď U . Define amean m on LUCpGq by mpfq “

ş

Gfpxq|ξpxq|2dt. Then, for each f P LUCpGq and

a P G,

|mpLaf ´ fq| ď 2 supt}Lyf ´ f} : y P V u

`}f}8p2 ` 3ε ´ |xT pξ b ηq, La´1ξ b ηy|2q.

Proof. Let δ “ supt}Lyf ´ f} : y P V u. Since }η}2 “ 1, we get

mpfq “

ż

G

fpxq|ξpxq|2}η}

22dx

“

ż

GˆG

fpxq|ξpxqηpyq|2dpx, yq

“

ż

GˆG

fpxq|pξ b ηqpx, yq|2dpx, yq.

This formula and the definition of δ imply

ˇ

ˇ

ˇ

ˇ

mpfq ´ |V |´1

ż

V

ż

GˆG

fpt´1yq|pξ b ηqpx, yq|2dpx, yqdt

ˇ

ˇ

ˇ

ˇ

ď δ.

Now, this double integral equals

1

|V |

ż

G

fpyq

ˆż

V ˆG

|pξ b ηqpty, xq|2dpt, xq

˙

dy.

Similarly, it is shown that

ˇ

ˇ

ˇ

ˇ

mpLafq ´1

|W |

ż

G

fpyq

ˆż

WˆG

|pLa´1ξ b ηqpty, xq|2dpt, xq

˙

dy

ˇ

ˇ

ˇ

ˇ

ď δ.

Now, put

gpyq “

ˆż

WˆG

|pLa´1ξ b ηqpty, xq|2dpt, xq

˙1{2

and

hpyq “

ˆż

V ˆG

|pξ b ηqpty, xq|2dpt, xq

˙1{2

.

Notice next thatş

Ghpyq2dy “ |V | and

ş

Ggpyq2dy “ |W |. Indeed,

ż

G

hpyq2dy “ }η}

22

ż

V

ż

G

|ξptyq|2dy dt “ }ξ}

22 ¨ }η}

22 ¨ |V | “ |V |,

and similarly for g. Since

| |V |´1

´ |W |´1

| “ ´p|V |´1

´ |W |´1

|q ă ε |V |´1,


we conclude that

|mpLaf ´ fq| ď 2δ `

ˇ

ˇ

ˇ

ˇ

ż

G

fpyq

ˆ

gpyq2

|V |´

hpyq2

|W |

˙

dy

ˇ

ˇ

ˇ

ˇ

ď 2δ ` }f}8|W |´1

ż

G

|gpyq2

´ hpyq2|dy

`}f}8| |V |´1

´ |W |´1

|

ż

G

hpyq2dy|

ď 2δ ` }f}8

ˆ

ε ` |W |´1

ż

G

|gpyq2

´ hpyq2|dy

˙

.

To finish the proof, by the preceding inequality, it remains to show that

ż

G

|gpyq2

´ hpyq2|dy ď |W | p2 ` 2ε ´ |xT pξ b ηq, La´1ξ b ηy|q

2 .

To that end, we are going to find a lower bound forş

Ggpyqhpyqdy and then apply

Lemma 5.3.5. Notice first that, by definition of g and h,

gpyq “ Δpyq´1{2

}La´1ξ b η|WyˆG}2

and

hpyq “ Δpyq´1{2

}ξ b η|V yˆG}2 ,

where Δ denotes the modular function of G. We claim that

}pξ b ηq|V yˆG}2 ě }T pξ b ηq|V yˆG}2 ě }T pξ b ηq|WyˆGs}2 .

The first inequality is clear since }T } ď 1. For the second we are going to showthat

T rpξ b ηq|V yˆGs|WyˆG “ T pξ b ηq|WyˆG,

which requires some effort. Consider non-empty compact subsets C of GzV y andD of Wy and functions σ P L2pCq and τ P L2pDq. Define u P ApGq by upxq “

xλpxqσ, τy, x P G. If upxq ‰ 0, then

H ‰ xC X D Ď xrpGzV q X x´1W sy

and hence x R U . Thus supp u X suppT “ H and hence u ¨ T “ 0 since supp uis compact. It follows that xT, uvy “ 0 for all v P ApGq and therefore xT pσ b

ϕq, τ b ψy “ 0 for all ϕ, ψ P L2pGq. Since C and D are arbitrary non-emptycompact subsets of GzV y and Wy, respectively, a simple approximation argumentgives that xT pσ b ϕq, τ b ψy “ 0 for all ϕ, ψ P L2pGq whenever σ P L2pGzV yq andτ P L2pWyq. This in turn implies that T pσbϕq “ 0 almost everywhere on WyˆGfor every ϕ P L2pGq whenever σ P L2pGzV yq. Returning to ξ and η, we thereforeobtain

T pξ b ηq|WyˆG “ T rpξ b ηq|V yˆGs|WyˆG.


We can now estimateż

G

gpyqhpyqdy “

ż

G

}La´1ξ b η|WyˆG}2 ¨ }ξ b η|V yˆG}2

ě

ż

G

}La´1ξ b η|GˆWy}2 ¨ }T pξ b ηq|WyˆG}2

“

ż

G

Δpyq´1

ż

WyˆG

|pLa´1ξ b ηqpT pξ b ηqqpt, xq|dpt, xq dy

“

ż

G

ż

WˆG

|pLa´1ξ b ηqpT pξ b ηqqpt, xq|dpty, xqdy

ě |W | ¨ |xT pξ b ηq, La´1ξ b ηy| .

This inequality combined with Lemma 5.3.5 yieldsˆż

G

|gpyq2

´ hpyq2|dy

˙2

ď p|V | ` |W |q2

´ 4

ˆż

G

gpyqhpyqdy

˙2

ď |W |2p2 ` εq2 ´ 4|W |

2¨ |xT pξ b ηq, La´1ξ b ηy|

2

ď 4|W |2p1 ` 2ε ´ |xT pξ b ηq, La´1ξ b ηy|

2q.

Now using 1 ` s ď p1 ` s{2q2 and setting s “ 2ε ´ |xT pξ b ηq, La´1ξ b ηy|2, weconclude that

ż

G

|gpyq2

´ hpyq2|dy ď |W |

`

2 ` 2ε ´ |xT pξ b ηq, La´1ξ b ηy|2˘

,

which completes the proof of the lemma. �

Let M denote the set of all means on LUCpGq, the space of all left uniformlycontinuous functions on G. For any mean m on either LUCpGq or L8pGq andx P G, we put

dpm,xq “ supt|mpLxf ´ fq| : f P LUCpGq, }f}8 ď 1u.

Lemma 5.3.7. Suppose that there exists c ą 0 with the following property. Forany finitely many (not necessarily distinct) y1, . . . , yq P G, q P N, there existsm P M such that

řqj“1 dpm, yjq ď cq. Then, given x1, . . . , xn P G, n P N, there

exists m P M such that dpm,xjq ď c for j “ 1, . . . , n.

Proof. To each m P M we associate the n-tuple pdpm,xjqqnj“1 P Rn. Let Cdenote the convex hull of the set of all these n-tuples. We claim that for each ε ą 0,there exists ptjqnj“1 P C with }ptjqnj“1}8 ď c ` ε.

Towards a contradiction, assume that for some ε ą 0 no such n-tuple ptjqnj“1

exists. Then, by the separation theorem for convex sets, there exists psjqnj“1 P

Rn such that }psjqnj“1}1 “ 1 andřn

j“1 sjtj ě c ` ε for all ptjqnj“1 P C. Since

C Ď r0,8qn, we can assume that sj ě 0 for all j. Moreover, since C is bounded,a straightforward approximation argument shows that we can assume that sj “

pj{q P Q for j “ 1, . . . , n, whereřn

j“1 pj “ q andřn

j“1 sjtj ě c ` ε{2 for all

ptjqnj“1 P C. Now define y1, . . . , yq P G by

y1 “ . . . “ yp1“ x1, yp1`1 “ . . . “ yp1`p2

“ x2, . . . ,

y1`řn´1

j“1 pj“ . . . “ yq “ xn.


Now, by hypothesis, there exists m P M withřq

i“1 dpm, yiq ď cq. However, bydefinition of psjqnj“1,

1

q

qÿ

i“1

dpm, yiq “1

q

nÿ

j“1

pjdpm,xjq “

nÿ

j“1

sjdpm,xjq ě c `ε

2.

This contradiction shows that for each ε ą 0 there exists ptjqnj“1 P C with }ptjqnj“1}8

ď c ` ε.Since C is the convex hull of all vectors dpm,xjqnj“1, m P M, there exist

mk P M and αk ě 0, k “ 1, . . . , r, such that

tj “

rÿ

k“1

αkdpmk, xjq, j “ 1, . . . , n, andrÿ

k“1

αk “ 1.

Finally, put m “řr

k“1 αkmk. Then m P M and

dpm,xjq ď sup

#

rÿ

k“1

αk|mkpLxjf ´ fq| : f P LUCpGq, }f}8 ď 1

+

ď

rÿ

k“1

sup

αk|mkpLxjf ´ fq| : f P LUCpGq, }f}8 ď 1

(

“

rÿ

k“1

αkdpmk, xjq “ tj ď c ` ε

for all j. Since ε ą 0 is arbitrary, it follows from the compactness of M that thereexists m P M with dpm,xjq ď c for j “ 1, . . . , n. �

Proposition 5.3.8. Suppose that c satisfies the hypothesis of Lemma 5.3.7.Then, given x1, . . . , xn P G and ε ą 0, there exists a mean rm on L8pGq such that

dprm,xjq ď c ` ε, 1 ď j ď n.

Proof. Let m be any mean on LUCpGq and choose u P L1pGq such thatu ě 0 and }u}1 “ 1. Since u ˚ L8pGq Ď LUCpGq, we can define rm on L8pGq byrmpfq “ mpu ˚ fq, f P L8pGq. Then rm is a mean on L8pGq and

rmpLxjf ´ fq “ mppu ˚ δxj

´ uq ˚ fq, 1 ď j ď n.

Let δ “ ε{2. There exists v P L1pGq with v ě 0 and }v}1 “ 1 such that

}pu ˚ δxj´ uq ˚ v ´ pu ˚ δxj

´ uq}1 ă δ

for j “ 1, . . . , n. Moreover, there exist yi P G and αi P R, i “ 1, . . . , r, such thatαi ě 0,

řri“1 αi “ 1 and

›

›

›

›

›

˜

u ´

rÿ

i“1

αiδyi

¸

˚ pδxj˚ v ´ vq

›

›

›

›

›

1

ď δ


for j “ 1, . . . , n. Then, for f P L8pGq with }f}8 ď 1 and all j “ 1, . . . , n, it followsthat

|rmpLxjf ´ fq| “ |mppu ˚ δxj

´ uq ˚ fq|

ď |mppu ˚ δxj´ uq ˚ v ˚ fq| ` }pu ˚ δxj

´ uq ˚ pδe ´ vq ˚ f}8


´ uq ˚ pδe ´ vq}1

ď δ ` |mppu ˚ δxj´ uq ˚ v ˚ fq|

ď ε `

ˇ

ˇ

ˇ

ˇ

ˇ

m

˜˜

rÿ

i“1

αiδyi

¸

˚ pδxj˚ v ´ vq ˚ f

¸ˇ

ˇ

ˇ

ˇ

ˇ

ď ε `

rÿ

i“1

αi|mpδyi˚ pδxj

˚ v ´ vq ˚ fq|

“ ε `

rÿ

i“1

αi|mppδyixjy´1i

´ δeq ˚ δyi˚ v ˚ fq|

ď ε `

rÿ

i“1

αidpm, yixjy´1i q.

Applying Lemma 5.3.6, taking the elements yixjy´1i , 1 ď j ď n, 1 ď i ď r, we

find m P M such that dpm, yixjy´1i q ď c for all 1 ď j ď n and 1 ď i ď r. Thus

|rmpLxjf ´ fq| ď c ` ε for j “ 1, . . . , n, as required. �

Employing all the preceding results, we can now complete the proof of Theorem5.1.10 by showing the following

Theorem 5.3.9. Let G be a nondiscrete locally compact group and suppose thatApGq is closed in MpApGqq. Then G is amenable.

Proof. Since ApGq is closed in MpApGqq, the original norm and the multipliernorm on ApGq are equivalent. So there exists a constant C ą 0 such that

supt}uv}ApGq : v P ApGq, }v}ApGq ď 1u ě C }u}ApGq

for all u P ApGq. By Proposition 5.3.10, it suffices to find a constant c ă 2 with thefollowing property: for any finitely many x1, . . . , xn P G, there exists a mean m onLUCpGq with

řnj“1 dpm,xjq ď cn. Let

ε “ C2{200 and c “ 2 ` 3ε ´ C2

{64 ă 2.

For each relatively compact neighbourhood V of e in G we are going to constructa mean mV on LUCpGq such that

nÿ

j“1

|mV pLx´1jf ´ fq| ď 2 ¨ supt}Lyf ´ f}8 : y P V u ` cn}f}8

for all f P LUCpGq. Let V be a neighbourhood basis of the identity consisting ofcompact sets. Since y Ñ Lyf is continuous with respect to the } ¨ }8-norm, takingthen a w˚-cluster point of the net pmV qV PV , we will obtain a mean m on LUCpGq

with the desired properties.Thus fix x1, . . . , xn P G and a neighbourhood V of e. We choose neigh-

bourhoods U and W of e such that U´1W Ď V and |W | ě |V |{p1 ` εq and


u P P 1pGq X ApGq with supp u Ď U . Then, by Corollary 5.3.4, there existsv P P 1pGq X ApGq such that

nÿ

j“1

}vLxju}ApGq ą

cn

4.

Let cj “ }vLxju}ApGq, 1 ď j ď n, and let ξ, η P L2pGq such that upxq “ xλpxqξ, ξy

and vpxq “ xλpxqη, ηy for all x P G. For each j there exists Sj P V NpGq such that}Sj} ď 1 and xSj , vLxj

uy “ cj . Clearly,

xLx´1jSJ , pLx´1

jvquy “ xSj , vLxj

uy.

Now define w P ApGq by

wpxq “ |W |´1

|xW X V | “ |W |´1

p1V ˚ 1W qpxq.

Then w “ 1 on U and }w}ApGq ď |W |´1{2|V |1{2 ď 1 ` ε.

For 1 ď j ď n, let Tj “ p1 ` εq´1w ¨ pLx´1jSjq. Then }Tj} ď 1 and, since

suppu Ď U and w “ 1 on U ,

xTjpη b ξq, Lx´1jη b ξy “ xTj , pLx´1

jvquy

“1

1 ` εxw ¨ pLx´1

jSjq, pLx´1

jvquy

“1

1 ` εxLx´1

jSj , pLx´1

jvquy

“1

1 ` εxSj , vpLx´1

juqy

“cj

1 ` εě

cj2.

We now define a mean mV on LUCpGq by

mV pfq “

ż

G

fpxq|ηpxq|2dx, f P LUCpGq.

It follows from Lemma 5.3.6 that

|mV pLxjf ´ fq| ď 2 supt}Lyf ´ f} : y P V u ` }f}8

´

2 ` 3ε ´cj4

¯

for all f P LUCpGq. Sinceřn

j“1 cj ě cn{8,

nnÿ

j“1

c2j ě

˜

nÿ

j“1

cj

¸2

ěn2c2

64

and thereforenÿ

j“1

|mV pLxjf ´ fq| ď 2 supt}Lyf ´ f} : y P V u ` n}f}8

ˆ

2 ` 3ε ´c2

256

˙

“ 2 supt}Lyf ´ f} : y P V u ` dn}f}8.

This completes the proof of the theorem. �

Proposition 5.3.10. Suppose that c has the property of Lemma 5.3.7. If c ă 2,then G is amenable.

5.4. COMPLETELY BOUNDED MULTIPLIERS 179

Proof. We are going to show that given x1, . . . , xn P G and ε ą 0, there existsa mean rm on L8pGq such that dprm,xjq ď c ` 2ε for j “ 1, . . . , n. It then followsfrom [236, Proposition 4.21] that G is amenable.

Let m be any mean on LUCpGq. Choose u P L1pGq such that u ě 0 and}u}1 “ 1, and define rm on L8pGq by rmpfq “ mpu ˚ fq, f P L8pGq. Then rm is amean on L8pGq and

rmpLxjf ´ fq “ mppu ˚ δxj

´ uq ˚ fq, 1 ď j ď n.

There exists v P L1pGq with v ě 0 and }v}1 “ 1 such that›

›

›

›

›

˜

u ´

rÿ

i“1

αiδyi

¸

˚ pδxj˚ v ´ vq

›

›

›

›

›

1

ď ε

for j “ 1, . . . , n. Using these two inequalities, it follows for f P L8pGq with }f}1 ď 1and j “ 1, . . . , n,

|rmpLxjf ´ fq| “ |mppu ˚ δxj

´ uq ˚ f |


´ uq ˚ pδe ´ vq ˚ f}8


´ uq ˚ pδe ´ vq}1

ď ε ` |mppu ˚ δxj´ uq ˚ v ˚ fq|

ď 2ε `

ˇ

ˇ

ˇ

ˇ

ˇ

m

˜˜

rÿ

i“1

αiδyi

¸

˚ pδxj˚ v ´ vq ˚ f

¸ˇ

ˇ

ˇ

ˇ

ˇ

ď 2ε `

rÿ

i“1

αi|mpδyi˚ pδxj

˚ v ´ vq ˚ fq|

“ 2ε `

rÿ

i“1

αi|mppδyixjy´1i

´ δeq ˚ δyi˚ v ˚ fq|

ď 2ε `

rÿ

i“1

αidpm, yixjy´1i q.

Now, by Lemma 5.3.7, there exists m P M such that dpm, yixjy´1i q ď c for all

1 ď j ď n and 1 ď i ď r. Thus |rmpLxjf ´ fq| ď c ` 2ε for j “ 1, . . . , n, as

required. �

5.4. Completely bounded multipliers

Let G be a locally compact group. We have seen in Theorem 5.1.10 that everymultiplier of the Fourier algebra ApGq is given by an element of BpGq if and onlyif the group G is amenable. Of course, there is genuine interest in knowing howin the nonamenable case elements of MpApGqqzBpGq can arise. Note that theproofs of Nebbia and of Losert for discrete and nondiscrete groups, respectively,are highly nonconstructive. It turned out that there is a subalgebra of MpApGqq,the algebra of all so-called completely bounded multipliers, which is much easieraccessible than MpApGqq itself. In this section we study these completely boundedmultipliers, and in the subsequent section we show that they are closely related touniformly bounded representations of G in Hilbert spaces.

Definition 5.4.1. Let G be a locally compact group. A multiplier u ofApGq is called completely bounded if the associated σ-weakly continuous map Mu :


V NpGq Ñ V NpGq satisfying MupλGpxqq “ upxqλGpxq for all x P G is com-pletely bounded. The space of completely bounded multipliers of ApGq is denotedMcbpApGqq.

The proof of the following simple fact is left to the reader.

Lemma 5.4.2. The space McbpApGqq of completely bounded multipliers of ApGq

is a Banach space with the norm

}u}McbpApGqq “ }Mu}cb.

We now first present a theorem which identifies those elements of MpApGqq

which are completely bounded. This result will prove extremely useful in the sequel.

Theorem 5.4.3. Let G be a locally compact group and K “ SUp2q, and let fbe a multiplier of ApGq. Then the following conditions are equivalent.

(i) f is a completely bounded multiplier of ApGq.(ii) For every locally compact group H, f ˆ 1H is a multiplier of ApG ˆ Hq.(iii) f ˆ 1K is a multiplier of ApG ˆ Kq.

Moreover, if any one of these conditions is satisfied, then

}f}McbpApGqq “ }f ˆ 1K}MpApGˆKqq “ supH

}f ˆ 1H}MpApGˆHqq,

where the supremum is taken over all locally compact groups H.

Proof. (i) ñ (ii) Let f P McbpApGqq and let H be a closed subgroup ofG. Then Mf is completely bounded and hence there exists a σ-weakly continuous

operator ĂMf on V NpG ˆ Hq “ V NpGq pbV NpHq such that

ĂMf pT b Sq “ Mf pT q b S, T P V NpGq, S P V NpHq.

In particular, for x P G and y P H,

ĂMf pλGpxq b λHpyqq “ fpxqλGpxq b λHpyq.

By Proposition 5.1.2, (ii) ñ (i), this means that ĂMf P MpApG ˆ Hqq. Moreover,

}f ˆ 1H}MpApGˆHqq “ }ĂMf } ď }Mf }cb “ }f}McbpApGqq

(Lemma 5.4.2).The implication (ii) ñ (iii) being trivial, suppose that (iii) holds. It is well

known that for each n P N, the group K “ SUp2q has exactly one n-dimensionalirreducible representation. Therefore, V NpKq is isometrically isomorphic to thel8-direct sum ‘8

n“1MnpCq.Since f ˆ 1K P MpApG ˆ Kqq, Mfˆ1K is a σ-weakly continuous operator on

V NpGq pbV NpKq satisfying

Mfˆ1K pλGpxq b λKpyqq “ fpxqλGpxq b λKpyq

for all x P G and y P K. Since the sets tλGpxq : x P Gu and tλKpyq : y P Ku are totalin V NpGq and V NpKq), respectively, two consecutive approximation argumentsyield

Mfˆ1K pT b Sq “ Mf pT q b S, T P V NpGq, S P V NpKq.


By restricting Mfˆ1K to each of the components of the direct sum decomposition

V NpGq pbV NpKq “

8à

n“1

V NpGq b MnpCq,

and denoting by in the embedding of MnpCq into V NpKq, we recognize that }Mf b

in} ď }Mfˆ1K } for all n P N. This shows that Mf is completely bounded and alsothat }Mf }cb ď }Mfˆ1K }.

Concerning the last statement of the theorem, we have already seen that, forany locally compact group H, }f ˆ 1H}MpApGˆHqq ď }f}McbpApGqq. Combining thelast two inequalities, we get

supH

}f ˆ 1H}MpApGˆHqq ď }f}McbpApGqq “ }Mf }cb

ď }Mfˆ1K } “ }f ˆ 1K}MpApGˆKqq

ď supH

}f ˆ 1H}MpApGˆHqq.

This completes the proof of the theorem. �We have seen in Section 4.1 that MpApGqq is a dual Banach space. In passing

we apply Theorem 5.4.3 to show that the same is true of McbpApGqq.

Lemma 5.4.4. The unit ball of McbpApGqq is σpL8pGq, L1pGqq-closed in L8pGq.

Proof. Let pfαqα be a net in the unit ball of McbpApGqq which converges tosome f P L8pGq in the σpL8pGq, L1pGqq-topology, and let K “ SUp2q. Then, byTheorem 5.4.3, pfα ˆ 1Kqα is a net in the unit ball of MpApG ˆ Kqq converging tof ˆ 1K in the σpL8pGˆKq, L1pGˆKqq-topology. Now, by Lemma 5.1.4, the unitball of MpApG ˆ Kqq is closed in L8pG ˆ Kq in the σpL8pG ˆ Kq, L1pG ˆ Kqq-topology. Therefore we can assume that f ˆ 1K P MpApG ˆ Kqq, and this impliesthat f P McbpApGqq (Theorem 5.4.3). �

The proof of the following proposition uses the preceding lemma and is similarto the proof of Proposition 5.1.5.

Proposition 5.4.5. If E0 is the completion of L1pGq with respect to the norm

}f}E0“ sup

"ˇ

ˇ

ˇ

ˇ

ż

G

fpxqupxqdx

ˇ

ˇ

ˇ

ˇ

: u P McbpApGqq, }u}McbpApGqq ď 1

*

,

then E˚0 “ McbpApGqq. More precisely, every bounded linear functional ψ on E0 is

of the form

ψpfq “

ż

G


for some u P McbpApGqq, and then }φ} “ }u}McbpApGqq.

The next theorem provides the link between completely bounded multipliersand uniformly bounded representations which will be established in Section 5.5.

Theorem 5.4.6. Let G be a locally compact group and u a continuous functionon G. Then the following conditions are equivalent.

(i) u is a completely bounded multiplier of ApGq.(ii) There exist a Hilbert space H and bounded continuous maps ξ and η from

G into H such that, for all x, y P G,

upy´1xq “ xξpxq, ηpyqy.


Moreover, if these conditions are satisfied, then

}u}McbpApGqq “ inf t}ξ}8}η}8u,

where the infimum is taken over all such pairs pξ, ηq.

Proof. The proof of (i) ñ (ii) rests on a representation theorem for completelybounded maps on unital C˚-algebras [233, Theorem 7.4]. Given the completelybounded map Mu : V NpGq Ñ V NpGq Ď BpL2pGqq, there exist a Hilbert spaceH, a ˚-representation π : V NpGq Ñ BpHq and two bounded linear operators T1

and T2 from L2pGq to H such that MupSq “ T˚2 πpSqT1 for all S P V NpGq and

}u}McbpAGqq “ }Mu}cb “ }T1} ¨ }T2}. Let σ denote the unitary representation of G

associated to the ˚-representation π ˝ λG of L1pGq. We claim that

MupλGpxqq “ upxqλGpxq “ T˚2 σpxqT1

for all x P G. Fix x P G and let V denote the collection of all compact neigh-bourhoods of x in G, ordered by V1 ě V2 if V1 Ď V2. For each V P V , choose acontinuous nonnegative function fV with supp fV Ď V and

ş

GfV pxqdx “ 1. Then

the net pλGpfV qqV PV converges σ-strongly to λGpxq in V NpGq, and pπ˝λGpfV qqV PVconverges σ-strongly to σpxq in BpHq. Since MupλGpfV qq “ T˚

2 πpλGpfV qqT1 for ev-ery V and Mu is σ-weakly continuous, we conclude that MupλGpxqq “ T˚

2 σpxqT1.Now, fix any f0 P L2pGq with }f0}2 “ 1 and define maps ξ and η from G into

H by

ξpxq “ σpxqT1λGpx´1qf0 and ηpxq “ σpxqT2λGpx´1

qf0,

x P G. Then ξ and η are bounded and continuous, and for x, y P G, we have

xξpxq, ηpyqy “ xσpxqT1λGpx´1qf0, σpyqT2λGpy´1

qf0y

“ xT˚2 σpy´1xqT1λGpx´1

qf0, λGpy´1qf0y

“ upy´1xqxλGpy´1xqλGpx´1qf0, λGpy´1

qf0y

“ upy´1xq.

Moreover, }ξ}8}η}8 ď }T1} ¨ }T2} “ }u}McbpApGqq.(ii) ñ (i) We show that, given any v P ApGq, we have uv P ApGq and }uv}ApGq ď

}ξ}8}η}8. There exist f, g P L2pGq such that vpxq “ xλGpxqf, gy for all x P G and}v}ApGq “ }f}2}g}2. Choose an orthonormal basis peαqαPA ofH and, for each α P A,

define functions fα and gα in L2pGq by

fαpxq “ xξpx´1q, eαyfpxq and gαpxq “ xηpx´1

q, eαygpxq.

The seriesř

αPA }fα}22 andř

αPA }gα}22 converge. More precisely,

ÿ

αPA

}fα}22 “

ż

G

|fpxq|2

˜

ÿ

αPA

|xξpx´1, eαy|2

¸

dx

“

ż

G

|fpxq|2}ξpx´1

q}2dx

ď }ξ}28}f}

22,


and similarlyř

αPA }gα}22 ď }η}28}g}22. These inequalities implyÿ

αPA

}xλGp¨qfα, gαy}ApGq ď

ÿ

αPA

}fα}2}gα}2

ď

˜

ÿ

αPA

}fα}22

¸1{2˜ÿ

αPA

}gα}22

¸1{2

ď }ξ}8}η}8}f}2}g}2

“ }ξ}8}η}8}v}ApGq.

Now, convergence in ApGq being verified, we get for each x P G, by Parseval’sequation,

ÿ

αPA

xλGpxqfα, gαy “

ÿ

αPA

ż

G

fαpx´1yqgαpyq dy

“

ÿ

αPA

ż

G

xξpy´1xq, eαyfpx´1yqxηpy´1q, eαygpyq dy

“

ż

G

fpx´1yqgpyq

˜

ÿ

αPA

xξpy´1xq, eαyxeα, ηpy´1qy

¸

dy

“

ż

G

fpx´1yqgpyqxξpy´1xq, ηpy´1qy dy

“ upxqxλGpxqf, gy “ upxqvpxq.

Thus uv P ApGq and }uv}ApGq ď }ξ}8}η}8}v}ApGq and hence, by Theorem 5.4.3,}u}McbpAGqq ď }ξ}8}η}8.

Since we have seen in the proof of the implication (i) ñ (ii) that there exists arepresentation of u as in (ii) with }ξ}8}η}8 ď }u}McbpAGqq, it follows that

}u}McbpAGqq “ inf }ξ}8}η}8,

where the infimum is taken over all such pairs pξ, ηq. �Corollary 5.4.7. Let Gd be the group G with the discrete topology. Then

McbpApGqq “ McbpApGdqq X CpGq.

Proof. Let j : Gd Ñ G be the identity mapping. If u P McbpApGqq and ξ, ηare as in (ii) of Theorem 5.4.6, then

pu ˝ jqpy´1xq “ xξ ˝ jpxq, η ˝ jpyqy

for all x, y P Gd, and hence the implication (ii) ñ (i) of Theorem 5.4.6 shows thatu ˝ j P McbpApGdqq.

Conversely, let u P McbpApGdqqXCpGq and, as in Theorem 5.4.3, letK “ SOp2q

and f “ u ˆ 1K . Then f is continuous on G ˆ K and a multiplier of ApGd ˆ Kq.It follows that for any v P ApG ˆ Kq,

vf P ApGd ˆ Kq X CpG ˆ Kq “ ApG ˆ Kq.

Consequently, f is a multiplier of ApG ˆ Kq and hence u P McbApGqq by Theorem5.4.3. �

The preceding corollary is of course reminiscent of the fact that for any locallycompact group G, BpGq “ BpGdq X CpGq. The next corollary is the analogue forcompletely bounded multipliers of Lemma 5.1.1.


Corollary 5.4.8. Let H be a closed subgroup of G and let u P McbpApGqq.Then u|H P McbpApHqq and

}u|H}McbpApHqq ď }u}McbpApGqq.

Proof. Simply apply Lemma 5.1.1 and use the equivalence of (i) and (ii) inTheorem 5.4.6. �

We conclude this section with two results about completely positive multipliers.

Proposition 5.4.9. Let u be a continuous function on a locally compact groupG. Then the following conditions are equivalent.

(i) u is a completely positive multiplier of ApGq.(ii) u is a positive definite function.

Proof. (i) ñ (ii) Let n P N, x1, . . . , xn P G and α1, . . . , αn P C. Choose a unitvector ξ P L2pGq and put ξj “ αjλGpx´1

j qξ, 1 ď j ď n. Then, since by hypothesis

Mu “ m˚u is n-positive,

nÿ

i,j“1

αjαiupx´1j xiq “

nÿ

i,j“1

upx´1j xiqxαiξ, αjξy

“

nÿ

i,j“1

upx´1j xiqxλGpx´1

j xiqξi, ξjy

“

nÿ

i,j“1

xMupλGpxjq˚λGpxiqqξi, ξjy ě 0.

(ii) ñ (i) Since u P P pGq, u is a multiplier of ApGq. To prove that Mu iscompletely positive, it is sufficient to show that for any finitely many T1, . . . , Tn P

V NpGq and ξ1, . . . , ξn P L2pGq,

nÿ

i,j“1

xMupT˚j Tiqξi, ξjy ě 0.

Since the set tλGpxq : x P Gu spans a σ-strongly dense subset of V NpGq, it isenough to consider Ti P spanpλGpGqq. Then we find x1, . . . , xr P G such that eachTi, 1 ď i ď n, is of the form Ti “

řrk“1 αikλGpxkq, αik P C, 1 ď k ď r. Let

ηk “ λGpxkq

˜

nÿ

i“1

αikξi

¸

P L2pGq, 1 ď k ď r.

Then, for 1 ď i, j ď n,

xMupT˚j Tiqξi, ξjy “

rÿ

k,l“1

αjlαikupx´1l xkqxλGpx´1

l xkqξi, ξjy

and hence, by definition of ηk,

nÿ

i,j“1

xMupT˚j Tiqξi, ξjy “

rÿ

k,l“1

upx´1l xkqxηk, ηly.


Since the pr ˆ rq-matrices A “ pupx´1l xkqq1ďk,lďr and B “ pxηk, ηlyq1ďk,lďr are

positive definite, it follows thatrÿ

k,l“1

upx´1l xkqxηk, ηly “ tracepABq ě 0.

This proves (i). �Proposition 5.4.10. Let G be an amenable locally compact group and let u be

a positive multiplier of ApGq. Then u is completely positive.

Proof. Since Mu is positive, we have xMupT˚ ˚T qξ, ξy ě 0 for all T P V NpGq

and ξ P L2pGq. Thus, taking T “ λGpfq and ξ “ g, where f, g P CcpGq,ż

G

upxqpf˚˚ fqpxqpg ˚ rgqpx´1

qdx “

ż

G

upxqpf˚˚ fqpxqxλGpxqg, gydx

“

ż

G

pf˚˚ fqpxqxMupλGpxqqg, gydx

“ xMupλGpf˚˚ fqqg, gy,

which is ě 0. Since G is amenable, there exists a net pgαqα in CcpGq such thatpgα ˚ rgαqpyq Ñ 1 uniformly on compact subsets of G. It follows that

ż

G

upxqpf˚˚ fqpxqdx ě 0

for all f P CcpGq, and this shows that u is positive definite. �Corollary 5.4.11. Let G and H be locally compact groups.

(i) BpGq is contained in McbpApGqq and for any u P BpGq,

}u}McbpApGqq ď }u}BpGq.

(ii) If G is amenable, then

MpApGqq “ McbpApGqq “ BpGq,

and the three corresponding norms agree.(iii) If u P McbpApGqq and v P McbpApHq, then u ˆ v P McbpApG ˆ Hqq.

Proof. (i) Let K “ SUp2q and choose a representation π of G such that u is acoordinate function of π. It is clear that uˆ 1K P BpGˆKq and, identifying Hpπq

withHpπˆ1Kq in the usual manner, it is easily seen that }uˆ1K}BpGˆKq “ }u}BpGq.Because ApG ˆ Kq is an ideal in BpG ˆ Kq,

}u ˆ 1K}MpApGˆKqq ď }u ˆ 1K}BpGˆKq.

Now (i) follows from Theorem 5.4.3.(ii) If G is amenable, then MpApGqq “ BpGq with the same norm (Theorem

5.1.8). So (ii) follows from (i).(iii) Again, let K “ SUp2q. By Theorem 5.4.3, u ˆ 1H ˆ 1K and 1G ˆ v ˆ 1K

both belong to MpApG ˆ H ˆ Kqq, and hence so does their product u ˆ v ˆ 1K .Using Theorem 5.4.3, (iii) ñ (i), we conclude that u ˆ v P McbpApG ˆ Hqq. �

Remark 5.4.12. Let G be a nonamenable locally compact group and supposethat ApGq has an approximate identity which is bounded in the McbpApGqq-norm.Then McbpApGqqzBpGq ‰ H. In fact, if puαqα is such an approximate identity,then p}uα}ApGqqα is unbounded because G is nonamenable. It follows that the two


norms } ¨ }McbpApGqq and } ¨ }BpGq on BpGq are not equivalent and therefore BpGq

cannot be closed in McbpApGqq.

Corollary 5.4.13. Let G be a locally compact group. For a subset E of G,the following are equivalent.

(i) 1E P BpGq with }1E}BpGq “ 1.(ii) 1E P McbpApGqq with }1E}McbpApGqq “ 1.(iii) E is a coset of an open subgroup of G.

Proof. (ii) ñ (iii) Clearly, E is open in G. Replacing E by some translateif necessary, we can assume that e P E. We are going to show that then E is asubgroup of G. By Theorem 5.4.6, there exist a Hilbert space H and boundedcontinuous maps ξ, η : G Ñ H such that 1 “ }1E}McbpApGqq “ }ξ}8}η}8 and

1Epy´1xq “ xξpxq, ηpyqy, x, y P G.

Obviously, we can further assume that }ξ}8 “ }η}8 “ 1. The Cauchy-Schwarzinequality implies that, for all x, y P G, y´1x P E if and only if ηpyq “ ξpxq. Inparticular, as e P E, ξpeq “ ηpeq. Thus, with ξ “ ξpeq,

E “ tx P G : ξpxq “ ξu “ ty P G : ηpy´1q “ ξu.

Hence, if x, y P E, then

1Epyxq “ xξpxq, ηpy´1qy “ xξ, ξy “ 1,

so that yx P E. This shows that E is a subsemigroup of G. Finally, x P E impliesx´1 P E. Indeed, if x P E then e P x´1E and }1x´1E}McbpApGqq “ 1. Therefore,

by the foregoing, x´1E is a subsemigroup of G and hence x´1 “ xpx´1x´1q P

xpx´1Eq “ E. Thus E is a subgroup of G. �

5.5. Uniformly bounded representations and multipliers

In this section we investigate a class of representations of locally compactgroups in Hilbert spaces, the coefficient functions of which turn out to be com-pletely bounded multipliers. Moreover, if such a representation is not similar to aunitary representation, then its coefficient functions provide multipliers which arenot in BpGq.

Definition 5.5.1. A strongly continuous representation π of a locally compactgroup G in a Hilbert space is called uniformly bounded if

}π} :“ supxPG

}πpxq} ă 8.

Two uniformly bounded representations π1 in Hpπ1q and π2 in Hpπ2q are said to besimilar if there exists a bounded linear operator T : Hpπ1q Ñ Hpπ2q with boundedinverse T´1 : Hpπ2q Ñ Hpπ1q such that, for all x P G,

π2pxq “ Tπ1pxqT´1.

We start with a result which the reader might expect in view of what we havesaid above and the fact that MpApGqq “ BpGq when G is amenable.

Theorem 5.5.2. Let G be an amenable locally compact group. Then everyuniformly bounded representation of G is similar to a unitary representation.

5.5. UNIFORMLY BOUNDED REPRESENTATIONS AND MULTIPLIERS 187

Proof. Let π be a uniformly bounded representation of G on the Hilbert spaceHpπq. For ξ, η P Hpπq, define fξ,η : G Ñ C by

fξ,ηpxq “ xπpx´1qξ, πpx´1

qηy, x P G.

For x, y P G, we then have

|fξ,ηpxq ´ fξ,ηpyq| “ |xπpx´1qξ, πpx´1

qηy ´ xπpy´1qξ, πpy´1

qηy|

ď |xπpx´1qξ, πpx´1

qη ´ πpy´1qηy|

`|xπpx´1qξ ´ πpy´1

qξ, πpy´1qηy|

ď }π} ¨ }ξ} ¨ }πpx´1qη ´ πpy´1

qη}

`}πpx´1qξ ´ πpy´1

qξ} ¨ }π} ¨ }η}.

Since π is strongly continuous, this shows that fξ,η is a bounded continuous function.Let m be a left invariant mean on CbpGq and define β : Hpπq ˆ Hpπq Ñ C by

βpξ, ηq “ xm, fξ,ηy, ξ, η P Hpπq.

Clearly, β is a positive semidefinite sesquilinear form on Hpπq. Let |ξ|0 “ βpξ, ξq1{2,ξ P Hpπq. Then |ξ|0 ď }π} ¨ }ξ} and conversely

}ξ}2

ď }πpxq}2fξ,ξpxq ď }π}

2fξ,ξpxq

for all x P G, whence }ξ} ď }π} ¨ |ξ|0. Thus | ¨ |0 and } ¨ } are equivalent norms. Sinceβ is bounded, there exists S P BpHpπqq such that

βpξ, ηq “ xSpξq, ηy, ξ, η P Hpπq.

Because β is positive definite, S is a positive operator. Let T “ S1{2, then Tis invertible. In fact, the equivalence of the norms } ¨ } and | ¨ |0 implies thatT pHpπqq is closed in Hpπq and that T´1 : T pHpπqq Ñ Hpπq is bounded. Moreover,if η P Hpπq is such that η K T pHpπqq, then 0 “ xη, T pT pηqqy “ }T pηq}2, so thatη “ 0. Finally, the representation x Ñ τ pxq “ T ˝ πpxq ˝ T´1 is unitary. Indeed,since fπpxqξ1,πpxqη1 “ δx ˚ fξ1,η1 for all ξ1, η1 P Hpπq and x P G, we have

xτ pxqξ, τ pxqηy “ βpπpxqT´1ξ, πpxqT´1ηq

“ xm, fπpxqT´1ξ,πpxqT´1ηy

“ xm, δx ˚ fT´1ξ,T´1ηy

“ xm, fT´1ξ,T´1ηy

“ βpT´1ξ, T´1ηq

“ xξ, ηy

for all x P G and ξ, η P Hpπq. �

Lemma 5.5.3. Let G be a locally compact group and π a strongly continuousuniformly bounded representation of G on the Hilbert space Hpπq. Then the rep-resentation π b λG of G on Hpπq bL2pGq is similar to the unitary representationx Ñ I b λGpxq, where I denotes the identity operator on Hpπq. More precisely,there exists an invertible operator T P BpHpπq bL2pGqq such that

pπ b λGqpxq “ T ppI b λpxqqqT´1, x P G,

and }T } “ }T´1} “ }π}.


Proof. The Hilbert spaces Hpπq b L2pGq and L2pG,Hpπqq are isometricallyisomorphic. Define T P BpL2pG,Hpπqq by pTξqpxq “ πpxqpξpxqq for ξ P L2pG,Hpπqq

and x P G. Since

}Tξ}2

“

ż

G

}πpxqpξpxqq}22 ď }π} ¨ }ξ}

22

for ξ P L2pG,Hpπqq, we have }T } ď }π}. Conversely, given ε ą 0, choose ξ0 P Hpπq

with }ξ0} “ 1, x0 P G and a relatively compact neighbourhood V of x0 such that}πpxqξ0} ě }π} ´ ε for all x P V and define ξ by ξpxq “ |V |´11V pxqξ0. It thenfollows that

}T pξq}2

“ |V |´1

ż

V

}πpxqξ0}22dx ě p}π} ´ εq2.

Since ε ą 0 is arbitrary, }T } ě }π}. The inverse of T is given by T´1pξqpxq “

πpx´1qpξpxqq, and then an analogous argument shows that }T´1} “ }π}. We nowhave, for ξ P L2pG,Hpπqq and x, y P G,

T pI b λGpxqqT´1pξqpyq “ πpyqpT´1

pξqpx´1yqq

“ πpyqpπpy´1xqpξpx´1yqq “ πpxqpξpx´1yqq

“ rpπpxq b λGpxqqpξqspyq.


Employing the preceding lemma and Theorem 5.4.6, we are now able to provethe first main result of this section.

Theorem 5.5.4. Let G be a locally compact group and π a strongly continuousuniformly bounded representation of G on the Hilbert space Hpπq. Then for ξ, η P

Hpπq, the coefficient function upxq “ xπpxqξ, ηy, x P G, is a completely boundedmultiplier of ApGq and

}u}McbpApGqq ď }π}2

¨ }ξ} ¨ }η}.

Proof. Let v P ApGq and vpxq “ vpx´1q, x P G. There exist f, g P L2pGq

such that

vpxq “ xλGpxqf, gy “ vpx´1q “ pf ˚ rgqpx´1

q,

where, as usual, rgpyq “ gpy´1q for y P G, and

}v}ApGq “ }f}2}g}2 “ }u}ApGq.

Let T P BpHpπq bL2pGqq be as in Lemma 5.5.3 and put rξ “ T´1pξ b fq andrη “ T˚pη b gq. Then, for all x P G,

upxqvpxq “ xpπpxq b λGpxqqpξ b fq, η b gy

“ xT pI b λGpxqqT´1pξ b fq, η b gy

“ xpI b λGpxqqrξ, rηy.

Thus uv is a coefficient function of a unitary representation. Therefore uv P BpGq

and

}uv}BpGq ď }rξ} ¨ }rη} ď }T´1} ¨ }ξ b f} ¨ }T˚

} ¨ }η b g}

ď }π}2

¨ }ξ} ¨ }η} ¨ }f}2 ¨ }g}2

“ }π}2

¨ }ξ} ¨ }η} ¨ }v}ApGq.

5.5. UNIFORMLY BOUNDED REPRESENTATIONS AND MULTIPLIERS 189

This shows that v Ñ uv is a bounded linear map from ApGq into BpGq. SinceupApGq X CcpGqq Ď BpGq X CcpGq Ď ApGq, it follows that even uApGq Ď ApGq.Consequently, u P MpApGqq and }u}MpApGqq ď }π}2}ξ} ¨ }η}.

Let now H be an arbitrary locally compact group and define a representationrπ of G ˆ H in Hpπq by rπpx, yq “ πpxq, x P G, y P H. Then

xrπpx, yqξ, ηy “ pu ˆ 1Hqpx, yq,

and applying the first part of the proof with G replaced by G ˆ H and π by rπ, weobtain that u ˆ 1H P MpApG ˆ Hqq and

}u ˆ 1H}MpApGˆHqq ď }π}2}ξ} ¨ }η}.

Theorem 5.4.3 and Theorem 5.4.6 yield that u P McbpApGqq and }u}McbpApGqq ď

}π}2 ¨ }ξ} ¨ }η}. �

Now the question arises whether Theorem 5.5.4 gives us with multipliers whichare not in BpGq. The next theorem provides an affirmative answer, at least whenthe representation π is cyclic. Recall that a representation π : G Ñ BpHpπqq iscalled cyclic if there exists a vector ξ P Hpπq such that the span of tπpxqξ : x P Gu

is dense in Hpπq. Unfortunately, the proof of the following theorem builds onheavy machinery, namely the positive solution to the similarity problem for cyclicrepresentations of C˚-algebras (see [112]).

Theorem 5.5.5. Let π be a strongly continuous uniformly bounded representa-tion of a locally compact group G on a Hilbert space Hpπq. Moreover, assume thatπ is cyclic. Then the following conditions are equivalent.

(i) All the coefficients of π are contained in BpGq.(ii) π is similar to a unitary representation.

Proof. (ii) ñ (i) is trivial because the sets of coefficient functions of twosimilar representations coincide.

Conversely, suppose that (i) holds and consider the sesquilinear map s : Hpπqˆ

Hpπq Ñ BpGq defined by

spξ, ηqpxq “ xπpxqξ, ηy, ξ, η P Hpπq.

We first show that s is separately continuous. For fixed η P Hpπq, the map ξ Ñ

spξ, ηq from Hpπq into BpHpπqq has a closed graph. Indeed, if pξnqn is a sequencein Hpπq such that

}ξn ´ ξ} Ñ 0 and }spξn, ηq ´ u}BpGq Ñ 0

for some ξ P Hpπq and u P BpGq, then

xπpxqξ, ηy “ limnÑ8

xπpxqξn, ηy “ limnÑ8

spξn, ηqpxq “ upxq

for all x P G. Thus spξ, ηq “ u and hence s is continuous in the first variable by theclosed graph theorem. The same argument applies to the second variable. Now,being separately continuous, s is a bounded bilinear map.

Since π is strongly continuous and uniformly bounded, we can associate to eachf P L1pGq an operator

rπpfq “

ż

G

fpxqπpxqdx P BpHpπqq,

where the integral converges in the strong operator topology.


It is easy to verify that rπ is a bounded representation of the convolution algebraL1pGq and that }rπ} “ }π}. Since

xrπpfqξ, ηy “

ż

G

fpxqxπpxqξ, ηydx

for all ξ, η P Hpπq and

}xπp¨qξ, ηy}BpGq ď }s} ¨ }ξ} ¨ }η},

it follows from the duality between C˚pGq and BpGq that

}rπpfq} ď }s} ¨ }f}C˚pGq, f P L1pGq.

Thus rπ extends to a bounded representation, also denoted rπ, of C˚pGq on Hpπq.Since π is cyclic, to is rπ. Therefore, by the solution to the similarity problem [112],there exists an invertible operator T P BpHpπqq such that

f Ñ rρpfq “ T rπpfqT´1

is a ˚-representation of C˚pGq. Now, let

ρpxq “ TπpxqT´1, x P G.

Then ρ is a strongly continuous representation of G and

rρpfq “

ż

G

fpxqρpxqdx, f P L1pGq.

Since }rρpfq} ď }f}C˚pGq ď }f}1 for all f P L1pGq, it follows that }ρpxq} ď 1 for allx P G. Indeed, if ξ, η P Hpπq are of norm one, then f Ñ xrρpfqξ, ηy is a boundedlinear functional of norm at most one and hence

ż

G

fpxqxρpxqξ, ηydx “

ż

G

fpxqϕpxqdx

for some ϕ P L8pGq with }ϕ}8 ď 1, which in turn implies that |xρpxqξ, ηy| ď 1 forall x P G. Consequently, ρ is a unitary representation. This completes the proof of(i) ñ (ii). �

It follows from Theorems 5.5.4 and 5.5.5 that ifG possesses a uniformly boundedcyclic representation which is not similar to a unitary representation, then

McbpApGqqzBpGq ‰ H.

Actually, it is sufficient that some quotient group of G admits such a representation.

Corollary 5.5.6. Let G be a locally compact group and N a closed normalsubgroup of G. If G{N admits a uniformly bounded cyclic representation which isnot similar to a unitary representation, then McbpApGqqzBpGq ‰ H.

Proof. Let ρ be a uniformly bounded representation of G{N which is notsimilar to a unitary representation and let π “ ρ ˝ q, where q : G Ñ G{N is thequotient homomorphism. Then π is a uniformly bounded representation which isnot similar to a unitary representation, as required. �

5.6. MULTIPLIER BOUNDED APPROXIMATE IDENTITIES IN ApGq 191

5.6. Multiplier bounded approximate identities in ApGq

We remind the reader that the Fourier algebra ApGq of a locally compact groupG has a bounded approximate identity if and only if G is amenable (Theorem 2.7.2).It is an extremely challenging and difficult problem to identify those groups G forwhich ApGq admits an approximate identity which is bounded in the cb-multipliernorm.

Suppose that H is a closed subgroup of G and that puαqα is an approximateidentity for ApGq such that }uα}McbpApGqq ď c ă 8 for all α. Then }uα|H}McbpApHqq

ď c and since ApHq “ ApGq|H , it follows that puα|Hqα is an approximate identityfor ApHq bounded in the cb-multiplier norm. In this section we are going to showthat the converse is true at least when H is a lattice in G (and G is second count-able). This result will be applied in Section 5.7 in case G “ SLp2,Rq and H “ F2,the free group on two generators.

Lemma 5.6.1. Let G be a locally compact group and let c ě 1. Then thefollowing three conditions are equivalent.

(i) There exists a net pwαqα in ApGq such that supα }wα}McbpApGqq ď c and

wα Ñ 1G in the σpL8, L1q-topology.(ii) There exists a net pvαqα in ApGq such that supα }vα}McbpApGqq ď c and

vα Ñ 1 uniformly on compact subsets of G.(iii) There exists an approximate identity puαqα in ApGq such that, for all α,

}uα}McbpApGqq ď c.

Proof. (i) ñ (ii) Let the net pwαqα satisfy (i) and choose f P CcpGq withf ě 0 and }f}1 “ 1. For each α, put

vαpxq “ pf ˚ wαqpxq “

ż

G

fpxyqwαpy´1qdy, x P G.

Let K Ď G be compact. Then the functions Lx´1f, x P K, form a compact subsetof L1pGq. Since wα Ñ 1G in the σpL8, L1q-topology and

supα

}wα}8 ď supα

}wα}McbpApGqq ď c,

the convergence is uniform on compact subsets of L1pGq. Thus

vαpxq “ xwα, Lx´1fy Ñ x1G, Lx´1fy “ pf ˚ 1Gqpxq “ 1

uniformly on K. Moreover, since f P CcpGq, f ě 0 and }f}1 “ 1, vα is containedin the σpL8, L1q-closed convex hull of left translates of wα. Because the unit ballof McbpApGqq is σpL8pGq, L1pGqq-closed (Lemma 5.4.4), it follows that

}vα}McbpApGqq ď }wα}McbpApGqq ď c.

So the net pvαqα satisfies the conditions in (ii).(ii) ñ (iii) Suppose that pvαqα is a net as in (ii). Again, choose f P CcpGq with

f ě 0 and }f}1 “ 1 and put uα “ f ˚ vα. Then, as in the proof of (i) ñ (ii), wehave }uα}McbpApGqq ď c for all α. Let w P ApGq X CcpGq and set

K “ supp f, L “ suppw and hα “ vα1K´1L.

Then, for each x P L,

uαpxq “

ż

K

fpyqvαpy´1xqdy “ pf ˚ hαqpxq,


and similarly, for x P L,

pf ˚ 1Gqpxq “ pf ˚ 1K´1Lqpxq.

By hypothesis, vα Ñ 1 uniformly on the compact sets and hence hα Ñ 1 uniformlyon the compact set K´1L. Since hα vanishes outside of K´1L, it follows thathα Ñ 1K´1L in L2pGq. This implies that f ˚ hα Ñ f ˚ 1K´1L in ApGq and, sincew P ApGq, this in turn yields that

}pf ˚ hαqw ´ pf ˚ 1K´1Lqw}ApGq Ñ 0.

By the above equations and since L “ suppw, we have

uαw “ pf ˚ vαqw Ñ pf ˚ 1Gqw “ w

in ApGq. Finally, since ApGq X CcpGq is dense in ApGq and

supα

}uα}MpApGqq ď supα

}uα}McbpApGqq ă 8,

a simple triangle inequality argument shows that }uαw ´ w}ApGq Ñ 0 for everyw P ApGq.

Conversely, (iii) ñ (ii) ñ (i). Indeed, (iii) ñ (ii) since the ApGq-norm dom-inates the L8-norm and since, given any compact subset K of G, there existsu P ApGq with u|K “ 1. Clearly, (ii) ñ (i) because CcpGq is dense in L1pGq. �

Corollary 5.6.2. Let G be a locally compact group and let c ě 1. Supposethat G satisfies the equivalent conditions of Lemma 5.6.1. Then there exists a netpuβqβ in ApGq X CcpGq with the following properties.

(1) }uβ}McbpApGqq ď c for all β.(2) }uβu ´ u}ApGq Ñ 0 for every u P ApGq.(3) uβ Ñ 1 uniformly on compact subsets of G.

Proof. Let pvαqα be a net in ApGq satisfying condition (ii) in Lemma 5.6.1.Choose f P CcpGq with f ě 0 and }f}1 “ 1, and for each α let uα “ f ˚ Ăvα. Thenin the proof of the implication (ii) ñ (iii) of Lemma 5.6.1 the net puαqα was shownto be an approximate identity for ApGq such that }uα}McbpApGqq ď c for all α.

LetK be a compact subset ofG containing the identity and let C “ K´1 supp f .Then, for each x P K, fpxyq “ 0 for all y P GzC and hence

|uαpxq ´ 1| “

ˇ

ˇ

ˇ

ˇ

ż

C

fpxyqpvαpx´1q ´ 1qdy

ˇ

ˇ

ˇ

ˇ

ď supyPC

|vαpy´1q ´ 1| ¨

ż

C

fpxyqdy

ď supyPC

|vαpy´1q ´ 1|.

Since, by condition (ii), vα Ñ 1 uniformly on the compact set C´1, it follows thatuα Ñ 1 uniformly on K.

Thus the net puαqα has the properties (1), (2) and (3), but uα may not havecompact support. Therefore, for each α and n P N, choose uα,n P ApGq X CcpGq

such that }uα,n ´ uα}ApGq ď 1{n. Then

}uα,n}McbpApGqq ď }uα,n ´ uα}McbpApGqq ` }uα}McbpApGqq

ď }uα,n ´ uα}ApGq ` }uα}McbpApGqq

ď c ` 1{n.

5.6. MULTIPLIER BOUNDED APPROXIMATE IDENTITIES IN ApGq 193

For each α and n, we now define

vα,n “c

c ` 1{nuα,n P ApGq X CcpGq.

It is straightforward to check that if the set of all β “ pα, nq is given the productordering, then the net pvβqβ in ApGq X CcpGq has properties (1), (2) and (3). �

A lattice Γ in a locally compact group G is a closed discrete subgroup for whichG{Γ admits a finite G-invariant measure. A locally compact group G which containsa lattice is necessarily unimodular. In the following, G is a second countable locallycompact group and Γ is a lattice in G. Then Γ is countable and there exists a Borelcross-section for the left cosets of Γ in G, that is, a Borel subset S of G with theproperty that each x P G has a unique decomposition x “ sa with s P S and a P Γ. Equip Γ with counting measure and let ν be a left invariant measure on G{Γ suchthat νpG{Γq “ 1. Then

ż

G

fpyqdμpyq “

ż

G{Γ

˜

ÿ

aPΓ

fpxaq

¸

dνpxΓq

defines a Haar measure on G and μpSq “ νpG{Γq “ 1. For every bounded function

f on Γ, define rf on G byrf “ 1S ˚ f ˚ 1S .

Then rf is a bounded continuous function on G since 1S P L1pGq, G is unimodularand 1S ˚ f is the bounded function on G given by

p1S ˚ fqpsaq “ fpaq, s P S, a P Γ.

We now show that if f P ApΓq (respectively, f P McbpApΓqq), then rf has thecorresponding property. The cb-statement builds on Theorem 5.4.6.

Proposition 5.6.3. Let f be a bounded function on Γ and let rf be as above.

(i) If f P ApΓq, then rf P ApGq and

} rf}ApGq ď }f}ApΓq.

(ii) If f P McbpApΓqq, then rf P McbpApGqq and

} rf}McbpApGqq ď }f}McbpApΓqq.

Proof. (i) There exist g, h P l2pΓq such that

f “ g ˚ h and }f}ApΓq “ }g}2}h}2.

Let g1 “ 1S ˚ g and h1 “ 1S ˚ h. Then g1, h1 P L2pGq and

rf “ 1S ˚ f ˚ 1S “ p1s ˚ gq ˚ ph ˚ 1Sq “ g1 ˚ h1 P ApGq.

Moreover,

} rf}ApGq ď }g1}2}h1}2 “ }g}2}h}2 “ }f}ApΓq.

This proves (i).(ii) Every y P G has a unique decomposition y “ σpyqγpyq with σpyq P S and

γpyq P Γ. For each x P G, define σx : S Ñ S by σxpsq “ σpxsq, s P S, and letlx : G{Γ Ñ G{Γ be the left translation yΓ Ñ xyΓ. Then, with q : S Ñ G{Γ themap s Ñ sΓ,

qpσxpsqq “ qpxsγpxyq´1

q “ xsΓ “ lxpqpsqq


for all s P S. Since q and lx are Borel isomorphisms, so is σx : S Ñ S. Since themeasure ν on G{Γ is left translation invariant, the above formula implies that μ|S

is σx-invariant for every x P G.

We next rewrite the function rf “ 1S ˚ f ˚ 1S in a suitable way. Since

p1S ˚ fqpsaq “ fpaq, a P Γ, s P S,

we get, for each x P G,

p1S ˚ fqpxq “ p1S ˚ fqpσpxqγpxqq “ fpγpxqq

and therefore

rfpxq “

ż

G

p1S ˚ fqpyq1Spx´1yqdμpyq “

ż

xS

fpγpyqqdμpyq

“

ż

S

fpγpxsqqdμpsq.

Now, since xs “ σxpsqγpxsq and ys “ σypsqγpysq for x, y P G and s P S, we have

yx´1σxpsq “ yspxsq´1σxpsq “ σypsqγpysqγpxsq

´1.

As σxpsq P S and γpysqγpxsq´1 P Γ, this means that γpyx´1σxpsqq “ γpysqγpxsq´1

and therefore, since μ|S is σx-invariant,

rfpyx´1q “

ż

S

fpγpyx´1sqqdμpsq

“

ż

S

fpγpyx´1σxpsqqdμpsq

“

ż

S

fpγpysqγpxsq´1

qdμpsq

for all x, y P G.Since f P McbpApΓqq, by Theorem 5.4.6 there exist a Hilbert space H and

bounded maps ξ, η : Γ Ñ H such that

}f}McbpApΓqq “ }ξ}8}η}8 and fpb´1aq “ xξpaq, ηpbqy

for all a, b P Γ. Define rξ, rη : G Ñ L2pS,H, μ|Sq by

rξpxqpsq “ ξpγpx´1sq´1

q and rηpxqpsq “ ηpγpx´1sq´1

q,

x P G, s P S. Then rξ and rη are bounded Borel measurable maps from G into Hsatisfying

supxPG

}rξpxq}L2 ď }ξ}8 and supxPG

}rηpxq}L2 ď }η}8.

Here we have used that μpSq “ 1. Moreover, the above formula for rf gives

rfpy´1xq “

ż

S

fpγpy´1sqγpx´1sq´1

qdμpsq

“

ż

S

xξpγpx´1sq´1

q, ηpγpy´1sq´1

qydμpsq

“

ż

S

xrξpxqpsq, rηpxqpsqydμpsq

“ xrξpxq, rηpyqy

5.7. EXAMPLES: FREE GROUPS AND SLp2,Rq 195

for all x, y P G. Since rf is continuous, Theorem 5.4.6, (iii) ñ (i) implies thatrf P McbpApGqq and

} rf}McbpApGqq ď }rξ}8}rη}8 ď }ξ}8}η}8 “ }f}McbpApΓqq.

This finishes the proof of (ii) �With the preceding proposition at hand, we can now deduce the main result of

this section.

Theorem 5.6.4. Let Γ be a lattice in a second countable locally compact groupG and let c ě 1. Then the following are equivalent.

(i) ApGq has an approximate identity puαqα such that }uα}McbpApGqq ď c forall α.

(ii) ApΓq has an approximate identity pvαqα such that }vα}McbpApΓqq ď c forall α.

Proof. As mentioned at the outset of this section, (i) ñ (ii) holds for anyclosed subgroup of an arbitrary locally compact group.

(ii) ñ (i) Let pvαqα Ď ApΓq be as in (ii), and as before let S be a Borelcross-section for the left cosets of Γ in G. For each α, let

uα “ 1S ˚ vα ˚ 1S P ApGq.

Since Γ is discrete and vα Ñ 1 pointwise on Γ, also vα Ñ 1 in the σpl8pΓq, l1pΓqq-topology.

To each g P L1pGq, we now associate the function T pgq on Γ defined by

T pgqptq “

ż

G

ż

S

gpxq1Spxyt´1qdydx, t P Γ.

Thenř

tPΓ |T pgqptq| ď }g}1 and hence T : g Ñ T pgq is a bounded linear map fromL1pGq into �1pΓq. By the definition of T ,

x1S ˚ f ˚ 1S , gy “ xf, T pgqy

for all f P �8pΓq and g P L1pGq. Thus 1S ˚f ˚ 1S “ T˚pfq, that is, f Ñ 1S ˚f ˚ 1S isthe adjoint of the map T . It follows that uα Ñ 1G in the σpL8pGq, L1pGqq-topology.Moreover, by Proposition 5.6.3(ii), we have

}uα}McbpApGqq ď }vα}McbpApGqq ď c

for all α. Now (i) follows from Lemma 5.6.1, (i) ñ (iii). �

5.7. Examples: Free groups and SLp2,Rq

Let G be a free group on finitely many generators a1, . . . , aN . Any element xof G has a unique representation as a finite product of ai and a´1

j , i, j “ 1, . . . , N ,

which does not contain two adjacent inverse factors, aia´1i or a´1

j aj . This repre-sentation is called the word of x. The number of factors in the word is called thelength of x and denoted |x|. For x, y P G, the word of xy can be found as follows.Take the words of x and y and delete the maximal number of products of the formaia

´1i of a´1

i ai. If this number of products is k, then the word for xy starts withthe first |x| ´ k factors in the word of x followed by the last |y| ´ k factors in theword of y.

For m P N0, let Em “ tx P G : |x| “ mu and let 1Emdenote the characteristic

function of Em. Note that the sets Em are finite. Actually, Em “ teu and it is


not difficult to show that the number of elements in Em equals 2Np2N ´ 1qm´1 form ě 1.

Lemma 5.7.1. Let G be a free group on finitely many generators and let α ą 0.Then the function x Ñ e´α|x| on G is positive definite.

Proof. By Schoenberg’s theorem (Section 1.2), it suffices to show that thefunction x Ñ |x| on G is negative definite, that is, for any finitely many x1, . . . , xn P

G and c1, . . . , cn P C withřn

j“1 cj “ 0, we have

nÿ

i,j“1

cicj |x´1i xj | ď 0.

Let A “ ta1, . . . , aNu be a set of free generators of G and let

M “ tpx, yq P G ˆ G : x´1y P Au

and

M “ tpx, yq P G ˆ G : py, xq P Mu “ tpx, yq P G ˆ G : x´1y P A´1u.

Let H be a Hilbert space with orthonormal basis tvpx,yq : px, yq P Mu, and for

px, yq P M put vpx,yq “ ´vpy,xq.Now consider an arbitrary element x of G and let x “ x1 ¨ . . . ¨ xn be the word

of x, that is, n “ |x| and xj P A Y A´1 for each j. Put y0 “ e and yk “ x1 ¨ . . . ¨ xk

for 1 ď k ď n. Then y´1k´1yk “ xk and hence pyk´1, ykq P M Y M . Define a map

f : G Ñ H byfpxq “ vpy0,y1q ` vpy1,y2q ` . . . ` vpyn´1,ynq.

Since x “ x1 ¨. . .¨xn is the word of x, y0, . . . , yn are distinct elements of G. Thereforethe elements vpyk´1,ykq of H are orthogonal and hence }fpxq}2 “ n “ |x|. The mainstep now is to show that

}fpyq ´ fpxq}2

“ |y´1x|, x, y P G.

Let n “ |x| and x “ x1 ¨ . . . ¨ xn as before and let m “ |y| and y “ y1 ¨ . . . ¨ ym bethe word of y. Then

y´1x “ y´1m y´1

m´1 ¨ . . . ¨ y´11 x1 ¨ . . . ¨ xn´1xn.

Let l P N0 be the number for which yl`1 ‰ xl`1, but yj “ xj for j ď l. Thenl ď mintn,mu and

y´1x “ y´1m ¨ . . . ¨ y´1

l`1xl`1 ¨ . . . ¨ xn

is the word of y´1x. In particular, |y´1x| “ m ` n ´ 2l. Now

fpxq “ vpe,x1q ` vpx1,x1x2q ` . . . ` vpx1¨...¨xn´1,x1¨...¨xnq

andfpyq “ vpe,y1q ` vpy1,y1y2q ` . . . ` vpy1¨...ÿm´1,y1¨...ÿmq.

Clearly, the first l terms in the expression for fpyq coincide with the first l termsin the expression for fpxq. Using that vps,tq “ ´vpt,sq for ps, tq P A Y A, we get

fpyq ´ fpxq “ vpy1¨...ÿl,y1¨...ÿl`1q ` . . . ` vpy1¨...ÿm´1,y1¨...ÿmq

`vpx1¨...¨xl`1,x1¨...¨xlq ` . . . ` vpx1¨...¨xn,x1¨...¨xn´1q.

Now the elements

y1 ¨ . . . ¨ ym, . . . , y1 ¨ . . . ¨ yl “ x1 ¨ . . . ¨ xl, . . . , x1 ¨ . . . ¨ xn


of G are all distinct. So the m ` n ´ 2l unit vectors in the decomposition offpyq ´ fpxq are orthogonal. It follows that

}fpyq ´ fpxq}2

“ m ` n ´ 2l “ |y´1x|,

as stated above.Let now x1, . . . , xn P G and c1, . . . , cn P C such that

řnj“1 cj “ 0. Then

nÿ

i,j“1

cicj |x´1i xj | “

nÿ

i,j“1

cicj }fpxiq ´ fpxjq}2

“

nÿ

i,j“1

cicj`

}fpxiq}2

` }fpxjq}2

´ 2xfpxiq, fpxjqy˘

“

nÿ

i“1

ci}fpxiq}2

¨

nÿ

j“1

cj `

nÿ

j“1

cj}fpxjq}2

¨

nÿ

i“1

ci

´2nÿ

i,j“1

cicj xfpxiq, fpxjqy

“ ´2

›

›

›

›

›

nÿ

i“1

cifpxiq

›

›

›

›

›

2

ď 0.

This shows that the function x Ñ |x| is negative definite and completes the proof.�

Lemma 5.7.2. Let k, l and m be nonnegative integers, and let f and g be func-tions on G with finite support contained in Ek and El, respectively. Then

}1Empf ˚ gq}2 ď }f}2}g}2

if |k ´ l| ď m ď k ` l and k ` l ´ m is even and

}1Empf ˚ gq}2 “ 0

for all other values of m.

Proof. We have, for all x P G,

pf ˚ gqpxq “ÿ

s,tPG,st“x

fpsqgptq “ÿ

|s|“k,|t|“l,st“x

fpsqgptq.

Since the word of st is obtained from the words of s and t by deleting an evennumber of factors, |st| can only attain the values

|k ´ l|, |k ´ l| ` 2, . . . , k ` l ´ 2, k ` l.

Thus }1Empf˚gq}22 “ 0 ifm is not one of these numbers. Assume now thatm “ k`l.

If |x| “ m, x can only in one way be written as a product x “ st, where |s| “ kand |t| “ l. In fact, s consists of the first k letters in the word of x, and t consistsof the last l letters in the word of x. Therefore

}1Empf ˚ gq}

22 “

ÿ

|s|“k,|t|“l,|st|“k`l

|fpsq|2|gptq|

2

ď

ÿ

|s|“k,|t|“l

|fpsq|2|gptq|

2ď }f}

22}g}

22.


Let now m be one of the numbers k` l´2, k` l´4, . . . , |k´ l|. Then m “ k` l´2p,where 1 ď p ď mintk, lu. If |x| “ m, x “ st, where |s| “ k and |t| “ l, then theword of x consists of the first k ´ p letters of s and the last l ´ p letters of t.

We define functions f 1 and g1 with supports in Ek´p and El´p, respectively, by

f 1pyq “

¨

˝

ÿ

|z|“p

|fpyzq|2

˛

‚

1{2

if |y| “ k ´ p and f 1pyq “ 0 otherwise

and

g1pyq “

¨

˝

ÿ

|z|“p

|fpz´1yq|2

˛

‚

1{2

if |y| “ l ´ p and g1pyq “ 0 otherwise.

Since any element x P G with |x| “ k can only in one way be written as x “ yx,where |y| “ k ´ p and |z| “ l ´ p, we have

}f 1}22 “

ÿ

|y|“k´p

¨

˝

ÿ

|z|“p

|fpyzq|2

˛

‚“ }f}22.

In the same way one gets }g1}2 “ }g}2.Let now x P G with |x| “ m. Let s1 consist of the first k ´ p letters of x, and

let t1 consist of the last l ´ p letters of x. Then x “ s1t1. Moreover, if x “ st, where|s| “ k and |t| “ l, then s “ s1z and t “ z´1t1 where |z| “ p. Consequently,

|pf ˚ gqpxq| “

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ÿ

|s|“k,|t|“l,st“x

fpsqgptq

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

“

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ÿ

|z|“p,|s1z|“k,|z´1t1|“l

fps1zqgpz´1t1q

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

“

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ÿ

|z|“p

fps1zqgpz´1t1q

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ď

¨

˝

ÿ

|z|“p

|fps1zq|2

˛

‚

1{2¨

˝

ÿ

|z|“p

|gpz´1t1q|2

˛

‚

1{2

“ f 1ps1

qg1pt1

q.

Now f 1ps1qg1pt1q “ pf 1 ˚ g1qpxq because x “ s1t1 is the unique splitting of x in aproduct of elements with |s1| “ k ´ p and |t1| “ l ´ p. Thus

|f ˚ g|1Empxq ď pf 1

˚ g1q1Em

pxq

for all x P G. Since pk ´ pq ` pl ´ pq “ m, it follows from the first part of the proofthat

}1Empf ˚ gq}2 ď }1Em

pf 1˚ g1

q}2}f 1}2}g1

}2 “ }f}2}g}2.

This finishes the proof of the lemma. �


Lemma 5.7.3. Let f be a function on G with finite support. Then

}λGpfq} ď

8ÿ

n“0

pn ` 1q}1Enf}2.

Proof. Since f “ř8

n“0 1Enf , it is enough to show that when the support of

f is contained in En, then }λGpfq} ď pn ` 1q}f}2. Thus let f be such a functionand let g P �2pGq be arbitrary and set gk “ 1Ek

g, k P N0. Then gk has support inEk and }g}22 “

ř8

k“0 }gk}22. Let

h “ f ˚ g “

8ÿ

k“0

f ˚ gk and hm “ 1Emh, m P N0.

Note that h P �2pGq and }h}22 “ř8

m“0 }hm}22. From Lemma 5.7.2, we get that

}1Empf ˚ gkq}2 ď }f}2}gk}2,

if |n ´ k| ď m ď n ` k and n ` k ´ m is even, and }1EMpf ˚ gkq}2 “ 0 otherwise.

Therefore

}hm}2 “

›

›

›

›

›

8ÿ

k“0

1Empf ˚ gkq

›

›

›

›

›

2

ď

8ÿ

k“0

}1Empf ˚ gkq}2

ď }f}2

¨

˝

m`nÿ

k“|m´n|,m`n´k even}gk}2

˛

‚.

Writing k “ m ` n ´ 2l, we get

}hm}2 ď }f}2

¨

˝

mintm,nuÿ

l“0

}gm`n´2l}2

˛

‚

ď }f}2

¨

˝

mintm,nuÿ

l“0

}gm`n´2l}22

˛

‚

1{2¨

˝

mintm,nuÿ

l“0

1tlu

˛

‚

1{2

ď pn ` 1q1{2

}f}2

¨

˝

mintm,nuÿ

l“0

}gm`n´2l}22

˛

‚

1{2


and therefore

}h}22 “

8ÿ

m“0

}hm}22

ď pn ` 1q}f}22

8ÿ

m“0

¨

˝

mintm,nuÿ

l“0

}gm`n´2l}22

˛

‚

“ pn ` 1q}f}22

nÿ

l“0

˜

8ÿ

m“l

}gm`n´2l}22

¸

“ pn ` 1q}f}22

nÿ

l“0

˜

8ÿ

k“n´l

}gk}22

¸

ď pn ` 1q}f}22

8ÿ

l“0

}gl}22

“ pn ` 1q2}f}

22}g}

22.

This proves that }f ˚ g}2 ď pn ` 1q}f}2}g}2 for every g P �2pGq, that is, }λGpfq} ď

pn ` 1q}f}2. �

Lemma 5.7.4. Let f be a function on G with finite support. Then

}λGpfq} ď 2

˜

ÿ

xPG

|fpxq|2p1 ` |x|q

4

¸1{2

.

Proof. It follows from Lemma 5.7.3 and the Cauchy-Schwarz inequality that

}λGpfq} ď

8ÿ

n“0

pn ` 1q}1Enf}2 “

8ÿ

n“0

1

n ` 1

`

pn ` 1q2}1En

f}2

˘

ď

˜

8ÿ

n“0

ˆ

1

n ` 1

˙2¸1{2˜

8ÿ

n“0

pn ` 1q4}1En

f}22

¸1{2

“

c

π2

6

˜

ÿ

xPG

|fpxq|2p1 ` |x|q

4

¸1{2

,

and hence the statement of the lemma sincea

π2{6 ď 2. �

Lemma 5.7.5. Let u be a function on G for which

supxPG

|upxq|p1 ` |x|q2

ă 8.

Then u is a multiplier of ApGq and

}u}MpApGqq ď 2 supxPG

|upxq| ¨ p1 ` |x|q2.

Proof. Let F pGq denote the space of functions on G with finite support. Forany v P F pGq Ď ApGq we have uv P F pGq Ď ApGq and, since λGpF pGqq is dense in


V NpGq “ ApGq˚,

}uv}ApGq “ sup t|xuv, λGpfqy| : f P F pGq, }λGpfq} ď 1u

“ sup t|xv, λGpufqy| : f P F pGq, }λGpfq} ď 1u

ď }v}ApGq ¨ supt}λGpufq} : f P F pGq, }λGpfq} ď 1u.

Now, by Lemma 5.7.4,

}λGpufq} ď 2

˜

ÿ

xPG

|upxq|2|fpxq|

2p1 ` |x|q

4

¸1{2

.

Put c “ supxPG |upxq| ¨ p1 ` |x|q2 and observe that }λGpfq} ě }f ˚ δe}2 “ }f}2. Itfollows that

}λGpufq} ď 2c }f}2 ď 2c }λGpfq}

for every f P F pGq and hence

}uv}ApGq ď 2c }v}ApGq.

Since F pGq is dense in ApGq, we conclude that uApGq Ď ApGq and }u}MpApGqq ď 2c,as stated. �

Theorem 5.7.6. Let G be a free group. Then there exists a net puαqα offunctions in ApGq with finite support such that for each v P ApGq:

(i) }uαv}ApGq ď }v}ApGq for every α.(ii) }uαv ´ v}ApGq Ñ 0.

Proof. We assume first that G is finitely generated. By Lemma 5.7.1, thefunction uγ , γ ą 0, defined by uγpxq “ e´γ|x|, x P G, is positive definite. For eachγ ą 0 and n P N, put

vγ,npxq “ e´γ|x| if |x| ď n and vγ,npxq “ 0 if |x| ą n.

Then vγ,n P F pGq and by Lemma 5.7.5,

}vγ,n ´ uγ}MpApGqq ď 2 sup|x|ąn

e´γ|x|p1 ` |x|q

2.

Since e´γmp1 ` mq2 Ñ 0 as m Ñ 8, it follows that }vγ,n ´ uγ}MpApGqq Ñ 0 asn Ñ 8 and, in particular,

}vγ,n}MpApGqq Ñ }uγ}MpApGqq “ uγpeq “ 1.

Thus, setting uγ,n “ }vγ,n}´1MpApGqq

vγ,n, we have

}uγ,n}MpApGqq “ 1 and }uγ,n ´ uγ}MpApGqq Ñ 0 as n Ñ 8.

Now let v “řr

j“1 cjδxjP F pGq and let n ě maxt|xj | : 1 ď j ď ru. Then

}uγv ´ v}ApGq ď }puγ ´ uγ,nqv}ApGq ` }uγ,nv ´ v}ApGq

ď }uγ ´ uγ,n}MpApGqq}v}ApGq

`

rÿ

J“1

|cj | ¨ }uγ,npxjqδxj´ δxj

}ApGq

“ }uγ ´ uγ,n}MpApGqq}v}ApGq `

rÿ

j“1

|cj | ¨ |e´γ|xj |´ 1|.


Since }uγ,n ´ uγ}MpApGqq Ñ 0 for each γ as n Ñ 8 and e´γ|x| Ñ 1 as γ Ñ 0 foreach x, it follows that }uγv ´ v}ApGq Ñ 0 as γ Ñ 0 and therefore

limγÑ0,nÑ8

}uγ,nv ´ v}ApGq “ 0.

Finally, since F pGq is dense in ApGq and }uγ,n}MpApGqq “ 1, a simple triangleinequality argument shows that uγ,nv Ñ v as γ Ñ 0 and n Ñ 8 for all v P ApGq.

Now drop the hypothesis that G be finitely generated and let H denote thecollection of all finitely generated subgroups of G. For H P H, the mapping

ApHq Ñ ČApHq, u Ñ ru,

where rupxq “ upxq for x P H and rupxq “ 0 for x P GzH, is an isometric embedding

of ApHq into ApGq, and YHPHČApHq is dense in ApGq. Moreover, by the first partof the proof, each ApHq has an approximate identity with multiplier norm boundedby one, and for u P ApHq,

}ru}MpApGqq “ sup t}ruv}ApGq : v P ApGq, }v}ApGq ď 1u

ď sup t}Ăuw}ApGq : w P ApHq, }w}ApHq ď 1u

“ sup t}uw}ApHq : w P ApHq, }w}ApHq ď 1u

“ }u}MpApHqq.

Using these facts, another simple triangle inequality argument completes the proofof the theorem. �

We can now apply the results of Section 4.6 to deduce from Theorem 5.7.6 thatthe Fourier algebra of SLp2,Rq has a multiplier bounded approximate identity.

Corollary 5.7.7. ApSLp2,Rqq possesses an approximate identity which isbounded by one in the multiplier norm.

Proof. It is well-known that the two matricesˆ

1 20 1

˙

and

ˆ

1 02 1

˙

generate a closed subgroup H of G “ SLp2,Rq which is isomorphic to F2. Moreover,G{H admits a finite G-invariant measure. Thus H is a lattice in SLp2,Rq. Bythe preceding theorem, ApHq has a approximate identity bounded by one in themultiplier norm. The statement now follows from Theorem 5.6.4. �


The fundamental Lemma 5.1.2, which gives various characterizations of multi-pliers of ApGq, can be found in [27]. The fact that MpApGqq “ BpGq when G isamenable (Theorem 5.1.8) was independently shown in [76] and [242].

All the results about completely bounded multipliers and uniformly boundedrepresentations given in Sections 5.4 and 5.5, are due to de Canniere and Haagerup[27], and our presentation follows very closely that of [27]. De Canniere andHaagerup were interested in when McbpApGqq, the space of completely boundedmultipliers of ApGq, is strictly larger than BpGq. It was shown by Cowling [37]that the coefficient functions of uniformly bounded representations define multipli-ers of the Fourier algebra. In [27, Theorem 2.2] (Theorem 5.5.4) a slightly sim-plified proof was given, and it was moreover shown that these multipliers of ApGq


are even completely bounded. This raises the problem of characterizing those uni-formly bounded representations which provide multipliers not belonging to BpGq.Within the class of cyclic representations, Theorem 5.5.5 [27, Theorem 2.3] iden-tifies the ones in question as precisely those which are not similar to a unitaryrepresentation. The proof, however, is fairly difficult as it uses the solution, due toHaagerup [112], of the so-called similarity problem. As pointed out in [27, Section2], many semisimple Lie groups (e.g. SLp2,Rq, SLpn,Cq and SUpn, 1q, n ě 2) ad-mit uniformly bounded irreducible representations which are not similar to unitaryrepresentations and thus lead to multipliers in McbpApGqqzBpGq.

In [111], Haagerup proved that if G is a free group on at least two generators,then C˚

λ pGq, the reduced group C˚-algebra of G, has the metric approximationproperty, although it is not nuclear. As an application, he obtained that ApGq hasan approximate identity which is bounded by 1 in the multiplier norm.

CHAPTER 6

Spectral Synthesis and Ideal Theory

Let A be a regular and semisimple commutative Banach algebra with Gelfandspectrum σpAq and Gelfand representation A Ñ C0pσpAqq, a Ñ pa. We remind thereader that associated to any closed subset E of σpAq are two distinguished idealsof A, namely

IpEq “ ta P A : pa “ 0 on Eu

and

jpEq “ ta P A : pa has compact support disjoint from Eu.

Then IpEq and jpEq are the largest and the smallest ideal of A with zero set E.

The set E is called a spectral set or set of synthesis if jpEq “ IpEq, and one saysthat spectral synthesis holds for A if every closed subset of σpAq is a set of synthesis.

Moreover, E is called a Ditkin set if a P ajpEq for every a P IpEq. The main purposeof this chapter is to study spectral sets and Ditkin sets and the spectral synthesisproblem for Fourier algebras of locally compact groups.

A famous theorem of Malliavin [207] states that spectral synthesis fails forApGq whenever G is any nondiscrete abelian locally compact group. Using thisand a deep theorem of Zelmanov [293] ensuring the existence of infinite abeliansubgroups of infinite compact groups, we prove in Section 6.2 that for an arbitrarylocally compact group G, under a mild additional hypothesis, spectral synthesisholds for ApGq if and only if G is discrete.

One of the most interesting problems in the ideal theory of a commutativeBanach algebra is to identify the closed ideals with bounded approximate identities.For Fourier algebras this problem is treated in Section 6.5. Improving on earlierresults for abelian groups and employing operator space techniques, it is shownthat for an amenable group G, the closed ideals in ApGq with bounded approximateidentities are in one-to-one correspondence with the closed sets in the coset ring ofG. Moreover, for a closed subgroup H of G, the ideal IpHq has an approximateidentity with norm bounded by 2.

Suppose that G is amenable and let H be a closed normal subgroup of G. InSection 6.6 we prove the existence of a bijection e : I Ñ epIq between the set of allclosed ideals of ApG{Hq and the set of all closed ideals of ApGq, which are invariantunder translation by elements of H. Moreover, we show that a closed ideal I ofApG{Hq has a bounded approximate identity if and only if the same is true of epIq.

After collecting some basic material and proving that closed subgroups are setsof synthesis (Section 6.1), we proceed in Sections 6.2 and 6.3 to establish variousinjection and projection theorems for spectral sets and Ditkin sets as well as theirlocal variants. Given a closed subgroup H and a closed normal subgroup N of G,these results relate spectral sets and Ditkin sets for ApHq and ApG{Nq, respectively,to such sets for ApGq.

205

206 6. SPECTRAL SYNTHESIS AND IDEAL THEORY

6.1. Sets of synthesis and Ditkin sets

Let G be a locally compact group and ApGq the Fourier algebra of G. Recallthat a closed subset E of G “ σpApGqq is a set of synthesis (or spectral set ) if

jpEq “ IpEq and E is a Ditkin set if u P ujpEq for each u P IpEq. Both notionsadmit local variants as follows. The set E is said to be a set of local synthesis(or local spectral set) if IpEq X CcpGq Ď jpEq, and E is called a local Ditkin set if

u P ujpEq for every u P IpEq X CcpGq. Clearly, every local Ditkin set is a localspectral set. Notice that H is a set of synthesis since ApGq X CcpGq is dense in

ApGq. On the other hand, H is a Ditkin set if and only if u P uApGq for everyu P ApGq.

Lemma 6.1.1. Let E be an open and closed subset of G. Then E is a set of localsynthesis. If, in addition, E has the property that u P uApGq for each u P IpEq,then E is a set of synthesis.

Proof. Let T P V NpGq such that suppT Ď E, and let u P IpEq X CcpGq.Choose v P ApGq such that v “ 1 on supp u. Since E is open in G, supp uXsuppT “

H and hence u ¨ T “ 0. Thus

xT, uy “ xT, uvy “ xu ¨ T, vy “ 0.

Now suppose that there exists a net pvαqα in ApGq such that uvα Ñ u. Then, asin the previous case, u ¨ T “ 0 and hence

xT, uy “ limα

xT, uvαy “ limα

xu ¨ T, vαy “ 0,

as required. �

We shall introduce some notions which allow us to treat spectral set and Ditkinsets and their local variants to some extent simultaneously.

Definition 6.1.2. Let X be an ApGq-invariant linear subspace of V NpGq. Aclosed subset E of G is called an X-spectral set or set of X-synthesis if for any T P Xsuch that supp T Ď E we have T P IpEqK.

The set E is called an X-Ditkin set for ApGq if for every T P X and u P IpEq

there exists a net pvαqα in jpEq such that

xvα ¨ T, uy Ñ xT, uy.

Remark 6.1.3. (a) We shall frequently use the following simple fact. Supposethat E is X-Ditkin. Then, given T P X and u P IpEq, there exists v P jpEq suchthat xT, uy “ xT, vuy. To see this, choose v “ 0 when xT, uy “ 0, and if xT, uy ‰ 0,notice that txT, vuy : v P jpEqu “ C.

(b) Every X-Ditkin set is an X-spectral set. Indeed, if T P X is such thatsupp T Ď E and u P IpEq, then taking v P jpEq as in (a), it follows that xT, uy “

xT, vuy “ 0 since vu P jpEq and supp T Ď E.

We next identify the proper choices of X to recover the original notions. For

that, let UCp pGq denote the closed linear span of

tu ¨ T : T P V NpGq, u P ApGqu.

Then UCp pGq is the norm closure of UCcp pGq, the set of operators in V NpGq withcompact support. To explain the notation, we briefly mention that when G is

6.1. SETS OF SYNTHESIS AND DITKIN SETS 207

abelian, UCp pGq is precisely the C˚-algebra of bounded uniformly continuous func-

tions on the dual group pG of G. Furthermore, UCp pGq is a C˚-subalgebra of V NpGq

(see [174] and [102] for more details).

Lemma 6.1.4. Let E be a closed subset of G. Then

(i) E is of local synthesis if and only if E is of UCcp pGq-synthesis.(ii) E is of synthesis if and only if E is of V NpGq-synthesis.

Proof. We show (i), the proof of (ii) being similar (in fact, easier).

Suppose first that E is of local synthesis, and let T P UCcp pGq such thatsupp T Ď E. Choose v P ApGq X CcpGq such that v “ 1 in some open neigh-bourhood V of supp T . Then u ´ uv “ 0 on V for all u P ApGq, and this impliesv ¨ T ´ T “ 0 (Proposition 2.5.3(ii)). Now, since supp T Ď E, we have xT, uy “ 0for all u P IpEq X CcpGq. It follows that, for u P IpEq,

xT, uy “ xv ¨ T, uy “ xT, vuy “ 0,

since vu P IpEq X CcpGq.

Conversely, suppose that E is of UCcp pGq-synthesis, and let u P IpEq X CcpGq.We have to show that xT, uy “ 0 whenever T P V NpGq annihilates jpEq. Choose

v P ApGq XCcpGq such that v “ 1 on supp u. Then v ¨T P UCcp pGq, and hence v ¨Tannihilates IpEq. Thus xT, uy “ xT, vuy “ 0. �

Lemma 6.1.5. Let E be a closed subset of G. Then

(i) E is a local Ditkin set if and only if E is UCcp pGq-Ditkin.(ii) E is a Ditkin set if and only if E is V NpGq-Ditkin.

Proof. Suppose first that E is a Ditkin set (respectively, a local Ditkin set),

and let u P IpEq and T P V NpGq (respectively, T P UCcp pGq). If T P UCcp pGq, thenchoose v P ApGq X CcpGq such that v ¨ T “ T . Now, by hypothesis, there exists anet pvαqα in jpEq such that

vαu Ñ u and vαpuvq Ñ uv

in ApGq, respectively. It follows that

xvα ¨ T ´ T, uy “ xT, vαu ´ uy Ñ 0

in the first case, whereas in the second case

xvα ¨ T ´ T, uy “ xvα ¨ pv ¨ T q ´ v ¨ T, uy “ xT, vαuv ´ uvy Ñ 0.

Conversely, suppose that E is UCcp pGq-Ditkin, and let T P V NpGq and u P IpEq X

CcpGq. Choose v P ApGq XCcpGq such that v “ 1 on supp u. Since v ¨T P UCcp pGq,there exists a net pvαqα in jpEq such that

xvα ¨ pv ¨ T q, uy Ñ xv ¨ T, uy,

and hence, since vu “ u,

xT, vαuy Ñ xT, uy.

Thus xT, uy “ 0 whenever T annihilates ujpEq, as required. The proof that E isDitkin if it is V NpGq-Ditkin is even simpler. �

Proposition 6.1.6. Let G be a locally compact satisfying u P uApGq for everyu P ApGq, and let E be a closed subset of G.


(i) If E is a UCcp pGq-spectral set, then E is a spectral set.

(ii) If E is UCcp pGq-Ditkin, then E is a Ditkin set.

Proof. Choose a net puαqα in ApGq X CcpGq such that }uαu ´ u} Ñ 0.

Suppose first that E is UCcp pGq-spectral, and let u P IpEq and T P V NpGq

such that supp T Ď E. Then, since uα ¨ T P UCcp pGq and supp uα ¨ T Ď E, we havethat xuα ¨ T, uy “ 0. Hence xT, uy “ 0 since xT, uαuy Ñ xT, uy.

Now, let E be UCcp pGq-Ditkin and let T P V NpGq and u P IpEq. We have toshow that there exists v P jpEq such that xT, uy “ xT, vuy. We can assume that

xT, uy ‰ 0. Since uα ¨ T P UCcp pGq, for each α there is a vα P jpEq such that

xuα ¨ T, uy “ xuα ¨ T, vαuy.

Since xT, uαuy Ñ xT, uy, xT, uy ‰ 0 and uαvα P jpEq, we must have that txT, vuy :v P jpEqu “ C, whence xT, uy “ xT, vuy for some v P jpEq. �

Proposition 6.1.6 shows that if ApGq has an approximate identity in the weakestpossible sense, then local spectral sets (local Ditkin sets) are necessarily spectralsets (Ditkin sets).

Remark 6.1.7. Let X “ pX˚q˚ be a dual Banach space and let Y be a w˚-closed linear subspace of X and

KY “ tu P X˚ : xf, uy “ 0 for all f P Y u.

Then Y “ pKY qK. The inclusion Y Ď pKY qK being clear, suppose there existsg P pKY qKzY . Then, by the Hahn-Banach theorem, we find u P X˚ such thatxu, fy “ 0 for all f P Y , but xu, gy ‰ 0. So u P KY and g R pKY qK, a contradiction.

Lemma 6.1.8. Let E be a closed subset of the locally compact group G and letV NEpGq be the w˚-closure of the linear span of the set tλGpxq : x P Eu. Then thefollowing are equivalent:

(i) E is a set of synthesis for ApGq.(ii) For any T P V NpGq, supp T Ď E implies that T P V NEpGq.

Proof. We apply the above remark with X “ V NpGq, X˚ “ ApGq andY “ V NEpGq. Then

IpEq “ tu P ApGq : xu, λGpxqy “ 0 for all x P Eu “ KV NEpGq,

and hence V NEpGq “ pKV NEpGqqK “ IpEqK.(i) ñ (ii) Let T P V NpGq be such that supp T Ď E. Then xT, uy “ 0 for all

u P ApGq X CcpGq with supp u X E “ H (Proposition 2.5.3(ii)). It follows from (i)and the above remark that T P jpEqK “ JpEqK “ IpEqK “ V NEpGq.

(ii) ñ (i) Suppose that E fails to be a set of synthesis. Then JpEq is a properclosed subspace of IpEq. Choose u P IpEqzJpEq. Then, by the Hahn-Banachtheorem, there exists T P V NpGq such that T P JpEqK and xT, uy ‰ 0. Proposition2.5.3(ii) implies that suppT Ď E, but T R IpEqK “ V NEpGq. This contradicts(ii). �

The following theorem together with Theorem 6.1.12 already provides us witha wealth of sets of synthesis for ApGq.

Theorem 6.1.9. Let G be a locally compact group and H a closed subgroup ofG. Then H is a set of synthesis for ApGq.

6.1. SETS OF SYNTHESIS AND DITKIN SETS 209

Proof. The statement follows from Lemma 6.1.8. �

Corollary 6.1.10. Let H be a closed subgroup of G and a P G. Then thecoset aH is a set of synthesis.

Proof. We only have to note that, for any closed subset E of G, the mapu Ñ Lau is an isometric algebra isomorphism mapping IpEq and jpEq onto IpaEq

and jpaEq, respectively. �

Theorem 6.1.11. Let G be a locally compact group and X an ApGq-invariantlinear subspace of V NpGq. Suppose that E1 and E2 are closed subsets of G suchthat E1 X E2 is an X-Ditkin set. Then E1 Y E2 is an X-spectral set if and only ifboth E1 and E2 are X-spectral sets.

Proof. Suppose first that E1 and E2 are X-spectral sets, and let T P X besuch that suppT Ď E1 Y E2 and u P IpE1 Y E2q. Since E1 X E2 is X-Ditkin, thereexists v P jpE1 X E2q such that xT, uy “ xv ¨ T, uy. Since v has compact supportdisjoint from E1 X E2, there are compact sets F1 and F2 such that

supppv ¨ T q “ F1 Y F2 and Fj Ď EjzpE1 X E2q, j “ 1, 2.

Now there exist vj P ApGq X CcpGq, j “ 1, 2, such that vj “ 1 on a neighbourhoodof Fj and supp v1 X supp v2 “ H. Then pv1 ` v2qv ¨ T “ v ¨ T since v1 ` v2 “ 1 onsome neighbourhood of F1 Y F2. Moreover,

supppvjvq ¨ T Ď Ej and pvjvq ¨ T P X pj “ 1, 2q,

since X is ApGq-invariant. Since E1 and E2 are of X-synthesis, it follows thatxpvjvq ¨ T, uy “ 0 for j “ 1, 2, an d hence xv ¨ T, uy “ 0, as was to be shown.

Conversely, suppose that E1YE2 is ofX-synthesis, and let T P X with suppT Ď

E1 and u P IpE1q be given. As above, since E1 X E2 is an X-Ditkin set, xT, uy “

xT, vuy for some v P jpE1 X E2q. Since supppv ¨ T q is a compact set containedin E1zE2, there exists w P ApGq so that w “ 1 on a compact neighbourhood ofsupppv ¨ T q and w “ 0 on E2. It follows that

wu P IpE1 Y E2q and v ¨ T “ pwvq ¨ T.

Now, since v ¨ T P X, supppv ¨ T q Ď E1 Y E2 and E1 Y E2 is of X-synthesis,

xT, uy “ xv ¨ T, uy “ xv ¨ T,wuy “ 0.

This shows that E1 is an X-spectral set. In the same manner it is shown that E2

is an X-spectral set. �

Theorem 6.1.12. Let G and X be as in Theorem 6.1.11, and let E and F beclosed subsets of G.

(i) If both E and F are X-Ditkin sets, then E Y F is an X-Ditkin set.(ii) If E YF and E XF are X-Ditkin sets, then E and F are X-Ditkin sets.

Proof. (i) Given u P IpE Y F q and T P X, there exist v P JpEq such thatxT, uy “ xT, uvy and then w P JpF q such that xT, uvy “ xT, puvqwy. Thus vw P

JpE Y F q andxT, uy “ xT, upvwqy,

as required.(ii) Suppose that E X F and E Y F are both X-Ditkin sets, and let T P X

and u P IpEq. Since E X F is X-Ditkin, there exists v P jpE X F q such that


xT, uy “ xT, uvy. Let C “ F Xsupppvuq, a compact set disjoint from E. Thus thereexists w P ApGq X CcpGq such that w “ 0 on a neighbourhood of E and w “ 1on C. So, in particular, w P JpEq. Let u1 “ vu ´ vwu, then u1 P IpE Y F q sincew “ 1 on C. Since E Y F is an X-Ditkin set, there exists v1 P JpE Y F q such thatxT, u1y “ xT, u1v1y. It follows that pv ´ vwqv1 ` vw P JpEq and

xT, uy “ xT, vuy “ xT, u1y ` xT, uvwy

“ xT, u1v1y ` xT, uvwy

“ xT, uppv ´ vwqv1` vwqy.

This proves that E is X-Ditkin, and similarly for F . �

The following corollary strengthens the second statement of Lemma 6.1.1.

Corollary 6.1.13. Suppose that ApGq has an approximate identity, and letE be an open and closed subset of G. Then E is a Ditkin set.

Proof. Since, by hypothesis, the empty set is a Ditkin set for ApGq, the claimfollows by applying Theorem 6.1.12 to E and F “ GzE. �

6.2. Malliavin’s theorem for ApGq

As mentioned in the introduction to this chapter, according to Malliavin’s the-orem, spectral synthesis holds for the Fourier algebra ApGq of a locally compactabelian group G if and only if G is discrete. In this section, we extend this resultto general locally compact groups.

Lemma 6.2.1. Let K be a compact normal subgroup of G and q : G Ñ G{Kthe quotient homomorphism. Let E be a closed subset of G{K. If q´1pEq is a localspectral set for ApGq, then E is a local spectral set for ApG{Kq.

Proof. Let u P IpEq X CcpG{Kq and ε ą 0 be given, and let u1 “ u ˝ q P

ApGq. Then, since K is compact, u1 P Ipq´1pEqq X CcpGq. Hence there existsv1 P jpq´1pEqq such that }u1 ´ v1}ApGq ď ε. Now define v on G{K by

vpxKq “

ż

K

v1pxkq dk “

ż

K

Rkv1pxq dk,

where dk denotes normalized Haar measure of K. Then v P ApG{Kq and

}u ´ v}ApG{Kq “

›

›

›

›

ż

K

Rkpu1 ´ v1q dk

›

›

›

›

ApGq

ď }u1 ´ v1}ApGq ď ε.

Then v has compact support since C “ supp v1 is compact and supp v Ď qpCq.Moreover, v vanishes in a neighbourhood of E. Indeed, since C X q´1pEq “ H,there exists a symmetric neighbourhood V of e in G such that V C X q´1pEq “ H

and hence C X V q´1pEq “ H. Since v1 “ 0 on V q´1pEq, v vanishes on theneighbourhood qpV q´1pEqq of E. Thus v P jpEq, as required. �

Proposition 6.2.2. Let G be a nontrivial connected locally compact group.Then local spectral synthesis fails for ApGq.

Proof. Towards a contradiction, assume that local spectral synthesis holdsfor ApGq. Since G ‰ teu, G has a proper compact subgroup K such that G{K isa Lie group. By Lemma 6.2.1, local spectral synthesis holds for ApG{Kq. Now thenontrivial connected Lie group G{K contains a closed nondiscrete abelian subgroup

6.3. INJECTION THEOREMS FOR SPECTRAL SETS AND DITKIN SETS 211

H (a one-parameter group). Then local spectral synthesis holds for ApHq as well(Corollary 6.3.4). This contradicts Malliavin’s theorem. �

Theorem 6.2.3. Let G be a locally compact group. Then

(i) Local spectral synthesis holds for ApGq if and only if G is discrete.(ii) Spectral synthesis holds for ApGq if and only if G is discrete and u P

uApGq for every u P ApGq.

Proof. Suppose first that local spectral synthesis holds for ApGq. Then, Gmust be totally disconnected. Fix a compact open subgroup K of G. We have toshow that K is finite. Towards a contradiction, assume that K is infinite. Then,by a theorem of Zelmanov [293, Theorem 2], K contains an infinite abelian closedsubgroup H. Now local spectral synthesis holds for ApHq, and hence so doesspectral synthesis since H is abelian. This contradicts Malliavin’s theorem. ThusG is discrete. Conversely, if G is discrete, then local spectral synthesis holds forApGq by Lemma 6.1.1.

For (ii), notice first that if synthesis holds for ApGq, then G is discrete by (i)

and u P uApGq for each u P ApGq. Indeed, denoting by E the zero set of u, we

have hpuApGqq “ E and therefore uApGq “ IpEq. Again, the converse follows fromLemma 6.1.1. �

6.3. Injection theorems for spectral sets and Ditkin sets

The subject of this section is to establish so-called injection theorems, whichconstitute an additional method to produce sets of synthesis and Ditkin sets.

Remark 6.3.1. Let H be a closed subgroup of G and let r : ApGq Ñ ApHq

denote the restriction map (Section 2.3).(1) For any ApGq-invariant linear subspace X of V NpGq and closed subgroup

H of G, let

XH “ r˚´1pXq.

Then XH is an ApHq-invariant linear subspace of V NpHq. Indeed, if v P ApHq andS P XH , then choosing u P ApGq with rpuq “ v,

r˚pv ¨ Sq “ u ¨ r˚

pSq P X,

and hence v ¨ S “ r˚´1pu ¨ r˚pSqq P r˚´1pXq.(2) If S P XH and supp S Ď E for some closed subset E ofH, then supp r˚pSq Ď

E. In fact, let x P supp r˚pSq. Then there exists a net puαqα in ApGq such that

r˚prpuαq ¨ Sq “ uα ¨ r˚

pSq Ñ λGpxq “ r˚pλHpxqq.

Hence rpuαq ¨ S Ñ λHpxq.

Lemma 6.3.2. Let H be a closed subgroup of G. Then

r˚pUCcp pHqq “ UCcp pGq X V NHpGq.

Proof. First, let T P UCcp pHq. There exists w P ApHq X CcpHq such thatw ¨ T “ T . Next, choose u1 P ApGq with u1|H “ w and u2 P ApGq X CcpGq such


that u2 “ 1 on the compact set supp w, and put v “ u1u2. Then v P ApGq XCcpGq

and v|H “ w. For all u P ApGq, it follows that

xr˚pT q, uy “ xr˚

pw ¨ T q, uy “ xw ¨ T, rpuqy

“ xT,wrpuqy “ xT, rpvuqy

“ xr˚pT q, vuy “ xv ¨ r˚

pT q, uy.

Thus r˚pT q “ v ¨ r˚pT q, which belongs to UCcp pGq X V NHpGq.

Conversely, let S P UCcp pGq X V NHpGq and let T P V NpHq such that r˚pT q “

S. Since supp S is compact, there exists v P ApGq X CcpGq such that v “ 1 on

a neighbourhood of supp S. Then v ¨ S “ S and rpvq ¨ T P UCcp pHq. Moreover,r˚prpvq ¨ T q “ S. Indeed, for all u P ApGq,

xr˚prpvq ¨ T q, uy “ xrpvq ¨ T, rpuqy “ xT, rpvuqy

“ r˚pT q, vuy “ xS, vuy

“ xv ¨ S, uy “ xS, uy.

This shows that UCcp pGq X V NHpGq Ď r˚pUCcp pHqq. �

Theorem 6.3.3. Let G be a locally compact group and let X be an ApGq-invariant linear subspace of V NpGq. Let H be a closed subgroup of G and E aclosed subset of H. Then E is an X-spectral set for ApGq if and only if it is anXH-spectral set for ApHq.

Proof. Suppose first that E is of X-synthesis and let S P XH “ r˚´1pXq besuch that supp S Ď E. Then r˚pSq P X and supp r˚pSq Ď E by Remark 6.3.1(2).Then, by hypothesis, for every u P IpEq,

0 “ xr˚pSq, uy “ xS, rpuqy.

Since rpIpEqq “ tv P ApHq : v|E “ 0u, it follows that E is of XH-synthesis.Conversely, suppose that E is ofX-synthesis, and let T P X such that supp T Ď

E. Since H is a set of synthesis for ApGq by Theorem 6.1.9, T annihilates IpHq.So there exists a unique S P XH such that r˚pSq “ T . Clearly, supp S Ď E alsoand hence, by hypothesis, xT, uy “ xS, rpuqy “ 0 for all u P IpEq, as required. �

Corollary 6.3.4. Let H be a closed subgroup of G and E a closed subset of H.Then E is a spectral set (local spectral set) for ApGq if and only if E is a spectralset (local spectral set) for ApHq.

Proof. Since V NpGqH “ r˚´1pV NpGqq “ V NpHq, the statement aboutspectral sets follows from Lemma 6.1.4 and Theorem 6.3.3, taking X “ V NpGq.Turning to local spectral sets, recall from Lemma 6.3.2 that

UCcp pHq “ r˚´1pUCcp pGq X V NHpGqq “ r˚´1

pUCcp pGqq “ UCcp pGqH .

Now apply Theorem 6.3.3 with X “ UCcp pGq and Lemma 6.1.4. �

Proposition 6.3.5. Let X be an ApGq-invariant linear subspace of V NpGq.Let H be a closed subgroup of G and E a closed subset of H. If E is an X-Ditkinset for ApGq, then E is an XH-Ditkin set for ApHq.

Proof. It suffices to show that given u P ApHq such that u|E “ 0 and S P XH ,there exists v P ApHq which vanishes in a neighbourhood of E in H and satisfies

6.3. INJECTION THEOREMS FOR SPECTRAL SETS AND DITKIN SETS 213

xS, uy “ xS, vuy. Choose u1 P ApGq extending u. Since r˚pSq P X and E is X-Ditkin, there exists v1 P ApGq such that v1 “ 0 in a neighbourhood of E in G andxr˚pSq, u1y “ xr˚pSq, v1u1y (Remark 6.1.3). Then

xS, uy “ xS, rpu1qy “ xr˚pSq, u1y “ xr˚

pSq, v1u1y

“ xS, rpv1quy,

and hence v “ rpv1q has the required property. �

Theorem 6.3.6. Let G be an amenable locally compact group, H a closed sub-group of G and E a closed subset of H. If E is a Ditkin set (local Ditkin set) forApHq, then it is a Ditkin set (local Ditkin set) for ApGq.

Proof. Let E be a Ditkin set for ApHq. We show that given u P IpEq andε ą 0, there exists v P jpEq such that }u ´ vu}ApGq ď C ε for some constant C ą 0.There exists w1 P ApHq such that w1 “ 0 in a neighbourhood of E in H and}u|H ´ w1pu|Hq} ď ε. Choose w2 P ApGq extending w1. By Theorem 6.3.3, E is aset of synthesis for ApGq. So there exists w3 P jpEq such that }w2 ´ w3} ď ε{}u}.Then

}rpu ´ w3uq} ď }rpuq ´ rpw2uq} ` }rpuqprpw2q ´ rpw3qq}

ď }rpuq ´ w1rpuq} ` }u}}w2 ´ w3} ď 2ε.

Now let w4 “ u ´ uw3 P ApGq. Since G is amenable, by Lemma 6.3.7 below thereexists w5 P jpHq such that

}w4 ´ w4w5} ď 3 }rpw4q} ` ε,

and hence }w4 ´ w4w5} ď 7ε. Finally, let

v “ w3 ` w5 ´ w3w5 P ApGq.

Then v has compact support since both w3 and w5 have compact support. More-over, v vanishes in a neighbourhood of E in G since w3 does so and w5 vanishes ina neighbourhood of H. Thus v P jpEq and

}u ´ vu}ApGq “ }w4 ´ w5w4}ApGq ď 7ε.

This shows that E is Ditkin set for ApGq. The proof of the local version is entirelyanalogous. �

Lemma 6.3.7. Let G be an amenable locally compact group and H a closedsubgroup of G. Given u P ApGq and ε ą 0, there exists v P ApGq such that vvanishes in a neighbourhood of H and satisfies

}u ´ vu} ď 3 }u|H} ` ε.

Proof. Since

}u|H} “ inf t}u ` w} : w P IpHqu,

we find w P IpHq such that

}u ´ w} ď }u|H} ` ε{4.

Now H is a set of synthesis (Theorem 6.1.9) and since G is amenable, the idealIpHq has an approximate identity of norm bound 2. Hence there exists v P jpHq


with }v} ď 2 and }w ´ vw} ď ε{4. It follows that

}u ´ vu} ď }u ´ w} ` }w ´ wv} ` }vpw ´ uq}

ď }u|H} ` ε{2 ` ε{4 ` 2p}u|H} ` ε{4q

“ 3}u|H} ` ε,


6.4. A projection theorem for local spectral sets

Let G be a locally compact abelian group, H a closed subgroup of G and

r : pG Ñ pH the restriction map. Then a closed subset E of pH is a local spectralset for L1pHq if and only if r´1pEq is a local spectral set for L1pGq [241, Section7.3]. In this section we present the analogue, due to Lohoue [196], for Fourieralgebras of general locally compact groups and their quotient groups. The proofof Theorem 6.4.6 below in the abelian case is essentially based on the Poissonformula on the dual group of G. Since such a nice dual object is not available forgeneral locally compact groups, the approach has to be completely different andthe arguments used are naturally much more complicated.

Lemma 6.4.1. Let γ : G Ñ Cˆ be a continuous homomorphism from G intothe multiplicative group of nonzero complex numbers.

(i) Then γu P ApGq for every u P ApGq X CcpGq.(ii) Given a compact subset K of G, there exists a constant CpKq such that,

for all u P ApGq with supp u Ď K,

}γu} ď CpKq}u}.

Proof. Let K be a compact subset of G. Then we can find compact subsetsK1 and K2 of positive measure such that the function

w “ 1K1K2˚ |K2|

´11K2

is identically 1 on K. If u P ApGq is such that supp u Ď K, then γu “ γwu.Therefore, (i) follows once we have recognized that γw P ApGq. For that, let

v1 “ γ ¨ 1K1K2and v2 “ |K2|

´1γ´1¨ 1K2

.

Then v1, v2 P L2pGq and hence γw “ v1 ˚ v2 P ApGq. Moreover,

}v1}2 ď supxPK1K2

|γpxq| ¨ }1K1K2}2 “

1

|K1K2|1{2sup

xPK1K2

|γpxq|,

and, since γ is multiplicative,

}v2}2 ď supxPK2

1

|γpxq|¨ }1K2

}2 “1

|K2|1{2sup

xPK´12

|γpxq|.

Now, define CpKq by

CpKq “ |K1K2|´1{2

|K2|´1{2

¨ supxPK1K2

|γpxq| ¨ supxPK´1

2

|γpxq| ă 8.

Then }γw} ď }v1}2}v2}2 ď CpKq and hence

}γu} “ }pγwqu} ď CpKq}u}.

This establishes (ii). �

6.4. A PROJECTION THEOREM FOR LOCAL SPECTRAL SETS 215

Since, given any compact subset K of G, there exists u P ApGq X CcpGq suchthat u “ 1 on K, the statement of Lemma 6.4.1 can be rephrased by saying that γbelongs locally to ApGq. In particular, the modular function of G belongs locallyto ApGq.

Corollary 6.4.2. Let S P V NpGq with compact support, and let γ : G Ñ

Cˆ be a continuous homomorphism. Then there exists an operator, denoted γS,in V NpGq such that xγS, uy “ xS, γuy for all u P ApGq X CcpGq. Moreover,supppγSq “ suppS.

Proof. According to Lemma 6.4.1, we can define γS on ApGq X CcpGq byxγS, uy “ xS, γuy. Fix a compact neighbourhood K of suppS and let CpKq as givenby Lemma 6.4.1. Choose w P ApGq X CcpGq such that w “ 1 on a neighbourhoodof suppS and suppw Ď K. Then w ¨ S “ S and, for any u P ApGq X CcpGq,

|xγS, uy| “ |xS, γuy| “ |xw ¨ S, γuy| “ |xS, γpwuqy|

ď }S} ¨ }γpwuq} ď }S} ¨ CpKq ¨ }wu}

ď CpKq}S} ¨ }w} ¨ }u}.

Thus γS extends uniquely to a bounded linear operator on ApGq.For the last statement, let a P suppS and u P ApGq with u ¨ pγSq “ 0. Then,

for all v P ApGq X CcpGq,

xu ¨ S, vy “ xu ¨ pγSq, γ´1vy “ 0,

so that u ¨ S “ 0 and hence upaq “ 0. This shows that suppS Ď supppγSq. Thereverse inclusion follows by replacing γ with γ´1. �

Lemma 6.4.3. Let N be a closed normal subgroup of G and K a compact subsetof G. Then there exists a constant cpKq ą 0 with the following property: If u P ApGq

is such that supp u Ď K, then TN puq P ApG{Nq and

}TN puq}ApG{Nq ď cpKq}u}ApGq.

Proof. There exist f, g P L2pGq such that u “ f ˚ g and }u} “ }f}2}g}2.Choose a compact symmetric neighbourhood V of e and set v1 “ 1KV , v2 “ |V |´11Vand v “ v1 ˚ v2 P ApGq. Then v “ 1 on K and, for every x P G,

vpxqpf ˚ gqpxq “

ż

G

v1pxyqv2pyq dy ¨

ż

G

fpxzqgpzq dz

“

ż

G

ż

G

fpxyzqv1pxyqgpyzqv2pyq dzdy

“

ż

G

ż

G

pRzfqv1pxyqpRzg ¨ v2q py´1q dydz

“

ż

G

ppRzf ¨ v1q ˚ pRzg ¨ v2q q pxq dz.

Since the map TN commutes with vector-valued integration and TN is a homomor-phism from L1pGq onto L1pG{Nq, we get

TN pvpf ˚ gqqp 9xq “

ż

G

TN rpRzf ¨ v1q ˚ pRzg ¨ v2q s p 9xq dz

“

ż

G

TN pRzf ¨ v1q ˚ TN ppRzg ¨ v2q q p 9xq dz,

where of course in the last term ˚ denotes convolution on G{N .


We temporarily fix z P G and estimate the 2-norms of the functions TN pRzf ¨v1q

and TN ppRzg ¨v2q q. For the first one we get, using Weil’s formula and the Cauchy-Schwarz inequality,

}TN pRzf ¨ v1q}22 “

ż

G{N

ˇ

ˇ

ˇ

ˇ

ż

N

Rzfpxnqv1pxnq dn

ˇ

ˇ

ˇ

ˇ

2

d 9x

ď

ż

G{N

ˆż

N

|Rzfpxnqv1pxnq|dn

˙2

d 9x

ď

ż

G{N

ˆż

N

|Rzfpxnq|2dn ¨

ż

N

|v1pxnq|2dn

˙

d 9x

“ |KV | ¨

ż

G

|Rzfpxq|2dx “ |KV | ¨ }Rzf}

22.(6.1)

To treat the second function above, we note that the support of TN pΔ´1G ¨Rzg ¨ v2q

is contained in V N{N , we obtain

}TN ppRzg ¨ v2q q}22 “

ż

G{N

ˇ

ˇΔG{N p 9xqTN pΔ´1G ¨ Rzg ¨ v2qp 9xq

ˇ

ˇ

2d 9x

ď

˜

sup9xPqpV q

ΔG{N p 9xq

¸2

¨

ż

G{N

ˇ

ˇTN pΔ´1G ¨ Rzg ¨ v2qp 9xq

ˇ

ˇ

2d 9x

Observe next that, for every x P G,ż

N

|Δ´1G pxnq1V pxnq|

2dn “

ż

x´1V

Δ´2G pxnqdn

ď suptΔGptq´2 : t P V u.

Setting now

c1 “ suptΔG{N p 9xq : 9x P qpV qu and c2 “ sup tΔGptq´2 : t P V u,

and applying Weil’s formula and Cauchy-Schwarz again, we get

}TN pΔ´1G Rzg ¨ v2}

22 ď

ż

G{N

ˆż

N

|pΔ´1G Rzg ¨ v2qpxnq|dn

˙2

d 9x

ď

ż

G{N

ˆ

}LxRzg ¨ v2q|N }2 ¨

ż

N

|Δ´1G pxnq1V xnqdn|

2

˙

d 9x

ď

ż

G

|Rzg ¨ v2pxq|2dx ¨ sup

xPG

"ż

N

|Δ´1G pxnq1V pxnq|

2dn

*

“ c1c2 }Rzg ¨ v2}22.

Since v “ 1 on K and supp u Ď K, we can now estimate the norm of TN puq asfollows.

}TN puq} “

›

›

›

›

ż

G

rTN pRzf ¨ v1q ˚ TN ppRzg ¨ v2q qs dz

›

›

›

›

ď

ż

G

}TN pRzf ¨ v1q ˚ TN ppRzg ¨ v2q q}dz

ď

ż

G

}TN pRzf ¨ v1q}2 ¨ }TN ppRzg ¨ v2q }2dz.


Now, by (6.1) above

}TN pRzf ¨ v1q}22 ď |KV | ¨ }Rzf ¨ v1}

22,

and by definition of c1 and c2,

}TN pRzf ¨ v2q }22 ď c1c2 ¨ }Rzg ¨ v2}

22.

It follows that

}TN puq} ď |KV |1{2

pc1c2q1{2

ż

G

}Rzf ¨ v1}2}Rzg ¨ v2}2 dz

ď p|KV |c1c2q1{2

ˆż

G

}Rzf ¨ v1}22 dz

˙1{2ˆż

G

}Rzg ¨ v2}22 dz

˙1{2

ď p|KV |c1c2q1{2

}f}2}v1}2 ¨ }g}2}v2}2

“ |KV | ¨ |V |´1c1c2}u}ApGq

ď |K| ¨ c1c2}u}ApGq.

Since this inequality holds for all compact symmetric neighbourhoods of e, we canlet V Ñ teu. Then c1 Ñ ΔG{N p 9eq “ 1 and c2 Ñ 1. Consequently,

}TN puq} ď |K| ¨ }u}ApGq.

�

Lemma 6.4.4. Let S P V NpGq with compact support and let f, ϕ, ψ P ApGq X

CcpGq. Then

pf ˚ Sqpxqpϕ ˚ ψqpxq “

ż

G

rpfRzϕq ˚ pRzψ ¨ S q spxqdz

for all x P G.

Proof. Since right translation on ApGq is continuous, the map

z Ñ pLxfqRzpLxϕqRzψ

from G into ApGq is continuous with compact support. Therefore,

upxq “

ż

G

pLxfqRzpLxϕqpRzψq dz

defines an element of ApGq. Now,

pf ˚ Sqpxqpϕ ˚ ψqpxq “

Bż

G

pLxfqRzpLxϕqpRzψq dz, S

F

“

ż

G

xpLxfqRzpLxϕqpRzψq, Sy dz

“

ż

G

xLxpfRzϕq, pRzψq ¨ Sy dz

“

ż

G

rLxpfRzϕq ˚ pRzψ ¨ Sqspeqdz

“

ż

G

rpfRzϕq ˚ pRzψS q spxqdz.

�


Let N be a closed normal subgroup of G and let S P V NpGq with compactsupport. Let w,w1 P ApGqXCcpGq such that w “ 1 and w1 “ 1 on a neighbourhoodof suppS, and choose v P ApGqXCcpGq such that v “ 1 on suppwYsuppw1. Then,for every u P ApG{Nq,

xS,wpu ˝ qqy “ xS, pvwqpu ˝ qqy “ xw ¨ S, vpu ˝ qqy

“ xw1¨ S, vpu ˝ qqy “ xS,w1

pu ˝ qqy.

Moreover,

|xS,wpu ˝ qqy| ď }S} ¨ }w}ApGq}u ˝ q}BpGq “ }S} ¨ }w}ApGq ¨ }u}ApG{Nq.

Thus, if w “ 1 on a neighbourhood of suppS, then xS,wpu ˝ qqy does not dependon the choice of w and u Ñ xS,wpu ˝ qqy defines a bounded linear operator onApG{Nq, denoted φN pSq.

Proposition 6.4.5. Let G be an amenable locally compact group, N a closednormal subgroup of G and q : G Ñ G{N the quotient homomorphism. Then there

exists a (unique) contraction φN : UCp pGq Ñ UCpzG{Nq with the following property.If T P V NpGq has compact support and w P ApGq X CcpGq is such that w “ 1 ona neighbourhood of supp T , then

xφN pT q, uy “ xT,wpu ˝ qqy

for all u P ApG{Nq. Moreover φN satisfies:

(1) if S has compact support, then suppφN pSq Ď qpsuppSq;(2) for f P L1pGq, φN pλGpfqq “ λG{N pTN pfqq.

Proof. By the considerations above, we can define a linear map φN from

UCcp pGq into V NpG{Nq. Now, since G is amenable, for every compact subset K ofG and ε ą 0, there exists w P ApGq XCcpGq such that w “ 1 on K and }w} ď 1` ε.This implies that, for any ε ą 0,

}φN pT q} “ supt|xT,wpu ˝ qqy| : }u}ApG{Nq ď 1u ď p1 ` εq}T }.

Thus φN is a contraction on UCcp pGq and hence extends uniquely to a contraction,

also denoted φN , of UCp pGq into V NpGq.

We next show (1) for S P UCcp pGq. To verify this, let a P G be such that9a P suppφN pSq and, towards a contradiction, assume that aN X suppS “ H.Choose a compact neighbourhood V of suppS such that V N X aN “ H. ThenV N is closed in G and a R V N , and hence there exists u P ApGq XCcpGq such thatu ě 0, upaq ‰ 0 and u “ 0 on V N . Then TN puqp 9aq ą 0 and TN puqp 9xq “ 0 for all9x P qpV q. For every v P ApG{Nq X CcpG{Nq, it follows

xTN puq ¨ φN pSq, vy “ xφN pSq, TNpuqvy “ xS,wppTN puqvq ˝ qqy

“ xS,wpTN puq ˝ qqpv ˝ qqy “ 0,

since TN puq ˝ q vanishes on the neighbourhood q´1pqpV qq of suppS. Thus TN puq ˝

φN pSq “ 0 and hence TN puqp 9aq “ 0 by Proposition 2.5.3. This contradiction provesthat suppφN pSq Ď qpsuppSq.

It remains to show (2). For f P L1pGq and v P ApG{Nq, we have

xλG{N pTNfq, vy “

ż

G{N

TNfp 9xqvp 9xqd 9x “

ż

G

fptqpv ˝ qqptqdt.


On the other hand, if f P CcpGq and w “ 1 on a neighbourhood of supp f “

suppλGpfq, then

xφN pλGpfqq, vy “ xλGpfq, wpv ˝ qqy “

ż

G

fptqpv ˝ qqptqdt.

Continuity now implies (2). �

Theorem 6.4.6. Let G be a locally compact group and N a closed normalsubgroup of G. Let q : G Ñ G{N denote the quotient homomorphism and let E bea closed subset of G{N . Then E is a local spectral set for ApG{Nq if and only ifq´1pEq is a local spectral set for ApGq.

Proof. Suppose first that q´1pEq is a local spectral set for ApGq. Recall thatwe have to show xS, uy “ 0 for every u P IpEq X CcpG{Nq and S P V NpG{Nq withcompact support contained in E.

There exists w P ApGq X CcpGq such that φN pwq “ 1 on a neighbourhood ofsupp S. It follows from Lemma 6.4.1 that the assignment f Ñ xS, φN pwfqy, f P

ApGq, defines a bounded linear functional of ApGq. Thus there exists T P V NpGq

such that xT, fy “ xS, φN pwfqy for all f P ApGq. Then T has compact supportand supp T Ď q´1pEq. To verify this, assume first that f P ApGq vanishes in aneighbourhood of supp w. Then wf “ 0 and hence xT, fy “ xS, φN pwfqy “ 0.This shows that supp T Ď supp w, which is compact. Now, let f P ApGq X CcpGq

be such that f “ 0 in a neighbourhood of q´1pEq. Then, since supp f X q´1pEq “

H and supp f is compact, there exists a neighbourhood W of e in G such thatsupp f X Wq´1pEq “ H. It follows that φN pfq vanishes in a neighbourhood of E,an d this implies that xT, fy “ xS, φN pwqφN pfqy “ 0 since supp S Ď E.

Now choose v P ApGq X CcpGq such that v “ 1 on a neighbourhood of supp T .Then, since v ¨ T “ T , φN pwq ¨ S “ S and q´1pEq is a set of local synthesis,

xS, uy “ xφN pwq ¨ S, uy “ xS, φN pwquy

“ xS, φN pwpu ˝ qqqy “ xT, u ˝ qy

“ xv ¨ T, u ˝ qy “ xT, vpu ˝ qqy “ 0,

as was to be shown.For the converse, let E be a local spectral set for ApG{Nq. We have to show

that if u P Ipq´1pEqq X CcpGq and S P UCp pGq such that suppS Ď q´1pEq, thenxS, uy “ 0. Note first that we can assume suppS to be compact. In fact, letv P ApGq X CcpGq be such that v “ 1 on supp u and observe that xv ¨ S, uy “

xS, vuy “ xS, uy and supppv ¨ Sq Ď supp v X suppS.There exist nets pwαqα and pvαqα in ApGq X CcpGq such that the family of

functions pwα ˚ vαq |N forms an approximate identity for L1pNq. From Lemma6.4.3 we get

ż

N

pu ˚ Sqpnqpwα ˚ vαqpnq dn “

ż

N

„ż

G

pu ¨ Rzwαq ˚ puα ¨ LzS q pnqdz

j

dn

“

ż

G

„ż

N

pu ¨ Rzwαq ˚ puα ¨ LzS q pnqdn

j

dz.

Notice next that the last inner integral equals

φN pu ¨ Rzwαq ˚

”

Δ´1G{NφN pΔ´1

G uα ¨ LzSq

ı

p 9eq.


In fact, for every v P ApGq X CcpGq and T P UCp pGq,

φN pvq ˚

”

Δ´1G{NφN pΔ´1

G T q

ı

p 9eq “ xφN pvq, φN pT qy

““

pφN pvq ¨ qq ˚ T‰

p 9eq

“ xφN pvq ¨ q, T y

“

Bż

N

Lnv dn, T

F

“

ż

N

xLnv, T ydn

“

ż

N

pv ˚ T qpnqdn.

Now φN pu¨RzwαqPApG{NqXCcpG{Nq vanishes on E and the operator φN ppΔ´1G uαq¨

LzSq P V NpG{Nq has compact support contained in E. Since E is local spectralset, it follows that

ż

N

pu ¨ Rzwαq ˚ puα ¨ LzS q pnq dn “ 0

for each α. Because u˚ S is continuous on N and ppwα ˚ vαq |N qα is an approximateidentity for L1pNq, we conclude

xu, Sy “ pu ˚ Sqpeq “ limα

ż

N

pu ˚ Sqpnqpwα ˚ vαqpnq dn “ 0.

This completes the proof of the theorem. �

Corollary 6.4.7. Let G be a locally compact group and N a closed normalsubgroup of G. Suppose that both, ApGq and ApG{Nq, have approximate identities.Then a closed subset E of G{N is a set of synthesis for ApG{Nq if and only ifq´1pEq is a set of synthesis for ApGq. The conclusion in particular holds if G isamenable.

Proof. The statement follows from Theorem 6.4.6 and Lemma 5.1.1. �

Let f, g, ϕ and ψ be functions in L2pGq. Then it is straightforward to verifythat

pf ˚ gqpxqpϕ ˚ ψqpxq “

ż

G

ppfRzϕq ˚ pRzψ ˚ gq q pxq dz

for all x P G. The formula presented in the next lemma may be viewed as ananalogue.

6.5. Bounded approximate identities I: Ideals

Let G be a locally compact abelian group. The closed ideals in the groupalgebra L1pGq have been completely described by Liu, van Rooij and Wang [193]

in terms of the closed sets in the coset ring of the dual group pG of G. The mainpurpose of this section is to present a generalization to Fourier algebras ApGq ofamenable locally compact groups G. Amenability seems to be an indispensablehypothesis because if G is any locally compact group, then ApGq has a boundedapproximate identity if and only if G is amenable (Theorem 2.7.2). We are goingto prove that a closed ideal I of ApGq with bounded approximate identity is ofthe form I “ IpEq and characterize the closed subsets E of G “ σpApGqq which

6.5. BOUNDED APPROXIMATE IDENTITIES I: IDEALS 221

determine ideals with bounded approximate identities (Theorem 6.5.11). Our firstgoal is to show that this class of subsets of G contains all closed subgroups of G(Theorem 6.5.5). This will be accomplished using the operator space structure ofApGq. We shall also address the problem of finding the best possible norm bound.

Lemma 6.5.1. Let H be an amenable locally compact group. Then there existsa completely contractive projection P : BpL2pHqq Ñ V NpHq.

Proof. First recall that V NpHq is the commutant of the set tρHpxq : x P Hu.Let m be a left invariant mean on L8pHq. It will be convenient for us to regard mas a finitely additive measure on H. Now define P : BpL2pHqq Ñ BpL2pHqq to bethe weak operator converging integral

P pT q “

ż

H

ρHpsqTρHpsq˚dmpsq

for each each T P BpL2pHqq. That is,

xP pT qf, gy “ mps Ñ xρHpsqTρHpsq˚f, gy

for all f, g P L2pHq.From the invariance of m, it is easy to see that for each T P BpL2pHqq,

ρHptqP pT q “ P pT qρHptq

for all t P H. It follows that P pT q P V NpHq. It is also clear that if T P V NpHq,then P pT q “ T . We have that P is completely positive, since each amplification

P pnq : MnpBpL2pHqqq Ñ MnpV NpHqq

is given by the weak operator converging integral

P pnqprTijsq “

ż

H

diagpρHpsqqrTijsdiagpρHpsqq˚dmpsq,

for every rTijs P MnpBpL2pHqqq, where diagpρHpsqq denotes the n ˆ n diagonalmatrix with all diagonal entries equal to ρHpsq. Finally, since P pIq “ I, we getthat P is completely contractive. �

Assume that H is an amenable group. We claim that if M is a von Neumannalgebra on a Hilbert space E such that M is ˚-isomorphic to V NpHq, then there

exists a contractive completely positive expectation rP : BpEq Ñ M. To see this,let Φ : V NpHq Ñ M be a ˚-isomorphism. The Arveson-Wittstock ExtensionTheorem [70, Theorem 4.1.5] implies that Φ´1 : M Ñ V NpHq admits a completelycontractive extension Ψ : BpEq Ñ BpL2pHqq. Moreover, since ΨpIq “ I, Ψ is

completely positive [70, Corollary 5.1.2]. We now let rP “ Φ ˝ P ˝ Ψ, where P isthe projection constructed in Lemma 6.5.1.

Proposition 6.5.2. Let G be a locally compact group and H an amenableclosed subgroup of G. Then there exists a completely contractive projection fromV NpGq onto IpHqK.

Proof. We recall that V NHpGq is ˚-isomorphic to V NpHq. It follows fromLemma 6.5.1 and the discussion preceding this proposition that there exists a com-pletely contractive projection

rP : BpL2pGqq Ñ V NHpGq “ IpHq

K.

We now simply restrict rP to V NpGq. �


Lemma 6.5.3. Let A be a commutative normed algebra with approximate iden-tity of norm bound 1, and let C ą 0. Then, for any closed ideal I of A, the followingtwo conditions are equivalent.

(i) I has an approximate identity with norm bound C.(ii) There exists a projection P from A˚ onto IK commuting with the action

of A on A˚ such that }I ´ P } ď C.

Proof. (i) ñ (ii) Let BpA˚q denote the space of all bounded linear operatorsT : A˚ Ñ A˚ equipped with the w˚-operator topology. Thus a net pTαqα in BpA˚q

converges to T P BpA˚q if and only if xTαpϕq, xy Ñ xT pϕq, xy for every ϕ P A˚ andx P A. Then, for any r ą 0, the ball of radius r in BpA˚q is compact.

Now let puαqα be an approximate identity for I such that }uα} ď C for all α.For each α, define Pα : A˚ Ñ A˚ by

xPαpϕq, xy “ xϕ, x ´ uαxy, x P A,ϕ P A˚.

It is straightforward to check that, for each ϕ P A˚,

}Pα} ď 1 ` C and }I ´ Pα} ď C.

Thus, passing to a subnet if necessary, we can assume that Pα Ñ P for someP P BpA˚q in the w˚-operator topology. Then }I ´ P } ď C and, for x P I andϕ P A˚,

xP pϕq, xy “ limα

xϕ, x ´ uαxy “ 0,

so that P pA˚q Ď IK, and conversely, if ϕ P IK, then xP pϕq, xy “ xϕ, xy for allx P A, whence P |IK is the identity. Moreover, P 2 “ P since, for any ϕ P A˚ andx P A,

xP 2pϕq, xy “ lim

αxP pϕq, x ´ uαxy “ xP pϕq, xy

because uαx P I and P pϕq P IK. Finally, P commutes with the action of A on A˚.Indeed,

xP pa ¨ ϕq, xy “ limα

xa ¨ ϕ, x ´ uαxy “ limα

xϕ, ax ´ uαaxy

“ limα

xPαpϕq, axy “ xP pϕq, axy

“ xa ¨ P pϕq, xy

for every x P A and ϕ P A˚.(ii) ñ (i) It suffices to show that given u P I and ε ą 0, there exists v P I such

that }v} ď C and }u ´ uv} ď ε. Fix u and ε and, for any finite nonempty subset Fof A˚, let

IpF, εq “ tv P I : }v} ď C, |xϕ, uy ´ xϕ, uvy| ď ε for all ϕ P F u.

We are going to show that IpF, εq ‰ H for each such F and ε ą 0. Observe firstthat there exists w P A with }w} ď 1 and

}u ´ uw} ďε

2maxt}ϕ} : ϕ P F u.

Now define f : I˚ Ñ C by setting, for ψ P I˚,

xf, ψy “ xϕ,wy ´ xP pϕq, wy,

where ϕ P A˚ is such that ϕ|I “ ψ. Note that f is well defined since P projectsonto IK. Moreover, }f} ď C since }w} ď 1, }I ´ P } ď C and f or every ψ P I˚


there exists ϕ P A˚ such that ϕ|I “ ψ and }ϕ} “ }ψ}. Since the ball of radius C inI is w˚-dense in the ball of radius C in I˚˚, we find v P I such that }v} ď C and

|xpu ¨ ϕq|I , vy ´ xf, pu ¨ ϕq|Iy| ďε

2

for all ϕ P F . By definition of f and since P pu ¨ ϕq “ u ¨ P pϕq, this means that

|xϕ, uvy ´ xϕ, uwy ` xP pϕq, uwy| ďε

2

for all ϕ P F . By the choice of w and since uw P I and P pϕq P IK, it follows that,for each ϕ0 P F ,

|xϕ0, uy ´ xϕ0, uvy| ď |xϕ0, uy ´ xϕ0, uwy|

` |xϕ0, uwy ´ xϕ0, uvy ´ xP pϕ0q, uwy|

ď }ϕ0} ¨ε

2maxt}ϕ} : ϕ P F u`

ε

2ď ε.

This shows that IpF, εq ‰ H.We have seen so far that u lies in the w˚-closure of the set

J “

ď

tu ¨ IpF, εq : F Ď A˚ finite, ε ą 0u.

It follows that u lies in the norm closure in A of the convex hull of J . Therefore,given ε ą 0, there exist n P N, finite subsets F1, . . . , Fn of A˚, c1, . . . , cn ą 0,ε1, . . . , εn ą 0 and v1, . . . , vn P I such that

vj P IpFj , εjq, 1 ď j ď n,nÿ

j“1

cj “ 1, and

›

›

›

›

›

u ´

nÿ

j“1

cjuvj

›

›

›

›

›

ď ε.

Then the element v “řn

j“1 cjvj of I satisfies }v} ď C and }u ´ uv} ď ε. Thus (i)holds. �

Proposition 6.5.4. Let G be an amenable locally compact group and H aclosed subgroup of G. Then there exists a unital completely positive projectionP from V NpGq onto V NHpGq “ IpHqK such that P pu ¨ T q “ u ¨ P pT q for allT P V NpGq and u P ApGq.

Proof. It follows from Proposition 6.5.2 that there exists a projection Q :V NpGq Ñ IpHqK with }Q}cb “ }QpIq} “ }I} “ 1. Since G is amenable, the opera-tor space projective tensor product ApGq pbApGq admits an approximate diagonalof norm bounded by 1. Now Lemma 6.5.3 applies taking A “ ApGq, A˚ “ V NpGq

and Y “ V NHpGq “ IpHqK, now shows that there exists a completely contractiveprojection P : V NpGq Ñ IpHqK that commutes with the module action of ApGq

on V NpGq. Since P pIq “ I, P is also completely positive. �By Theorem 3.2.5, the preceding proposition applies to the Fourier algebra of

any amenable locally compact group.

Theorem 6.5.5. Let G be an amenable locally compact group and H a closedsubgroup of G. Then the ideal IpHq has an approximate identity with norm boundedby 2.

Proof. By Proposition 6.5.4 there is a completely contractive projection Pfrom V NpGq onto V NHpGq “ IpHqK, which commutes with the action of ApGq onV NpGq. We can now apply Lemma 6.5.3 to conclude the existence of an approxi-mate identity of IpHq with norm bounded by 2. �


The following standard facts will be used in the sequel. Let I and J be closedideals of a commutative Banach algebra A. Then

(1) If I and J both have bounded approximate identities, then so do I XJ andI ` J .

(2) If I and A{I have bounded approximate identities, then A has a boundedapproximate identity.

Lemma 6.5.6. Let G be an amenable locally compact group and let H and Kbe closed subgroups of G such that K Ď H and K is open in H. Then, for anya, b P G, the ideal IpaHzbKq has a bounded approximate identity.

Proof. Note first that aHzbK “ apHza´1bKq and Hza´1bK ‰ H only ifa´1b P H, and in this case Hza´1bK “ a´1bpHzKq. It therefore suffices to showthat IpHzKq has a bounded approximate identity. Let

J “ tu P ApHq : upxq “ 0 for all x P HzKu.

Since K is open in H, the map u Ñ u|K is an isometric isomorphism from J ontoApKq. As K is amenable, ApKq has a bounded approximate identity, and hence sodoes J . Let r : ApGq Ñ ApHq denote the restriction map. Then IpHzKq “ r´1pJq

and therefore IpHzKq has a bounded approximate identity provided that IpHq andr´1pJq{IpHq have bounded approximate identities. For IpHq this is guaranteed byTheorem 6.5.5. Now the map φ : u ` IpHq Ñ u|H is an isometric isomorphismbetween ApGq{IpHq and ApHq. Since J has a bounded approximate identity, sodoes φ´1pJq “ r´1pJq{IpHq, as required. �

Proposition 6.5.7. Let G be an amenable locally compact group and E P

RcpGq. Then

(i) E is a Ditkin set for ApGq.(ii) The ideal IpEq has a bounded approximate identity.

Proof. Note first that if F is a closed subset ofG such that IpF q has a boundedapproximate identity and F is a set of synthesis, then IpF q has an approximateidentity consisting of elements in jpF q. Consequently, F is a Ditkin set.

Assume first that E is of the form E “ aHzŤm

j“1 cjKj , where H is a closedsubgroup of G, m P N0, a, cj P G and the Kj are open subgroups of H, 1 ď j ď m.Then E is a set of synthesis. Indeed, this follows from Theorem 6.1.11 since aH isa set of synthesis (Corollary 6.1.10), H is a Ditkin set and E is open and closed inaH. To see that IpEq has a bounded approximate identity, consider the ideal

J “řm

j“1 IpaHzcjKjq

of ApGq. It follows from Lemma 6.5.6 and fact (1) mentioned prior to that lemmathat J has a bounded approximate identity. Now

ZpJq “Şm

j“1paHzcjKjq “ E,

and hence J “ IpEq since E is a set of synthesis. By the remark at the outset ofthe proof, E is a Ditkin set.

Next, let E “Ťn

i“1 Ei, where each Ei is of the above form. We prove byinduction on n that E is a Ditkin set and IpEq has a bounded approximate identity.The case n “ 1 having been dealt with in the preceding paragraph, let n ě 2and set F “

Ťn´1i“1 Ei, and suppose that (i) and (ii) hold for F . Clearly, then

IpEq “ IpF q X IpEnq has a bounded approximate identity. Moreover, En and F


are both Ditkin sets. Since the union of two Ditkin sets is a Ditkin set (Theorem6.1.12), E “ En Y F is a Ditkin set as well. �

Lemma 6.5.8. Let J be a closed ideal of ApGq and suppose that J has anapproximate identity puαqα such that }uα}ApGq ď C for all α. Then 1ZpJq and1GzZpJq belong to BpGdq. Moreover, }1GzZpJq}BpGdq ď C and }1ZpJq}BpGdq ď C`1.

Proof. Let u P J . Then

}uαu ´ u}8 ď }uαu ´ u}ApGq Ñ 0.

Thus puαqα converges to 1GzZpJq pointwise. Since the set tv P BpGdq : }v} ď Cu isclosed in the topology of pointwise convergence,

1GzZpJq P BpGdq and }1GzZpJq}BpGdq ď C.

Consequently, 1ZpJq “ 1G ´ 1GzZpJq P BpGdq and }1ZpJq}BpGdq ď C ` 1. �

Corollary 6.5.9. Let J be a closed ideal of ApGq with bounded approximateidentity. Then ZpJq P RcpGq.

Proof. By the preceding lemma, 1ZpJq P BpGdq and hence ZpJq P RpGdq byHost’s idempotent theorem. Since ZpJq is closed in G, ZpJq P RcpGq. �

Remark 6.5.10. As shown in [91], the conclusion of Corollary 6.5.9 holds un-der a somewhat weaker hypothesis on the approximate identity of the ideal J . Toexplain this, we recall the following definition. Let A be a commutative Banachalgebra with structure space ΔpAq. A net peαqα is called a bounded Δ-weak approx-imate identity if xϕ, eαa ´ ay Ñ 0 for every ϕ P ΔpAq and there exists a constantC ą 0 such that }eα} ď C for all α.

Now suppose that J is a closed ideal of ApGq possessing a bounded Δ-weakapproximate identity puαqα. Then the bounded net puαqα in BpGdq convergespointwise to the characteristic function 1GzZpJq. It follows from Corollary 3.2.3that 1GzZpJq P BpGdq and hence 1ZpJq P BpGdq. Then apply Cohen’s theorem asin the proof of Corollary 6.5.9.

Theorem 6.5.11. Let G be an amenable locally compact group and let I be aclosed ideal of ApGq. Then I has a bounded approximate identity if and only ifI “ IpEq for some subset E of G of the form

E “Ťn

i“1

´

aiHizŤmi

j“1 bijKij

¯

,

where ai, bij P G, Hi is a closed subgroup of G and Kij is an open subgroup of Hi

(n,mi P N0, 1 ď i ď n, 1 ď j ď mi).

Proof. If E is of the indicated form (equivalently, E P RcpGq), then IpEq hasa bounded approximate identity by Proposition. 6.5.7(ii). Conversely, let I be aclosed ideal of ApGq with bounded approximate identity and let E “ ZpIq. ThenE P RcpGq by Corollary 6.5.9 and E is a set of synthesis by Proposition 6.5.7(i).It follows that I “ IpEq and E is of the stated form. �

Corollary 6.5.12. Let G be an amenable locally compact group and I a closedideal of ApGq. Then I is operator amenable if and only if I “ IpEq for someE P RcpGq.


Proof. Since ApGq is an operator amenable completely contractive Banachalgebra, a closed ideal I of ApGq is operator amenable precisely when I has abounded approximate identity. Thus the statement follows from Theorem 6.5.11.

�

We now turn to the question of whether 2 is the best possible norm bound foran approximate identity of IpHq when H is a closed subgroup of G.

Lemma 6.5.13. Let H be a nonopen closed subgroup of G. Let P : V NpGq Ñ

IpHqK be a projection, and suppose that there exists a bounded net puαqα in BpGq

such that

xuα ¨ T, uy Ñ xP pT q, uy

for all T P V NpGq and all u P ApGq. Then P pT q “ 0 for all T P C˚λ pGq.

Proof. To establish the lemma, it suffices to prove that P pλGpgqq “ 0 forevery g P CcpGq. Since H is nonopen, |H| “ 0, and hence given ε ą 0, we find anopen subset V of G containing H X supp g such that |V | ď ε. Let f “ g ¨ 1GzV .Then

}P pλGpgqq ´ P pλGpfqq} ď }P } ¨ }λGpg ´ fq}

ď }P } ¨ }g ´ f} ď }P } ¨ |V | ¨ }g}8

ď ε}g}8}P }.

We now show that P pλGpfqq “ 0. Let K “ supp g X pGzV q, a compact subset ofGzH. Then

ż

K

pfuqpxquαpxqdx “

ż

G

pfuqpxquαpxqdx “ xuα ¨ λGpfq, uy Ñ xP pλGpfqq, uy

for all u P ApGq. Since ApGq is regular, there exists v P ApGq such that vpxq “ 0 forall x P H and vpxq “ 1 for all x P K. Then, since vu P IpHq and P pλGpfqq P IpHqK,

ż

K

pfuqpxquαpxqdx “

ż

K

fpxqpvuqpxquαpxqdx Ñ xP pλGpfqq, vuy “ 0.

Thus xP pλGpfqq, uy “ 0 for all u P ApGq, as required. �

Lemma 6.5.14. Let IL2pGq denote the identity of BpL2pGqq. There exists T P

C˚λ pGq such that }T } “ 1 and }IL2pGq ´ 2T } “ 1.

Proof. Choose any f P CcpGq, f ‰ 0, with f “ f˚, and let A (respec-tively, B) denote the closed subalgebra of BpL2pGqq generated by λGpfq and IL2pGq

(respectively, λGpfq). Then A and B are C˚-algebras, ΔpBq Ď ΔpAq and theGelfand homomorphisms are isometric isomorphisms between A and CpΔpAqq (re-spectively, B and C0pΔpBqq). Now, take any g P CpΔpAqq such that }g}8 “ 1,

supp g Ď ΔpBq and 0 ď g ď 1. Let T P B Ď C˚λ pGq such that pT “ g. Then }T } “ 1

and }IL2pGq ´2T } “ 1 since pIL2pGq ´2T qp“ 1ΔpAq ´2g and }1ΔpAq ´2g}8 “ 1. �

Theorem 6.5.15. Let G be a locally compact group and H a nonopen closedsubgroup of G. Then 2 is the best possible norm bound for an approximate identityif the ideal IpHq.

Proof. Towards a contradiction, assume that there exists an approximateidentity pvαqα in IpHq such that, for some constant c ă 2, }vα} ď c for all α. Afterpassing to a subnet if necessary, we may assume that the net pvαqα converges in


the w˚-topology of V NpGq˚. This gives rise to a projection P from V NpGq ontoIpHqK such that

xp1 ´ vαq ¨ T, uy Ñ xP pT q, uy

for all T P V NpGq and all u P ApGq. Lemma 6.5.13 now yields that P pC˚λ pGqq “

t0u.Let I denote the identity of BpV NpGqq. By Lemma 6.5.14, there exists T P

C˚λ pGq such that }T } “ 1 and }IL2pGq ´ 2T } “ 1. Since P pT q “ 0, it follows that

pI ´ P qpIL2pGq ´ 2T q “ ´2T.

Since }T } “ 1, we conclude that }I ´ P } ě 2. Thus, since c ă 2, there existS P V NpGq and u P ApGq such that }S} ď 1, }u} ď 1 and

|xS ´ P pSq, uy| ě 1 `c

2.

Then |xvα ¨ S, uy| ą c for sufficiently large α since

xp1 ´ vαq ¨ S, uy Ñ xP pSq, uy.

On the other hand, we have

|xvα ¨ S, uy| ď }vα} ¨ }S} ¨ }u} ď c.

This contradiction shows that 2 is indeed the best possible norm bound. �

We conclude this section by discussing the problem of when for a closed subsetE of G, the ideal IpEq has an approximate identity with norm bound 1. In viewof the preceding result, this norm condition appears to be very restrictive. Thefollowing proposition identifies these sets E.

Proposition 6.5.16. Let G be any locally compact group and E a closed subsetof G. Then the ideal IpEq has an approximate identity with norm bound 1 if andonly if E “ GzaH, where a P G and H is an amenable open subgroup of G.

Proof. Suppose first that E is of the indicated form. Then, since H isamenable, ApHq has an approximate identity with norm bound 1. On the otherhand, since H is open, the map v Ñ rv, where for v P ApHq, rv denotes the triv-ial extension of v to all of G, is an isometric isomorphism between ApHq andIpGzHq “ Ipa´1Eq. So Ipa´1Eq has an approximate identity of norm bound 1,and hence so does IpEq.

Conversely, assume that IpEq has an approximate identity with norm bound1. Then, as in the proof of Lemma 6.5.8, it follows that 1GzE P BpGdq and

}1GzE}BpGdq “ 1. Fix any a P GzE and let H “ a´1pGzEq. Then e P H and}1H}BpGdq “ 1, and these facts together imply that H is a subgroup of G. In-deed, this can be seen as follows. There exist a unitary representation π of Gd andξ, η P Hpπq such that 1Hpxq “ xπpxqξ, ηy for all x P Gd and }1H}BpGdq “ }ξ} ¨ }η}.So xξ, ηy “ }ξ} ¨ }η} and hence η “ ξ. Consequently, 1H is a positive definite func-tion, and this readily implies that H is a subgroup of G. Since E is closed, H isopen. Now, using again that ApHq is isometrically isomorphic to IpEq, we concludethat ApHq has a bounded approximate identity, whence H must be amenable. �

In case E is a closed subgroup of G, E “ H say, Proposition 6.5.16 shows thatIpHq has an approximate identity bounded by 1 if and only if H has index two inG and G is amenable.


6.6. Bounded approximate identities II

Let G be an amenable locally compact group and H a closed normal subgroupof G, and let q : G Ñ G{H denote the quotient homomorphism. We establish theexistence of a bijection e : I Ñ epIq between the set of closed ideals I of ApG{Hq

and the set of all closed ideals of ApGq which are left and right invariant undertranslation by elements of H. We start with the definition of epIq. For a closedideal I of ApG{Hq, put

epIq “ spantupv ˝ qq : u P ApGq, v P Iu.

Then epIq is a closed ideal of ApGq. Moreover, epIq is left and right H-translationinvariant since Lhpupv ˝ qqq “ pLhuqpv ˝ qq, and similarly for Rh, h P H.

Lemma 6.6.1. Let I be a closed ideal of ApGq such that Rhu P I for every u P Iand h P H. Then φHpv ¨ T q P rpIqK for every v P ApGq and T P IK.

Proof. It suffices to consider v P ApGq X CcpGq. Choose w P ApGq X CcpGq

such that w “ 1 on a neighbourhood of supp v Ě supppv ¨ T q. Then, for anyu P I X CcpGq

xφHpv ¨ T q, THuy “ xv ¨ T,wpTHu ˝ qqy “ xT, vpTHu ˝ qqy

“ xT, v ¨

ż

H

Rhu dhy

“

ż

H

xT, v ¨ Rhuy dh “ 0,

since T P IK and I is H-right translation invariant. Since I X CcpGq is dense in Iand rpIq is the closure of the set of all elements of the form Thu, u P I X CcpGq, itfollows that xφHpv ¨ T q, wy “ 0 for every w P rpIq. �

Remark 6.6.2. If H is a closed normal subgroup of G, THpApGq XCcpGqq is adense ideal in ApG{Hq. Indeed, since v¨THu “ THpupv˝qqq for any u P ApGqXCcpGq

and v P ApG{Hq, THpApGq X CcpGqq is an ideal in ApG{Hq. Clearly, for everyx P G there exists u P ApGq X CcpGq such that THup 9xq ‰ 0. Thus the hull ofTHpApGq XCcpGqq is empty and hence this ideal is dense in ApG{Hq since ApG{Hq

is Tauberian.

Lemma 6.6.3. Let I be a closed ideal of ApGq and T P V NpGq such thatφHpu ¨ T q P rpIqK for every u P ApGq. Then T P IK.

Proof. Since I X CcpGq is dense in I, it suffices to show that xT, uy “ 0 forevery u P I X CcpGq. Fix such a u and choose w P ApGq X CcpGq such that w “ 1in a neighbourhood of supp u Ě supppu ¨ T q. By hypothesis and the definition ofφH , we then have

0 “ xφHpu ¨ T q, Thvy “ xu ¨ T,wpTHv ˝ qqy

for all v P ApGqXCcpGq. Next, choose w1 P ApG{Hq such that w1 “ 1 on qpsuppuq.There exists a sequence pvnqv in ApGq XCcpGq such that THpvnq Ñ w1 in ApG{Hq.Then w1 ˝ q “ 1 on supp u and THpvnq ˝ q Ñ w1 ˝ q in BpGq. It follows that

xu ¨ T,wpTHpvnq ˝ qqy “ xT, uwpTHpvnq ˝ qqy “ xT, upTHpvnq ˝ qqy

Ñ xT, upw1˝ qqy “ xT, uy.

This shows that xT, uy “ 0. �

6.6. BOUNDED APPROXIMATE IDENTITIES II 229

For a closed ideal J of ApGq, let

rpJq “ qpJ X CcpGqq Ď ApG{Hq.

Then rpJq is a closed ideal of ApG{Hq. In fact, if u P J X CcpGq and v P ApG{Hq

and w P ApGq is such that w “ 1 on supp u, then pv ˝qqu “ pv ˝qHqwu P JŞ

CcpGq

and vqpuq “ qppv ˝ qquq.

Theorem 6.6.4. Let G be an amenable locally compact group and H a closednormal subgroup of G. Then the assignment e : I Ñ epIq is a bijection between theset of all closed ideals of ApG{Hq and the set of all H-translation invariant closedideals of ApGq. More precisely:

(i) For every closed ideal I of ApG{Hq, we have I “ rpepIqq.(ii) For every closed ideal J of ApGq with Rhu P J for all u P J and h P H,

we have J “ eprpJqq.

Proof. (i) For u P ApGq X CcpGq and v P I, we have upv ˝ qq P epIq X CcpGq

and

THpupv ˝ qqq “ vTHpuq P I.

Since the span of the elements on the left hand side is dense in rpepIqq, is follows thatrpepIqq Ď I. For the reverse inclusion, let v P I XCcpG{Hq and choose w P ApG{Hq

such that w “ 1 on supp v. There exists a sequence punqn in ApGq X CcpGq suchthat THpunq Ñ w in ApG{Hq. Then

v “ vw “ v ¨ limnÑ8

THpunq “ limnÑ8

pvTHpunqq

“ limnÑ8

THpunpv ˝ qqq P rpepIqq.

Since I X CcpG{Hq is dense in I, we infer that I Ď rpepIqq.(ii) Taking rpIq in (i), we have rpeprpIqqq “ rpIq. Moreover, I and eprpIqq are

both right H-translation invariant. Therefore, we only need to verify that if I and Jare two closed, right H-translation invariant ideals of ApGq such that rpIq “ rpJq,then I “ J . For that, it suffices to observe that the annihilators coincide. LetT P IK, then φHpu ¨ T q P rpIqK “ rpJqK by Lemma 6.6.1 for every u P ApGq.Lemma 6.6.3 implies T P JK. Thus IK Ď JK. �

We are now turning to the second issue of this section. We are going to showthat an ideal I of ApG{Hq has a bounded approximate identity (with bound C ą 0)if and only if the ideal epIq of ApGq has a bounded approximate identity (with boundC).

Lemma 6.6.5. Let G be an amenable locally compact group, H a closed normalsubgroup of G and I a closed ideal of ApG{Hq. If I has an approximate identitybounded by c, then epIq has an approximate identity of norm bounded by c.

Proof. Let u P epIq and ε ą 0 be given. There exist u1, . . . , un P ApGq andv1, . . . , vn P I such that

›

›

›

›

›

u ´

nÿ

j“1

ujpvj ˝ qq

›

›

›

›

›

ApGq

ď ε.


Since G is amenable, there exists u1 P ApGq with }u1}ApGq ď 1 and›

›

›

›

›

nÿ

j“1

ujpvj ˝ qq ´ u1¨

nÿ

j“1

ujpvj ˝ qq

›

›

›

›

›

ApGq

ď ε.

Since a bounded approximate identity is a multiple approximate identity, thereexists v1 P I such that }v1}ApG{Hq ď c and, for 1 ď j ď n,

}vj ´ vjv1}ApG{Hq ď ε

˜

1 `

nÿ

i“1

}ui}ApGq

¸´1

.

Let w “ u1pv1 ˝ qHq P epIq. Then }w}ApGq ď }v1 ˝ q}BpGq ď c and

}u ´ uw}ApGq ď

›

›

›

›

›

u ´

nÿ

j“1

ujpvj ˝ qq

›

›

›

›

›

ApGq

`

›

›

›

›

›

nÿ

j“1

ujpvj ˝ qq ´ u1

nÿ

j“1

ujpvj ˝ qq

›

›

›

›

›

ApGq

`

›

›

›

›

›

pu1´ wq

nÿ

j“1

ujpvj ˝ qq

›

›

›

›

›

ApGq

`

›

›

›

›

›

wpu ´

nÿ

j“1

ujpvj ˝ qqq

›

›

›

›

›

ApGq

ď εp2 ` cq `

›

›

›

›

›

pu1´ wq

nÿ

j“1

ujpvj ˝ qq

›

›

›

›

›

ApGq

ď εp2 ` cq `

›

›

›

›

›

u1

nÿ

j“1

ujrpvj ˝ qq ´ pvj ˝ qqpv1˝ qqs

›

›

›

›

›

ApGq

ď εp2 ` cq `

nÿ

j“1

}uj} ¨ }vj ´ vjv1}ApG{Nq

ď εp3 ` cq.

Since ε ą 0 is arbitrary, the statement of the lemma follows. �

It is easy to verify that hpepIqq “ q´1phpIqq and RcpG{Hq Ď RcpGq. Therefore,it follows from the much more difficult Theorem 6.5.11 that the ideal epIq has abounded approximate identity whenever I has one. However, Theorem 6.5.11 doesnot provide any bound for the norm.

Remark 6.6.6. (i) Let J be a closed ideal of ApGq such that Lhu P J for everyu P J and h P H. Then Rhu P J for every u P J and h P H. This follows from (ii)

of the preceding theorem replacing J by the ideal rJ “ tru : u P Ju. In fact, since

pRhruqpxq “ uph´1x´1q “ pLh´1uqpx´1q “ ČLh´1upxq,

rJ is right H-translation invariant. Consequently, rJ “ eprp rJqq is two-sided H-translation invariant.


(ii) Let E be a closed subset of G{H. Then

epIpEqq “ Ipq´1pEqq and rpIpq´1

pEqqq “ IpEq.

The second equation follows from the first and Theorem 6.6.4(i). For the firstequation, if v P IpEq then upv˝qq vanishes on q´1pEq for every u P ApGq, and henceepIpEqq Ď Ipq´1pEqq. Conversely, if u P Ipq´1pEqq X CcpGq, then THu P IpEq.This shows that rpIpq´1pEqq Ď IpEq, and since Ipq´1pEqq is right H-translationinvariant, Theorem 6.6.4(i) yields

Ipq´1pEqq “ eprpIpq´1

pEqqq Ď epIq.

Corollary 6.6.7. Let G and H be as in Theorem 6.6.4 and let J be a closedideal of ApGq. Then eprpJqq coincides with the closed linear span of the set tRhu :u P J, h P Hu, which is the smallest H-translation invariant closed ideal of ApGq

containing J .

Proof. Let L denote the ideal in question. In view of Theorem 6.6.4, itsuffices to show that rpLq “ rpJq. For that, since J Ď L, it is enough to verify thatrpLq Ď rpJq.

Let v P LXCcpGq and ε ą 0 be given and choose a compact neighbourhood Kof supp v. By Lemma 6.4.3 there exists a constant c ą 0 such that }qpuq}ApG{Hq ď

c}u}ApGq for every u P ApGq with supp u Ď K. Since ApGq is regular, we findw P ApGq with w “ 1 on supp v, suppw Ď K and }w}ApGq ď 2 Corollary 3.2.6. Bydefinition of L there exist u1, . . . , un P J and h1, . . . , hn P H such that

›

›

›

›

›

v ´

nÿ

j“1

Rhjpujq

›

›

›

›

›

ApGq

ďε

2c.

For j “ 1, . . . , n, let wj “ ujRh´1j

pwq. Then

›

›

›

›

›

qpvq ´

nÿ

j“1

qpRhjpwqq

›

›

›

›

›

ApG{Hq

“

›

›

›

›

›

q

˜

v ´

nÿ

j“1

Rhjpwqq

¸›

›

›

›

›

ApG{Hq

ď c ¨

›

›

›

›

›

˜

v ´

nÿ

j“1

Rhjpujq

¸

w

›

›

›

›

›

ApGq

ď 2c ¨

›

›

›

›

›

v ´

nÿ

j“1

Rhjpujq

›

›

›

›

›

ApGq

ď ε.

Since qpRhjpwjqq P rpJq and ε ą 0 was arbitrary, we conclude that qpvq P rpJq.

This shows that rpLq Ď rpJq. �

Proposition 6.6.8. For ϕ P UCp pGq˚ and T P V NpGq, there exists a uniqueelement of V NpGq, denoted σGpϕqpT q, such that

xϕ, u ¨ T y “ xu, σGpϕqpT qy

for all u P ApGq. The map σG : ϕ Ñ σGpϕq is an isometric algebra isomorphism

form UCp pGq˚ onto HomApGqpV NpGqq. Moreover, for every φ P HomApGqpV NpGqq

and T P UCp pGq, we have

xσ´1G pφq, T y “ xφpT q, 1Gy.


Proof. It is clear that σG is a linear map and that }σGpϕq} ď }ϕ}. To showthat σG is actually an isometry and surjective, let φ P HomApGqpV NpGqq be given

and define ϕ P UCp pGq˚ by

xϕ, T y “ xφpT q, 1Gy, T P UCp pGq.

Then }ϕ} ď }φ} and, for u P ApGq and T P V NpGq, since u ¨ T P UCp pGq,

xu, σGpϕqpT qy “ xϕ, u ¨ T y “ xφpu ¨ T q, 1Gy,

“ xu ¨ φpT q, 1Gy, “ xφpT q, uy,

“ xu, φpT qy.

This shows that σGpϕq “ φ and }σGpϕq} ě }ϕ}. Since σG is bijective, it also follows

that σ´1G pφqpT q “ xφpT q, 1Gy, for every φ P HomApGqpV NpGqq and T P UCp pGq.

To show that σG is multiplicative, let ϕ, ψ P UCp pGq˚, T P V NpGq and u P

ApGq. Then, by definition of the product in UCp pGq˚ and since ψ ¨pu¨T q “ u¨pψ ¨T q,

xv, u ¨ σGpψqpT qy “ xvu, σGpψqpT qy “ xψ, pvuq ¨ T y

“ xvu, ψ ¨ T y “ xv, , u ¨ pψ ¨ T qy

for all v P ApGq, and hence

xu, σGpϕ ¨ ψqpT qy “ xϕ ¨ ψ, u ¨ T y “ xϕ, ψ ¨ pu ¨ T qy

“ xϕ, u ¨ pψ ¨ T qy “ xϕ, u ¨ pσGpψqpT qqy

“ xu, σGpϕqpσGpψqpT qqqy.

Thus σG is an algebra homomorphism. This completes the proof. �

Note that, for ϕ P UCp pGq˚ and T P UCp pGq, σGpϕqpT q “ ϕ ¨T , by definition ofϕ ¨ T and Proposition 6.6.8.

Lemma 6.6.9. The adjoint map φ˚H : UCpzG{Hq˚ Ñ UCp pGq˚ is an isometric

algebra homomorphism.

Proof. To start with, let T P V NpGq with compact support, and choosew P ApGq X CcpGq such that w “ 1 in a neighbourhood of suppT . Then, for allu, v P ApG{Hq,

xu ¨ φHpT q, vy “ xT,wpuv ˝ qqy “ xT,w2puv ˝ qqy

“ xwpu ˝ qq ¨ T,wpv ˝ qqy

“ xφH rwpu ˝ qq ¨ T s, vy,

since supprwpu ˝ qq ¨ T s Ď suppT . This implies, for every ψ P UCpzG{Hq˚ andu P ApG{Hq,

xu, φH rφ˚Hpψq ¨ T sy “ xφ˚

Hpψq ¨ T,wpu ˝ qqy

“ xφ˚Hpψq, wpu ˝ qq ¨ T y

“ xψ, φH rwpu ˝ qq ¨ T sy

“ xψ, u ¨ φHpT qy

“ xu, ψ ¨ φHpT qy.


For any two ϕ, ψ P UCpzG{Hq˚, we then obtain by the definition of multiplication

in UCpzG{Hq˚ and in UCp pGq˚,

xφ˚Hpϕq ¨ φ˚

Hpψq, T y “ xφ˚Hpϕq, φ˚

Hpψq ¨ T y “ xϕ, φH rφ˚Hpψq ¨ T sy

“ xϕ, ψ ¨ φHpT qy “ xϕ ¨ ψ, φHpT qy

“ xφ˚Hpϕ ¨ ψq, T y.

Since UCcpzG{Hq is dense in UCpzG{Hq, we conclude that φ˚H is an algebra homo-

morphism. �

Theorem 6.6.10. Let I be a closed ideal of ApG{Hq and suppose that thereexists a projection P from V NpG{Hq onto IK such that P commutes with theaction of ApG{Hq on V NpG{Hq. Then Q “ σG ˝ φ˚

H ˝ σ´1G{HpP q is a projection of

V NpGq onto epIqK.

Proof. Let T P V NpGq, v P ApG{Hq and u P ApGq X CcpGq. Then

xQpT q, upv ˝ qqy “ xσGrpφ˚Hpσ´1

G{HpP qqqpT qs, upv ˝ qqy

“ xφ˚Hpσ´1

G{HpP qq, upv ˝ qq ¨ T y

“ xσ´1G{HpP q, φHrupv ˝ qq ¨ T sy

“ xσ´1G{HpP q, pvTHuq ¨ φHpT qy

“ xP rpvTHuq ¨ φHpT qs, 1Gy,

“ xpvTHuq ¨ P pφHpT qq, 1Gy,

“ xP pφHpT qq, vTHuy.

This last term is zero if v P I since vTHu P I, P P HomApG{HqpV NpG{Hqq and P

maps into IK. Since epIq is the closed linear span of elements of the form upv ˝ qq,where v P I and u P ApGq X CcpGq, we conclude that QpT q P epIqK.

It remains to show thatQpT q “ T for every T P epIqK. For such T and u P ApGq

we have φHpu ¨ T q P rpepIqqK “ IK by Theorem 6.6.4 and Lemma 6.6.9 and henceP pφHpu ¨ T qq “ φHpu ¨ T q. On the other hand, using P P HomApG{HqpV NpG{Hqq,Propositions 6.6.8, 6.4.5 and choosing w P ApGq X CcpGq such that w “ 1 on aneighbourhood of supp u, we have for any v P ApG{Hq,

xupv ˝ qq, QpT qy “ xφ˚Hpσ´1

G{HpP qq, upv ˝ qq ¨ T y

“ xσ´1G{HpP q, φHpupv ˝ qq ¨ T qy

“ xP pφHpupv ˝ qq ¨ T qq, 1G{Hy,

“ P pv ¨ φHpu ¨ T qq, 1G{Hy,

“ xv ¨ P pφHpu ¨ T qq, 1G{Hy,

“ xv ¨ φHpu ¨ T q, 1G{Hy,

“ xφHpu ¨ T q, vy “ xu ¨ T,wpv ˝ qqy

“ xupv ¨ qq, T y.

Since every u P ApGq X CcpGq can be written as upv ˝ qq for some v P ApG{Hq, itfollows that QpT q “ T . �



Let G be a locally compact abelian group. The first example of a set of nonsyn-

thesis in pG “ σpL1pGqqq was given by L. Schwartz [258]. He proved that the unitsphere S2 Ď R3 “ σpL1pR3qq fails to be a set of synthesis by explicitly constructinga bounded linear functional on L1pR3q which annihilates jpS2q, but does not anni-hilate IpS2q. Actually, Schwartz’s proof works for Sd´1 Ď σpL1pRdqq for all d ě 3.In contrast, Herz [121] has shown that the circle S1 is a set of synthesis for L1pR2q.On the other hand, spectral synthesis holds for L1pGq whenever G is a compactabelian group. That compactness of G is necessary for spectral synthesis to hold forL1pGq, is the content of Malliavin’s celebrated theorem [207]. A somewhat moreaccessible proof of Malliavin’s theorem was given by Varopoulos [278] using tensorproduct methods. The ana logue of Malliavin’s theorem for general locally com-pact groups, which was presented in Section 6.3, was shown in [155]. The proofgiven here follows [155] and utilizes structure theory of locally compact groups,Malliavin’s theorem and a deep theorem of Zelmanov on the existence of infiniteabelian groups in infinite compact groups [293]. It seems to be unknown whether

there exist locally compact groups G at all, for which the condition u P uApGq isnot satisfied.

The fact that a closed subgroup of a locally compact groupG is a set of synthesisfor ApGq (Theorem 6.1.9), is due to Takesaki and Tatsuuma [272]. For amenableG, the same is true for the Figa-Talamanca-Herz algebras AppGq [123].

Section 6.4, which treats Lohoue’s projection theorem for local spectral sets, isan elaboration of the original article [196]. However, it should be pointed out thatLohoue accomplishes his result for Figa-Talamanca-Herz algebras AppGq, 1 ă p ă

8, rather than just Fourier algebras. In this general setting the proof turns out tobe technically more complicated.

Several authors have studied the problem of which closed ideals of ApGq havea bounded approximate identity ([92], [193], [154], [79], [80], [45] and [46]). Theinvestigations departed from Leptin’s theorem (Theorem 3.2.5) stating that ApGq

itself has a bounded approximate identity (actually, then one of norm bound 1)precisely when G is amenable. However, Proposition 6.5.16, which has been provedin [79], shows that proper closed ideals of ApGq rarely have approximate identitiesof norm bound 1. As shown in Theorem 6.5.5, if G is amenable then the idealIpHq admits an approximate identity with norm bound 2 for any closed subgroupH of G. Moreover, if H is nonopen, 2 is the best possible such bound (Theorem6.5.15). These results are due to Forrest and Spronk [92], and the first one evenholds for Figa-Talamanca-Herz algebras. The most general earlier result statingthe same ([154, Theorem 3.4]) required G to have the H-separation property (seeChapter 7 for the definition). As mentioned at the outset of Section 6.5, the closedideals in the group algebra L1pGq of a locally compact abelian group G have beencompletely described by Liu, van Rooij and Wang [193] in terms of the closed

sets in the coset ring of the dual group pG of G. Theorem 6.5.11, which gives theanalogous description for ideals with bounded approximate identities in ApGq foramenable groups G, is the main result of [85].

The material in Section 6.6 is taken from [45], and our presentation follows[45]. Delaporte and Derighetti have actually shown the results in the more generalcontext of algebras AppGq, 1 ă p ă 8, of any amenable locally compact groupG. However, proofs are less technical when p “ 2. Although amenability of G is


employed several times, it was conjectured in [45] that Theorem 6.6.4 should betrue for nonamenable groups as well.

We conclude these notes by mentioning another line of investigation, which wasextensively pursued in recent years. Generalizing the notion of spectral set, Warner[286] introduced the concept of a weak spectral set. If A is an arbitrary semisimpleand regular commutative Banach algebra, a closed subset E of its Gelfand spaceσpAq in called a weak spectral set if there exists some n P N such that an P jpEq forevery a P IpEq. The smallest such n is then called the characteristic of E. Warnerwas motivated by the union problem, that is, the question of whether the union oftwo sets of synthesis for L1pGq is again a set of synthesis. This problem is still open,even for �1pZq. In contrast, Warner showed that the union of two weak spectralsets is a weak spectral set. Another motivation for studying weak spectral sets isVaropoulos’ result [277] stating that the sphere Sd´1 in Rd “ σpL1pRdqq is a weakspectral set with characteristic t

d`12 u. Subsequently, weak spectral sets have been

investigated by several authors for commutative Banach algebra, especially Fourieralgebras ([150], [152], [153] [160], [225], [226], [228]). We mention just oneparticular issue and otherwise refer to the literature. Modifying Varopoulos’ proof[278] of Malliavin’s theorem, it was shown in [230] that weak spectral synthesis failsfor any nondiscrete abelian group. This result was extended to arbitrary locallycompact groups in [150] and also in [228].

If A is an arbitrary semisimple and regular commutative Banach algebra andE1 and E2 are closed subsets of the Gelfand space σpAq of A such that E1 X E2

is a set of synthesis, then E1 Y E2 is a set of synthesis if and only if both E1 andE2 are sets of synthesis [151, Theorem 5.2.5]. This general result covers the caseX “ V NpGq of Theorem 6.1.12.

CHAPTER 7

Extension and Separation Propertiesof Positive Definite Functions

The Hahn-Banach extension theorem asserts that if E is a normed linear spaceand F is a closed linear subspace of E, then each continuous linear functional onF extends to a continuous linear functional on E. From this it follows that givenx P EzF , there exists a continuous linear functional φ on E such that φ “ 0 on Fand φpxq ‰ 0 (the Hahn-Banach separation theorem). In this chapter we addressthe analogous properties for positive definite functions on locally compact groups.

Let G be an arbitrary locally compact group and H a closed subgroup of G. Wehave shown in Section 2.6 that the restriction map u Ñ u|H from ApGq into ApHq

is surjective. The corresponding problem for Fourier-Stieltjes algebras is muchmore delicate. To fix terminology, let us say that G has the extension propertyif for every closed subgroup H, each ϕ P P pHq admits an extension φ P P pGq

(equivalently, BpHq “ BpGq|H). The largest class of locally compact groups sharingthis extension property, is formed by the groups with small conjugation invariantneighbourhoods of the identity, the so-called SIN-groups. The converse implicationis true for connected Lie groups and for compactly generated nilpotent groups.More precisely, a connected Lie group has the extension property only if it is a directproduct of a vector group and a compact group. On the other hand, there existsa compactly generated 2-step solvable group which has the extension property, butfails to be a SIN-group. These results are developed in Sections 7.3 and 7.4.

In this situation, it is reasonable to pose the above problem also for a fixedsubgroup of G. We establish various results concerning subgroups which are ex-tending in the obvious sense and which prove helpful in identifying all the extendingsubgroups in specific examples. For instance, if G is a nilpotent locally compactgroup and H is a closed subgroup of G which is topologically isomorphic to eitherR or Z, then H is extending in G. In addition, we present a theorem, due to Cowl-ing and Rodway [39], providing a necessary and sufficient condition for a functionu P BpNq to belong to BpGq|N when N is a closed normal subgroup of G (Section7.2).

The second topic pursued in this chapter is a separation property of positivedefinite functions. A closed subgroup H of G is called a separating subgroup if foreach x P GzH there exists φ in

PHpGq “ tφ P P pGq : φphq “ 1 for all h P Hu

with φpxq ‰ 1. Accordingly, G is said to have the separation property if everyclosed subgroup of G is separating. The interest and importance of the separationproperty arise from the fact that it is useful in the study of the ideal theory ofFourier algebras. In Section 7.5 we first show that a neutral closed subgroup ofany locally compact group is separating, and consequently every SIN-group has the

237

238 7. EXTENSION AND SEPARATION PROPERTIES

separation property. The converse fails to be true. In fact, the so-called Fell group,which is 2-step solvable and totally disconnected, turns out to have the separationproperty, even though it is not an SIN-group. Moreover, we consider a number ofillustrative examples, for each of which we identify all the separating subgroups.Similar to the extension property, it turns out that for G, an almost connectedgroup or a compactly generated nilpotent group, validity of the separation propertyimplies that G is an SIN-group. Actually, for both classes of locally compact groups,the separation property for cyclic subgroup already forces the group to have smallinvariant neighbourhoods. Concerning a single closed subgroup H of G, the mostinteresting result asserts that if G has an almost connected open normal subgroup,then H is separating in G (if and) only if H is neutral in G. When G is a connectednilpotent group, the only separating subgroups of G are the normal subgroups.

7.1. The extension property: Basic facts

We start by introducing the properties which will be the subject of the followingfour sections.

Definition 7.1.1. Let G be a locally compact group and H a closed subgroupof G.

(i) We say that H is extending in G if for every ϕ P P pHq there exists φ P P pGq

such that φ|H “ ϕ.(ii) The group G is said to have the extension property if each closed subgroup

of G is extending.

Since every function in BpHq is a finite linear combination of continuous posi-tive definite functions, the restriction map u Ñ u|H from BpGq to BpHq is surjectivewhenever H is extending in G. As will be seen in Lemma 7.1.11, the converse isalso true.

In this section we collect a number of fundamental facts about extensibilitywhich will be used subsequently. The first part of the following lemma is shown in[212, (32.43)].

Lemma 7.1.2. Let H be an open subgroup of the locally compact group G. ThenH is extending in G. More precisely, given ϕ P BpHq, there exists φ P BpGq suchthat φ|H “ ϕ and }φ} “ }ϕ}.

Proof. Recall that φ can be defined by φpxq “ ϕpxq for x P H and φpxq “ 0otherwise. Let now ϕpxq “ xπpxqξ, ηy, x P G, where π is a unitary representation ofH and ξ, η P Hpπq are such that }ϕ} “ }ξ} ¨ }η}. Then, realizing the induced repre-

sentation indGH π on the Hilbert space �2pG{H,Hpπqq, we have Hpπq Ď HpindGH πq

and φpxq “ xindGH πpxqξ, ηy for all x P G. Thus }φ} ď }ξ} ¨ }η} “ }ϕ} and hence}φ} “ }ϕ}. �

Lemma 7.1.3. Let H and N be closed subgroups of G such that N is normal,H X N “ teu, HN is closed in G and H is topologically isomorphic to HN{N . IfHN{N is extending in G{N , then H is extending in G.

Proof. Let ϕ P P pHq and define ψ : HN{N Ñ C by ψphNq “ ϕphq, h P H.Then ψ is positive definite, and since the map h Ñ hN is a homeomorphism fromH onto HN{N , ψ is continuous. By hypothesis, ψ extends to some ρ P P pG{Nq.Then φ : G Ñ C, defined by φpxq “ ρpxNq, is in P pGq and extends ϕ. �

7.1. THE EXTENSION PROPERTY: BASIC FACTS 239

Lemma 7.1.4. Let H be a closed subgroup of G and let ϕ be a character of H.If ϕ extends to some continuous positive definite function on G, then there existsφ P expP pGqq such that φ|H “ ϕ.

Proof. Let PϕpGq denote the set of all functions in P pGq which extend ϕ,and suppose first that the set PϕpGq has an extreme point φ. Then φ P expP pGqq.Indeed, if ψ P P pGq is such that ψ ď c φ for some c ą 0, then ψ|H ď c φ|H “ c ϕand hence ψ|H “ ϕ since ϕ is extremal.

Therefore it remains to show that PϕpGq has an extreme point. For that, letK “ tφ P P pGq : φpeq ď 1u and consider the convex subset

Kϕ “ tφ P K : φ|H “ φpeqϕu

of K. Recall that if ρ P P 1pGq and h P G, then ρpxhq “ ρpxqρphq for all x P Gif and only if |ρphq| “ 1 by Proposition 1.4.16(v). Applying this to ρ “

1φpeq

φ,

where φ P K and φ ‰ 0, we get that a function φ P K belongs to Kϕ if and only ifφpxhq “ φpxqφphq for all x P G and h P H. This equation is equivalent to

ż

G

φpxhqfpxqdx “ ϕphq

ż

G

φpxqfpxqdx

for all f P L1pGq and h P H. This in turn implies that Kϕ is closed in K in thew˚-topology. Since K Ď P pGq Ď L8pGq is w˚-compact, Kϕ is w˚-compact andhence by the Krein-Milman theorem is the closed convex hull of its set of extremepoints. Now, any nonzero extreme point φ of Kϕ must satisfy φpeq “ 1 and hencebelongs to PϕpGq. Thus PϕpGq has extreme points, and such extreme points areextreme points of P pGq by the first paragraph of the proof. �

We next observe that that extensibility of a positive definite function on aclosed subgroup can be expressed in terms of representations.

Lemma 7.1.5. Let H be a closed subgroup of a locally compact group G and letϕ P P 1pHq. Then ϕ extends to some φ P P pGq if and only if πϕ is a subrepresen-tation of π|H for some representation π of G.

Proof. If πϕ is a subrepresentation of π|H , then there exists an isometriclinear mapping U from Hpπϕq into Hpπq such that Uπϕphq “ πphqU for all h P H.It follows that, for some η P Hpπϕq and all h P H,

ϕphq “ xπϕphqη, ηy “ xUπϕphqη, Uηy “ xπphqUη, Uηy.

Thus φ, defined by φpxq “ xπpxqUη, Uηy for x P G, is a continuous positive definitefunction extending ϕ.

Conversely, let φ P P pGq be an extension of ϕ and let π “ πφ. Then thereexists ξ P Hpπq such that φpxq “ xπpxqξ, ξy for all x P G. On the other hand,ϕphq “ xπϕphqη, ηy for some cyclic vector η of πϕ and all h P H. Let L denote the


linear span of all πϕphqη, h P H. Forřn

j“1 cjπϕphjqη P L, hj P H, cj P C, we have

›

›

›

nÿ

j“1

cjπϕphjqη›

›

›

2

“

nÿ

j,k“1

cjckxπϕph´1k hjqη, ηy “

nÿ

j,k“1

cjckϕph´1k hjq

“

nÿ

j,k“1

cjckφph´1k hjq “

nÿ

j,k“1

cjckxπphjqξ, πphkqξy

“

›

›

›

nÿ

j“1

cjπphjqξ›

›

›

2

.

Thus we can define a linear mapping from L into Hpπq bynÿ

j“1

cjπϕphjqη Ñ

nÿ

j“1

cjπphjqξ,

hj P H, cj P C, n P N, and this mapping is isometric. Since L is dense in Hpπϕq,this mapping extends uniquely to an isometric linear mapping U from Hpπϕq intoHpπq. Finally, U satisfies

Uπϕph1qπϕph2qη “ πph1h2qξ “ πph1qpUπϕph2qηq

for all h1, h2 P H. This implies that Uπϕphq “ πphqU for all h P H, as required. �

Corollary 7.1.6. Let H be a closed subgroup of G which is either compact orcontained in the centre of G. Then H is extending in G.

Proof. Let ϕ P P pHq and let π “ indGH πϕ, where πϕ is the cyclic represen-tation associated with ϕ. In both cases it is known that π|H contains πϕ as asubrepresentation. Actually, if H is contained in the centre of G, then π|H is amultiple of πϕ. So the statement follows from Lemma 7.1.5. �

Lemma 7.1.7. Let H and N be closed subgroups of the locally compact group Gsuch that N is normal and H Ě N . If H is extending in G, then H{N is extendingin G{N .

Proof. Let q : G Ñ G{N denote the quotient homomorphism and let ϕ P

P pH{Nq. Then ϕ˝q P P pHq and hence there exists φ P P pGq such that φ|H “ ϕ˝q.Let L “ tx P G : φpxq “ φpequ. Then L is a closed subgroup of G, φ is constanton cosets of L, and L clearly contains N . Therefore φ is of the form φ “ ψ ˝ q forsome ψ P P pG{Nq. For h P H, we have

ψphNq “ φphq “ ϕ ˝ qphq “ ϕphNq,

so that ψ extends ϕ. �

Lemma 7.1.8. Let G be a locally compact group and N a closed normal subgroupof finite index in G. If H is a closed subgroup of G such that H X N is extendingin N , then H is extending in G.

Proof. Let ψ P P pHq, K “ H XN and ϕ “ ψ|K . By hypothesis, there exists

φ P P pNq such that φ|K “ ϕ. We claim that the representation ρ “ indGN πφ hasthe property that ρ|H ě πψ.

To see this, note first that since H{K is finite,

πψ ď indHK πϕ “ indHKpπφ|K q ď indHKpπφ|Kq.

7.1. THE EXTENSION PROPERTY: BASIC FACTS 241

On the other hand, for any representation τ of N ,

indHKpτ |Kq ď

´

indGN τ¯

|H .

Indeed, since K “ H X N and G{N is finite, it is easily verified that the range ofthe map

T : HpindHKpτ |Kqq Ñ tF : G Ñ Hpτ qu,

defined by Tξpxq “ 0 if x R HN and Tξpxq “ τ py´1qξphq if x “ hy with h P H andy P N , is contained in

HpindGN τ q “ HppindGN τ q|Hq,

and that T provides an intertwining operator between the representations indHKpτ |Kq

and pindGN τ q|H . Thus πψ ď pindGN πφq|H , and an application of Lemma 7.1.5 givesthat ψ extends to some element of P pGq. �

Corollary 7.1.9. Let G1 be a closed subgroup of finite index in the locallycompact group G. Then G has the extension property if and only if G1 has theextension property.

Proof. Because G1 is extending in G, we only have to show the “if part”.So suppose that G1 has the extension property, and let N be the largest normalsubgroup of G which is contained in G1. Then N has finite index in G. If H is anyclosed subgroup of G, then by hypothesis, H X N is extending in G1 and hence inN . Now, Lemma 7.1.8 shows that H is extending in G. �

Lemma 7.1.10. Let H be a closed subgroup of G and let ϕ, ψ P P pHq be suchthat ϕ ´ ψ P P pHq. If ϕ extends to some element of P pGq, then the same is trueof ψ.

Proof. Write ϕphq “ xπϕphqξ, ξy, h P H, for some cyclic vector ξ P Hpπϕq. ByLemma 7.1.5, there exists a representation π of G in a Hilbert space H containingHpπϕq such that πϕphqη “ πphqη for all h P H and η P Hpπϕq. Thus, for allh P H, ϕphq “ xπphqξ, ξy. Since ϕ ´ ψ is positive definite on H, there exists apositive operator T on Hpπϕq, commuting with all operators πϕphq, h P H, suchthat ψphq “ xTπphqξ, ξy. Let S be the positive square root of T . Then S commuteswith all πϕphq and we have

ψphq “ xTπϕphqξ, ξy “ xSπϕphqSξ, ξy “ xπϕphqSξ, Sξy

“ xπphqSξ, Sξy.

So the positive definite function x Ñ xπpxqSξ, Sξy on G extends ψ. �With the aid of the preceding lemma we can now show that if the restriction

map BpGq Ñ BpHq is surjective, then H is extending in G.

Lemma 7.1.11. Let H be a closed subgroup of G and let ϕ P P pHq. If ϕ extendsto some function in BpGq, then ϕ also extends to some function in P pGq.

Proof. Let first ϕ be an arbitrary hermitian function in BpHq, that is, ϕphq “

ϕph´1q for all h P H, and let ψ P BpGq be an extension of ϕ. Write ψ “ ψ1 ` iψ2,where ψ1 and ψ2 are hermitian functions in BpGq. For h P H, we then haveϕphq “ ψ1phq ` iψ2phq and ϕ and ψ1 are hermitian on H, and hence iψ2 has to behermitian on H. Thus, since ψ2 is hermitian,

iψ2phq “ iψ2ph´1q “ ´iψ2ph´1q “ ´iψ2phq


for all h P H. So ψ2 “ 0 on H and the hermitian function ψ1 is itself an extensionof ϕ.

Now suppose that ϕ is positive definite. Since ψ1 is hermitian, it can be writtenas ψ1 “ ψ` ´ ψ´, where ψ` and ψ´ are in P pGq. For h P H, we then haveψ`phq “ ϕphq `ψ´phq. Since ψ`|H ´ϕ P P pHq and ψ`|H has the positive definiteextension ψ`, it follows from Lemma 7.1.10 that ϕ has an extension in P pGq. �

7.2. Extending from normal subgroups

In Section 7.1 we have identified a few simple cases of extending closed sub-groups H of a locally compact group G. For arbitrary H, the problem appearsto be untractable. However, the situation improves when H is normal in G. Thetheorem of Cowling and Rodway, mentioned in the introduction to this chapter, ispresented below (Theorem 7.2.3). We start with a special and considerably simplercase, namely that of a character of an abelian normal subgroup, which will actuallyturn out to be much easier applicable.

Proposition 7.2.1. [Douady’s observation] Let N be an abelian closed normalsubgroup of G and χ a character of N . Suppose that χ extends to some φ P P pGq.Then the stability group

Gχ “ tx P G : χpx´1yxq “ χpyq for all y P Nu

of χ is open in G.

Proof. There exist a representation π of G and a vector ξ P Hpπq such thatφpxq “ xπpxqξ, ξy for all x P G. Then }ξ} “ φpeq1{2 “ χpeq1{2 “ 1 and

1 “ |χpyq| “ |xπpyqξ, ξy| ď }πpyqξ} ¨ }ξ} “ 1

for all y P N . It follows that πpyqξ and ξ are linearly dependent, and then πpyqξ “

χpyqξ. For x P G, put ξx “ πpxqξ. Then

πpyqξx “ πpxqrπpx´1yxqξs “ πpxqrχpx´1yxqξs “ χpx´1yxqξx.

This means that ξx is an eigenvector of πpyq with eigenvalue χpx´1yxq.Towards a contradiction, assume now that Gχ is not open in G. Since π is

strongly continuous, there exists a neighbourhood U of the identity in G such that}πpxqξ ´ ξ} ă

?2 for all x P U . By assumption we find x P U and y P N such

that χpx´1yxq ‰ χpyq. Then ξx and ξ are both eigenvectors of πpyq with distincteigenvalues and therefore they must be orthogonal. Since both are unit vectors, itfollows that }ξx ´ ξ} “

?2. This contradiction completes the proof. �

Since the characters of an abelian locally compact group separate the elementsof the group, Proposition 7.2.1 implies:

Corollary 7.2.2. Let N be an abelian closed normal subgroup of G. If N isextending in G and G{N is connected, then N is contained in the centre of G.

Let N be a closed normal subgroup of the locally compact group G. Recallthat G acts on N by inner automorphisms and hence on spaces of functions onG and on representations in the obvious manner. For instance, if u P BpNq andx P G, then x ¨ u P BpNq is defined by x ¨ upnq “ upx´1nxq for n P N . It is clearthat }x ¨ u}BpNq “ }u}BpNq. The following theorem now provides a necessary andsufficient condition for a function u in BpNq to be extensible to some element ofBpGq.

7.2. EXTENDING FROM NORMAL SUBGROUPS 243

Theorem 7.2.3. Let N be a closed normal subgroup of the locally compactgroup G and let u P BpNq. Then u extends to some function in BpGq if and onlyif

}x ¨ u ´ u} Ñ 0 as x Ñ e,

and in this case

}u} “ inft}v} : v|N “ uu.

Proof. Note first that if v P BpGq, then v|N P BpNq and }v|N } ď }v}. More-over, if vpxq “ xπpxqξ, ηy, where }v}BpGq “ }ξ} ¨ }η}, and xα Ñ e, then

xα ¨ vpxqvpxq “ xπpxqπpxαqξ, πpxαqηy ´ xπpxqξ, ηy

and hence

}xα ¨ v ´ v} ď }πpxαqξ ´ ξ} ¨ }πpxαqη} ` }πpxαqη ´ η} ¨ }ξ} Ñ 0

and therefore

}xα ¨ pv|N q ´ v|N }BpNq ď }xα ¨ v ´ v} Ñ 0.

The proof of the converse is much more complicated. It suffices to show that forany u P BpNq satisfying }x ¨ u ´ u} Ñ 0 as x Ñ e and any ε ą 0, there existsv P BpGq such that }v}BpGq “ }u} and }v|N ´ u} ď ε. In fact, this can be seenas follows. Choose v1 P BpGq with }v1} “ }u} and }v1|N ´ u} ď ε{2, and thenconstruct inductively a sequence pvnqn in BpGq such that, for n ě 2,

›

›

›

›

›

vn|N ´

˜

u ´

n´1ÿ

j“1

vj |N

¸›

›

›

›

›

ďε

2n

and

}vn} “

›

›

›

›

›

u ´

n´1ÿ

j“1

vj |N

›

›

›

›

›

.

Then }vn} ď 1{2n´1 for n ě 2 and hence the series v “ř8

n“1 vn converges in BpGq

and satisfies v|N “ u and

}v} ď }u} `

8ÿ

n“2

}vn} “ }u} `

8ÿ

n“2

ε

2n´1“ }u} ` ε.

Given u and ε, we find a neighbourhood U of e in G such that

(7.1) }x ¨ u ´ u} ăε

2

for all x P U and

(7.2) }Lnu ´ u} ăε

2

for all n P U X N . Choose a compact symmetric neighbourhood V of e in G suchthat V 2 Ď U . Let Haar measures on G, N and G{N be normalized so that Weil’sformula holds. Take a nonnegative continuous functions f on G with supp f Ď V


such that

1 “

ż

G{N

ˆż

N

fpxnqdn

˙2

d 9x

“

ż

G{N

ˆż

N

fpxnqdn

˙ˆż

N

fpxmqdm

˙

d 9x

“

ż

G{N

ˆż

N

fpxnqdn

˙ˆż

N

fpxnmqdm

˙

d 9x

“

ż

G

ˆż

N

fpxqfpxnqdn

˙

dx.(7.3)

We now define a function v on G by

vpyq “

ż

G

ˆż

N

fpyxqfpxnqupnqdn

˙

dx, y P G.

We claim that v P BpGq. To verify this, we write u in the form upxq “ xπpxqξ, ηy,where π is a unitary representation ofN and ξ, η P Hpπq are such that }u} “ }ξ}¨}η}.Then, for any y P G,

vpyq “

ż

G

ˆż

N

fpxqfpy´1xnqupnqdn

˙

dx

“

ż

G{N

ˆż

N

ż

N

fpxmqfpy´1xmnqxπpnqξ, ηydndm

˙

d 9x

“

ż

G{N

ˆż

N

ż

N

fpy´1xnqfpxmqxπpnqξ, πpmqηydndm

˙

d 9x

“

ż

G{N

Bż

N

fpy´1xnqπpnqξdn,

ż

N

fpxmqπpmqηdm

F

d 9x.

This formula shows that v is a coordinate function of the unitary representation ofG induced from π. Moreover,

}v} ď

›

›

›

›

ż

N

fpxnqπpnqξdn

›

›

›

›

¨

›

›

›

›

ż

N

fpxmqπpmqηdm

›

›

›

›

and, for any ω P Hpπq, using (3),

›

›

›

›

ż

N

fpxnqπpnqωdn

›

›

›

›

“

˜

ż

G{N

›

›

›

›

ż

N

fpxnqπpnqωdn

›

›

›

›

2

d 9x

¸1{2

ď }ω}

˜

ż

G{N

ˆż

N

fpxnqdn

˙2

d 9x

¸1{2

“ }ω}.

So }v} ď }ξ} ¨ }η} “ }u}.To finish the proof, it remains to show that

}v|N ´ u} ď ε.

7.2. EXTENDING FROM NORMAL SUBGROUPS 245

To that end, we first obtain an expression for v as follows. For any n P N ,

vpnq “

ż

G

ż

N

fpxqfpn´1xmqupmqdmdx

“

ż

G

ż

N

fpxqfpxpx´1n´1xqmqupmqdmdx

“

ż

G

ż

N

fpxqfpxmqupx´1nxmqdmdx

“

ż

G

ż

N

fpxqfpxmqrpxmq ¨ pLm´1uqspnqdmdx.(7.4)

The map from G ˆ N to BpNq defined by

px,mq Ñ fpxqfpxmqpxmq ¨ pLm´1uq

is continuous and has compact support. Therefore, the vector-valued integralż

G

ż

N

fpxqfpxmqrpxmq ¨ pLm´1uqsdmdx

exists in BpNq and equals v|N by the pointwise equality (7.4). From (7.4) we get

u “

ż

G

ż

N

fpxqfpxnqudndx,

and together with (7.4), this yields

}v|N ´ u} ď

ż

G

ż

N

fpxqfpxnq}pxnq ¨ pLn´1uq ´ u}dndx

ď

ż

G

ż

N

fpxqfpxnq}pxnq ¨ pLn´1uq ´ pxnq ¨ u}dndx

`

ż

G

ż

N

fpxqfpxnq}pxnq ¨ u ´ u}dndx

ď

ż

G

ż

N

fpxqfpxnq}Ln´1u ´ u}dndx

`

ż

G

ż

N

fpxqfpxnq}pxnq ¨ u ´ u}dndx.(7.5)

If fpxqfpxnq ‰ 0, then both x and xn are in V and hence xn P U and n´1 P

V ´1V X N Ď U X N . Thus, by (7.1) and (7.2),

}pxnq ¨ u ´ u} ă ε{2 and }Ln´1u ´ u} ă ε{2.

Using (7.5), it follows that

}v|N ´ u} ď sup t}pxnq ¨ rLn´1u ´ us} : n P U X N, xn P Uu

` sup t}pxnq ¨ u ´ u} : n P U X N, xn P Uu

ď ε{2 ` ε{2 “ ε.

This completes the proof of the theorem. �We conclude this section with a result which at first glance might not be sur-

prising in view of the fact that open subgroups are always extending, but the proofturns out to be fairly involved and requires a clever inductive procedure.

Theorem 7.2.4. For an arbitrary locally compact group G, the connected com-ponent G0 of the identity is extending in G.


Proof. Since G{G0 is totally disconnected, we can choose an open subgroupH of G containing G0 such that H{G0 is compact. As H is extending in G, wecan henceforth assume that G{G0 is compact. Then G is a projective limit ofLie groups G{Kα (Section 1.2) and, with qα : G Ñ G{Kα denoting the quotienthomomorphism, BpG{Kαq ˝ qα Ď BpGq. For each α, let μα be normalized Haarmeasure on the compact normal subgroup G0 X Kα.

Now, let ϕ P P pG0q and let π be the associated cyclic representation. Since ϕis continuous at e, we have

|ϕpeq ´ pϕ ˚ μαqpeq| ă1

2}ϕ}

for sufficiently large α. Fix such an α and let K “ Kα, μ “ μα and q “ qα.Since K XG0 is a compact normal subgroup of G0, π decomposes into a direct sumπ “ π1‘π2, where π1|KXG0

is a multiple of the trivial 1-dimensional representationof K X G0 and π2|KXG0

is disjoint from π1|KXG0. Accordingly, ϕ can be written

as ϕ “ ϕ1 ` ϕ2, where ϕj is associated to πj , j “ 1, 2. Then ϕ ˚ μ “ ϕ1 and henceϕ2 “ ϕ ´ ϕ ˚ μ is a positive definite function of norm ă

12}ϕ}. Now ϕ1 “ ϕ ˚ μ is

constant on pK XG0q-cosets in G0 and hence defines an element ψ1 of P pG0K{Kq

byψ1pxKq “ ϕ1pxq, x P G0.

As G0K{K is open in G{K, ψ1 extends to some φ11 P P pG{Kq such that

}φ11} “ }ψ1} “ }ϕ1}.

Let φ1 “ φ11 ˝ q P P pGq. Then φ1|G0

“ ϕ1 and }φ1} “ }ϕ1}.Having constructed ϕ1 and φ1, we now apply the same procedure to ϕ ´ ϕ1

instead of ϕ to obtain ϕ2 P P pG0q such that pϕ ´ ϕ1q ´ ϕ2 P P pG0q and

}pϕ ´ ϕ1q ´ ϕ2} ă1

2}ϕ ´ ϕ1} ă

1

4}ϕ}

and φ2 P P pGq such that φ2|G0“ ϕ2 and }φ2} “ }ϕ2}.

Continuing this process, we construct inductively sequences pϕnqn in P pG0q

and pφnqn in P pGq satisfying φn|G0“ ϕn, }φn} “ }ϕn} and

}ϕ ´ pϕ1 ` . . . ` ϕnq} ă 2´n}ϕ}

for all n P N. It follows that the seriesř8

n“1 ϕn andř8

n“1 φn converge in BpG0q

and BpGq, respectively. Thenř8

n“1 ϕn “ ϕ and if we put φ “ř8

n“1 φn, thenclearly φ is continuous and positive definite and φ|G0

“ ϕ. �

Alternatively, the preceding theorem can also be shown by employing Theorem7.2.3. However, this would require projective limit arguments.

7.3. Connected groups and SIN-groups

So far, the only locally compact groups which we know to have the extensionproperty, are either discrete, or compact or locally compact abelian. There existsa natural class of groups comprising all these, which we now introduce and eachmember of which will turn out to have the extension property.

Definition 7.3.1. Let H be a closed subgroup of the locally compact groupG. Then G is said to be an rSINsH -group (a group with small H-invariant neigh-bourhoods) if G has a neighbourhood basis V of the identity such that h´1V h “ Vfor all V P V and all h P H. rSINsG-groups are simply referred to as SIN-groups.

7.3. CONNECTED GROUPS AND SIN-GROUPS 247

Theorem 7.3.2. Let G be a locally compact group and H a closed subgroup ofG such that G is an rSINsH-group. Then BpGq|H “ BpHq and, if u P BpHq, then

}u}BpHq “ inft}v}BpGq : v|H “ uu.

Proof. As in the proof of Theorem 7.2.3, it suffices to show that for anyu P BpHq and ε ą 0, there exists v P BpGq such that }v}BpGq ď }u}BpHq and}v|H ´ u}BpHq ă ε. Let C be a compact H-invariant neighbourhood of e in G.Then

|C| “

ż

G

1Cpxqdx “ ΔGphq

ż

G

1Cph´1xhqdx “ ΔGphq|C|

for every h P H. So ΔG|H “ 1 and also ΔH “ 1. Therefore, there exists anH-invariant measure on the quotient space G{H (see Section 1.3). We assume thatHaar measures on G and H are adjusted so that Weil’s formula

ż

G

fpxqdx “

ż

G{H

ˆż

H

fpxhqdh

˙

d 9x

holds for all f P CcpGq. Choose a compact neighbourhood V of e in G such that

(7.6) }Lh´1u ´ u} ă ε for all h P V ´1V X H.

Let f be a nonnegative continuous function on G such that supp f Ď V , fpxq “

fph´1xhq for all x P G and h P H, and

(7.7)

ż

G{H

ˆż

H

fpxhq2dh

˙

d 9x “ 1.

We define a function v on G by

vpyq “

ż

G

ˆż

H

fpyxqfpxhquphqdh

˙

dx.

Then, using that f is H-invariant and H is unimodular,

vph1q “

ż

G

ˆż

H

fpxqfph´11 xhquphqdh

˙

dx

“

ż

G

ˆż

H

fpxqfpxhh´11 quphqh

˙

dx

“

ż

G

ˆż

H

fpxqfpxhqLh´1uph1qdh

˙

dx

for every h1 P H. This pointwise equality is analogous to equation (7.3) in theproof of Theorem 7.2.3. Repeating the arguments in the proof of Theorem 7.2.3and using (7.6) and (7.7) and the fact that supp f Ď V , it is straightforward todeduce that }v|H ´ u} ă ε. Moreover, the proof that v P BpGq and }v} ď }u} isidentical to the corresponding part of the proof of Theorem 7.2.3. �

As an immediate consequence of Theorem 7.3.2 we state

Corollary 7.3.3. Every SIN-group has the extension property.

A locally compact group is called an IN-group if it possesses at least one compactconjugation invariant neighbourhood of the identity. It is worthwhile to mentionthat IN-groups need not have the extension property. As an example, let H de-note the 3-dimensional Heisenberg group and let D be a central discrete subgroupisomorphic to Z. Then G “ H{D has a compact invariant neighbourhood of the


identity, but it follows from Proposition 7.2.1 that G has a normal abelian subgroupof the form R ˆ T, which is not extending. We now investigate almost connectedlocally compact groups.

Proposition 7.3.4. Let G be a connected group and suppose that G possessesa descending sequence

R “ Rn Ě Rn´1 Ě . . . Ě R1 Ě R0 “ teu

of closed normal subgroups such that all the subquotients Rj{Rj´1, j “ 1, . . . , n,are abelian. If every closed normal subgroup of G is extending, then R is containedin the centre of G.

Proof. The proof essentially consists of repeatedly applying Proposition 7.2.1.We proceed by induction on k to show that Rk is contained in the centre ZpGq ofG.

To start with, observe first that R1 Ď ZpGq since R1 is an extending abelianclosed normal subgroup of G and G{R1 is connected. Similarly, for each 1 ď k ď n,Rk{Rk´1 Ď ZpG{Rk´1q. Now assume that Rk Ď ZpGq for some 1 ď k ď n´ 1. Weshow that then Rk`1 is contained in the centre of G. To that end, take any a P

Rk`1zRk and let H denote the closed subgroup of G generated by a and Rk. ThenH is abelian since Rk Ď ZpGq, and H is normal in G since Rk`1{Rk Ď ZpG{Rkq.By hypothesis, H is extending in G and therefore H Ď ZpGq. As a P Rk`1zRk isarbitrary, it follows that Rk`1 Ď ZpGq. This finishes the inductive step and thuscompletes the proof. �

We remind the reader that the radical of a connected Lie groupG is the maximalconnected solvable normal subgroup of G.

Corollary 7.3.5. Let G be a connected Lie group and suppose that everyclosed normal subgroup of G is extending. Then the radical of G is contained inthe centre of G.

Proof. The statement follows from Proposition 7.3.4, taking for R the radicalof G and for the descending sequence of normal subgroups the commutator seriesR Ě rR,Rs Ě rrR,Rs, rR,Rss Ě . . . of R. It is clear that this sequence satisfies thehypothesis of Proposition 7.3.4. �

Theorem 7.3.6. For a connected Lie group G, the following properties areequivalent.

(i) G has the extension property.(ii) G is of the form G “ V ˆ K, where V is a vector group and K is a

compact group.

Proof. (ii) ñ (i) A group of the form V ˆ K is a SIN-group and thereforehas the extension property by Corollary 7.3.3.

(i) ñ (ii) If (i) holds, then by Corollary 7.3.5 the radical R of G is containedin the centre of G, and G{R is a semisimple Lie group. If G{R is compact, then bythe structure theorem for connected central groups (see [110] or [60]), G is in facta direct product of a vector group and a compact group.

7.3. CONNECTED GROUPS AND SIN-GROUPS 249

So suppose that G{R is noncompact. Then G{R admits an Iwasawa decom-position G{R “ KAN , where K is compact, A is abelian and N is nilpotent.The group A normalizes N , but does not centralize N . Let q : G Ñ G{R de-note the quotient homomorphism. The group q´1pNq is extending in G and hencealso in q´1pANq. But q´1pNq is normal in q´1pANq and connected and solvableand therefore contained in the radical of q´1pANq. Proposition 7.3.4 shows thatq´1pNq Ď Zppq´1pANqq. It follows thatN “ q´1pNq{R is contained in the centre ofAN “ q´1pANq{R. This is a contradiction, and hence G{R has to be compact. �

In the proof of the following corollary and later in this chapter, we shall usethe next lemma.

Lemma 7.3.7. Let G be a projective limit of SIN-groups. Then G itself is anSIN-group.

Proof. Let U be an open neighbourhood of the identity e of G. Then, byhypothesis, there exists a compact normal subgroup K of G such that K Ď U andG{K is an SIN-group. Since K is compact and U is open, there exists an openneighbourhood W of e in G such that KW Ď U . Thus, replacing U by KW , wecan assume that U “ q´1pqpUqq, where q : G Ñ G{K denotes the quotient homo-morphism. Now, as G{K is SIN, we find a conjugation invariant neighbourhood Vof tKu in G{K contained in qpUq. Then q´1pV q Ď q´1pqpUqq “ U and q´1pV q isconjugation invariant. �

Corollary 7.3.8. Let G be an almost connected locally compact group. ThenG has the extension property (if and) only if G is an SIN-group.

Proof. We only need to show that if G has the extension property, then Gmust be an SIN-group. Since G{G0 is compact, G is a projective limit of Lie groupsG{Kα. Then, for each α, pG{Kαq0 is open in G{Kα and hence has finite index inG{Kα. Since G{Kα has the extension property, by Theorem 7.3.6 the connected Liegroup pG{Kαq0 is an SIN-group, and hence so is G{Kα because pG{Kαq0 has finiteindex in G{Kα. Finally, the preceding lemma shows that G is an SIN-group. �

The results of this section raise the question of whether an arbitrary locallycompact group which has the extension property must be an SIN-group. It turnsout that this is not the case. In fact, there exists an example, which we presentbelow, of a compactly generated, 2-step solvable locally compact group which hasthe extension property and nevertheless fails to be an SIN-group.

Example 7.3.9. Let G “ R�Z, where n P Z acts on R by t Ñ 2nt. Clearly, Ghas no compact invariant neighbourhood of the identity. We show that neverthelessG has the extension property. For this, we determine all the closed subgroups ofG.

We identify R and Z with the subgroups R ˆ t0u and t0u ˆ Z, respectively.Let H be a closed subgroup of G which is not contained in R. Then there existss P R, s ‰ 0, such that sHs´1 XZ ‰ t0u. So we may assume that H XZ “ mZ forsome m P N. Suppose that H ‰ mZ and take t P R, t ‰ 0, and k P Z such that


x “ tk P H. Then, as is easily verified,

xm“

˜

m´1ÿ

j“0

2jk

¸

tpmkq.

Since mZ Ď H, it follows that r “

´

řm´1j“0 2jk

¯

t P H. Again, since mZ Ď H, we

get that

p2mnr, 0q “ p0,mnqpr, 0qp0,mnq´1

P H X R

for all n P Z. Since r ‰ 0 and H X R is a closed subgroup of R, it follows thatR Ď H. Thus we have seen that every closed subgroup H of G either is containedin R or contains R or is conjugate to a subgroup of Z. In the first two cases H isextending in G since R is open in G, and in the third case it follows from Lemma7.1.2 that H is extending.

7.4. Nilpotent groups and 2-step solvable examples

In the preceding section we have seen that if G is a connected Lie group sat-isfying the extension property, then G has to be an SIN-group. The main themeof this section is to show that the same conclusion holds for compactly generatednilpotent locally compact groups. On the other hand, if H is a closed subgroup ofa nilpotent group G and H is topologically isomorphic to either R or Z, then H isextending in G. In addition, we determine all the extending subgroups of severalexamples of 2-step solvable groups.

Later in this section we shall exploit to some extent the structure of generalnilpotent locally compact groups, as developed in [99]. For any locally compactgroup G, let Gc denote the set of all compact elements of G (that is, elementswhich generate a relatively compact subgroup). Then G is said to be compact-freeif Gc “ teu. Suppose that G is nilpotent. Then Gc is a closed (normal) subgroupof G [99, Corollary 3.5.1 and Lemma 3.8], and G{Gc is compact-free and a Liegroup. If, in addition, G is compactly generated, then Gc is compact [99, Theorem9.7] and the subgroup G0G

c is open in G. We remind the reader that when G isdiscrete, Gc is simply the set of all elements of finite order, which is usually denotedGt and called the torsion subgroup of G.

We shall need below the following result on positive definite functions.

Lemma 7.4.1. Let ϕ P P 1pGq. Then, for all x, y P G,

1

2|1 ´ ϕprx, ysq| ď 1 ´ |ϕpxq| ` 1 ´ |ϕpyq|

`p1 ´ |ϕpxq|qp1 ´ |ϕpyq|q

`p1 ´ |ϕpxq|q1{2

` p1 ´ |ϕpyq|q1{2.

Proof. Recall first that

|ϕpaqϕpbq ´ ϕpabq|2

ď p1 ´ |ϕpaq|2qp1 ´ ϕpbq|

2q

7.4. NILPOTENT GROUPS AND 2-STEP SOLVABLE EXAMPLES 251

for all a, b P G (Proposition 1.4.16(iv)). Using several times this inequality as well

as |ϕpaq| ď 1 and ϕpa´1q “ ϕpaq for all a P G, we obtain, for x, y P G,

|1 ´ ϕprx, ysq| ď 1 ´ |ϕpxq|2

` |ϕpxq|2p1 ´ |ϕpyq|

2q

` |ϕpxqϕpyq|2

´ ϕpxqϕpyqϕpx´1y´1q|

` |ϕpxqϕpyqϕpx´1y´1q ´ ϕpxqϕpyx´1y´1

q|

` |ϕpxqϕpyx´1y´1q ´ ϕprx, ysq|

ď 2p1 ´ |ϕpxq|q ` 2p1 ´ |ϕpyq|q ` |ϕpxqϕpyq ´ ϕpyxq|

` |ϕpyqϕpx´1y´1q ´ ϕpyx´1y´1

q|

`|ϕpxqϕpyx´1y´1q ´ ϕprx, ysq|

ď 2p1 ´ |ϕpxq|q ` 2p1 ´ |ϕpyq|q

`p1 ´ |ϕpxq|2q1{2

p1 ´ |ϕpyq|2q1{2

` p1 ´ |ϕpyq|2q1{2

p1 ´ |ϕpx´1y´1|2q1{2

` p1 ´ |ϕpxq|2q1{2

p1 ´ |ϕpyx´1y´1q|2q1{2

ď 2p1 ´ |ϕpxq|q ` 2p1 ´ |ϕpyq|q

`p1 ´ |ϕpxq|2q1{2

p1 ´ |ϕpyq|2q1{2

` 2p1 ´ |ϕpxq|q1{2

` 2p1 ´ |ϕpyq|q1{2,

which implies the desired inequality. �

Before proceeding, we remind the reader that

teu “ Z0pGq Ď Z1pGq “ ZpGq Ď . . . Ď ZmpGq Ď . . .

denotes the ascending central series of G.

Lemma 7.4.2. Let H be a closed subgroup of the locally compact group G.Suppose that every character of H extends to some function in P pGq. Then

H X rH,Z2pGq0s Ď rH,Hs.

In particular, if G is connected and 2-step nilpotent, then H X rH,Gs Ď rH,Hs.

Proof. Towards a contradiction, assume there exists h0 P H X rH,Z2pGq0s

such that h0 R rH,Hs. There exists a character χ of H with χph0q ‰ 1. Byhypothesis, we find some φ P P pGq so that φ|H “ χ, and by Lemma 7.1.4 φcan be chosen in expP 1pGqq. Then φ|ZpGq is a character of ZpGq. Indeed, therepresentation πφ is irreducible and hence πφ|ZpGq is a multiple of a one-dimensionalrepresentation. Since |φphq| “ 1 for all h P H, we have by Lemma 7.4.1,

1

2|1 ´ φprh, xsq| ď 1 ´ |φpxq| ` p1 ´ |φpxq|q

1{2

for all h P H and x P G. Thus there exists an open neighbourhood V of the identityin G such that |1 ´ φprh, xsq| ă

?3 for all x P V and h P H.

Now let x P V X Z2pGq. Then the map h Ñ rh, xs is a homomorphism from Hinto ZpGq. Since φ|ZpGq is a character, it follows that the set

Γx “ tφprh, xsq : h P Hu

is a subgroup of T with the property that |z ´ 1| ă?3 for all z P Γx. As is well-

known, this implies Γx “ t1u. Finally, since for each h P H the map x Ñ rh, xs is ahomomorphism from Z2pGq into ZpGq, we conclude that φprh, xsq “ 1 for all h P H


and all x in the open subgroup of Z2pGq generated by V X Z2pGq. Consequently,φpyq “ 1 for all y P rH,Z2pGq0s. Since h0 P rH,Z2pGq0s, we have reached thecontradiction 1 “ φph0q “ χph0q. �

Corollary 7.4.3. Let G be a connected 2-step nilpotent locally compact groupand let H be an abelian closed subgroup of G. Suppose that H is σ-compact and thatHrH,N s is closed in G. Then H is extending in G if and only if H X rH,Gs “ teu.

Proof. The necessity of the condition follows from Lemma 7.4.2. Conversely,let N “ rH,Gs and suppose that H X N “ teu. By hypothesis, HN is closed inG and, since G is 2-step nilpotent, HN{N is contained in the centre of G{N . SoHN{N is extending in G{N and hence, since H X N “ teu, H is extending in Gby Lemma 7.1.3. �

Lemma 7.4.4. Let G be a semidirect product G “ N � A, where N is abelian,connected and contained in ZmpGq for some m P N. If G has the extension property,then N is contained in the centre of G.

Proof. We prove the lemma by induction on m. Thus, let N Ď Zm`1pGq forsome m and suppose that if H “ M �A is such that H has the extension propertyand M is abelian and connected and contained in ZmpHq, then M Ď ZpHq.

Let M “ rA,N s and H “ M � A. Then H has the extension property, M isabelian and connected and M Ď ZmpHq. So M Ď ZpHq, and since N is abelian,M is even central in G. Now, H “ AM is an extending subgroup of G. Since N isconnected and N Ď Z2pGq, Lemma 7.4.2 implies

rA,N s “ AM X rA,N s Ď H X rH,Z2pGq0s Ď rH,Hs “ rA,As.

Since A X N “ teu, it follows that rA,N s “ teu. Consequently, N is contained inthe centre of G since N is abelian. �

Corollary 7.4.5. Let G be a compact-free nilpotent locally compact group.Then G has the extension property if and only if the centre of G is open in G.

Proof. We only have to show that if G has the extension property then thecentre of G is open. The connected component G0 is open and G{G0 is torsion-free.Since G0 has the extension property, G0 “ V ˆ K, where V is a vector group andK is a compact group (Theorem 7.3.6). Now K, being a connected and nilpotentcompact group, K is abelian. Take any element a P GzG0 and let A be the subgroupof G generated by a. Since G0 is open and G{G0 is torsion-free, A X G0 “ teu andhence G0A “ G0 � A. Since G0 � A has the extension property and G0 is abelianand G is nilpotent, Lemma 7.4.4 applies and shows that G0 Ď ZpG0Aq. Sincea P GzG0 is arbitrary, it follows that G0 is contained in the centre of G. Thisfinishes the proof since G0 is open. �

Using the preceding preliminary results, we are now ready to deduce the ana-logue of Theorem 7.3.6 for compactly generated nilpotent groups.

Theorem 7.4.6. Let G be a compactly generated nilpotent locally compactgroup. Then G has the extension property if and only if G is an SIN-group.

Proof. We only have to show that if G has the extension property then Gis an SIN-group. We use the fact that a compactly generated nilpotent group is aprojective limit of Lie groups [127, Theorem 9]. Since each quotient group of G


has the extension property and a projective limit of SIN-groups is a SIN-group, wecan assume that G is a Lie group. Then G0 is open in G and G{G0 is a finitelygenerated nilpotent group. Let N Ď G be the pullback of the torsion subgrouppG{G0qt of G{G0. Then N{G0 “ pG{G0qt is finite, and G{N is torsion-free.

Since G0, being open in G, has the extension property, G0 “ V ˆ K, where Vis a vector group and K is a compact group (Theorem 7.3.6). Being a connected,nilpotent, compact group, K is abelian.

Now consider any a P GzN and let A denote the cyclic subgroup generated bya. Then, since G{N is torsion-free, G0A is actually a semidirect product G0 � A.Since G0A has the extension property, an application of Lemma 7.4.4 shows that acommutes with all elements of G0. As N{G0 is finite and G0 is abelian, it followsthat G acts as a finite group of inner automorphisms on G0. Thus G0 P rSINsG,and then G is a SIN-group since G0 is open in G. �

Example 7.3.9 shows that Theorem 7.4.6 does not hold for compactly generated2-step solvable groups.

We now turn to the second aim of this section, the identification of certainextending subgroups of general nilpotent groups and of all the extending subgroupsof a number of 2-step solvable groups.

Lemma 7.4.7. Let G be a connected and simply connected nilpotent Lie groupand H a closed subgroup which is topologically isomorphic to either R or Z. ThenH is extending in G.

Proof. Let g denote the Lie algebra of G, exp : g Ñ G the exponentialmap and z0 “ t0u Ď z1 . . . the ascending central series of g. Suppose first thatH is isomorphic to R and let H “ exp h and Zj “ exp zj, j “ 0, 1, . . .. Since h

is 1-dimensional, h X zj ‰ t0u implies that h Ď zj. Let j be minimal such thath Ď zj`1. Then h X zj “ t0u and hence H X Zj “ teu. Moreover, HZj is closed inG, HZj{Zj Ď Zj`1{Zj and, since G is σ-compact, H is topologically isomorphic toHZj{Zj . Lemma 7.1.3 now shows that H is extending in G.

Finally, if H is isomorphic to Z, then there exists a closed subgroup L of Gwhich contains H and is isomorphic to R. Since H is open in L and L is extending,H is extending as well. �

Applying Lemmas 7.4.7, 7.1.3 and 7.1.7 and structure theory of nilpotentgroups, it is possible to generalize Lemma 7.4.7 to arbitrary nilpotent locally com-pact groups as follows.

Theorem 7.4.8. Let G be a nilpotent locally compact group and let H be aclosed subgroup which is topologically isomorphic to either R or Z. Then H isextending in G.

Proof. Suppose first that H is topologically isomorphic to R. Then H Ď G0,and since G0 is extending in G by Theorem 7.2.4, we only have to observe that His extending in G0. To verify this, let K be the maximal compact normal subgroupof G0. Then G0{K is simply connected and HK{K is topologically isomorphic toR since H XK “ teu. Therefore HK{K is extending in G0{K by Lemma 7.4.7 andhence H is extending in G0 by Lemma 7.1.3.

Now let H be an infinite cyclic subgroup of G. Since it suffices to show thatH is extending in some compactly generated open subgroup of G containing H, wecan assume that G is compactly generated. Let L “ G0G

c, which is open in G


and almost connected since Gc is compact. We now distinguish the two cases thatH X L “ teu and H X L ‰ teu.

In the first case, H is extending in G by Lemma 7.1.3 since G{L is discrete. Inthe second case, let M “ HL. Then L has finite index in M and hence M is almostconnected. Since M is a projective limit of Lie groups, there exists a compactnormal subgroup K of M such that M{K is a Lie group. Then H X K “ teu

since H is infinite cyclic, and hence H is topologically isomorphic to HK{K. Thus,applying Lemma 7.1.3 again and passing to M{K, we can henceforth assume thatM is an almost connected nilpotent Lie group. So M0 has finite index in M , andby Lemma 7.1.8 it suffices to show that H XM0 is extending in M0. Consequently,we can assume that M is a connected nilpotent Lie group. Then M c is compactand M{M c is simply connected. Since H X M c “ teu, we can argue as before andapply Lemma 7.1.3 and Lemma 7.4.7 to conclude that H is extending in M , andhence in G. �

Applying Proposition 7.2.1, Lemma 7.4.7 and Lemma 7.4.2, we can now deter-mine the extending subgroups of the 3-dimensional Heisenberg group.

Example 7.4.9. We identify the 3-dimensional Heisenberg group G, as a set,with R3. Multiplication of G is then given by

px1, y1, z1qpx2, y2, z2q “ px1 ` x2, y1 ` y2, z1 ` z2 ` x1y2q,

pxj , yj , zjq P R3, j “ 1, 2. Then ZpGq “ tp0, 0, zq : z P Ru “ rG,Gs and N “

tp0, y, zq : y, z P Ru is an abelian normal subgroup of G isomorphic to R2. Let Hbe a proper closed subgroup of G. We are going to show that H is extending in Gif and only if H is isomorphic to either R or Z.

Assume first that H is isomorphic to R or Z. Then H is extending by Lemma7.4.7.

Conversely, suppose that H is extending and note that H X rH,Gs “ rH,Hs

by Lemma 7.4.2. Since ZpGq “ R, we can assume that H is not contained in ZpGq.Then rH,Gs is a nontrivial connected subgroup of ZpGq, hence equal to ZpGq, andtherefore H X ZpGq “ rH,Hs. We now distinguish the two cases H Ď N andH � N . If H Ď N , then H X ZpGq “ teu and so H is topologically isomorphic toHZpGq{ZpGq “ N{ZpGq “ R. This means that H is isomorphic to R or Z.

Finally, let H � N and, towards a contradiction, assume that H is isomorphicto neither R nor Z. It then follows as before that H X N ‰ teu. We now furthersplit the discussion into the two cases H0 Ď N and H0 � N . If H0 � N , so thatH0N “ G, then for any a P H X N , rH0, as is a non-trivial connected subgroupof ZpGq. Thus rH,Hs Ě ZpGq and hence H0 is normal in G. Also, dimH0 “ 2since H0 � N and H0 ‰ G. Now a 2-dimensional connected Lie group is abelian.Since H0 is open in H, H0 is extending in G. This contradicts Proposition 7.2.1. It

remains to consider the case that H � N , but H0 Ď N . Let rH denote the unique

minimal connected subgroup of G containing H. Then, as before, dim rH “ 2 andhence H is abelian. Also H XZpGq ‰ teu since H is not isomorphic to R or Z. Onthe other hand, by Lemma 7.4.2

H X ZpGq “ H X rH,Gs “ rH,Hs “ teu.

This is contradiction.


We continue with two lemmas which appear to be of interest in their own andwhich will be used to determine the extending subgroups of the ax ` b-group andthe motion group of the plane.

Lemma 7.4.10. Suppose that G has a closed normal subgroup N such that, withq : G Ñ G{N denoting the quotient homomorphism,

BpGq Ď BpG{Nq ˝ q ` C0pGq.

Let H be a noncompact closed subgroup of G. If H is extending, then H X N Ď

rH,Hs.

Proof. Towards a contradiction, suppose there exists h P H X N with h R

rH,Hs. Since hrH,Hs is not the identity of the abelian group H{rH,Hs, we canchoose a character χ of H such that χphq ‰ 1. By the hypothesis, χ extends tosome φ P P pGq and φ “ φ1 ` φ2, where φ1 is constant on cosets of N and φ2

vanishes at infinity. Then, since h P N ,

φpxhq “ φ1pxq ` φ2pxhq

for all x P G and, since |φphq| “ 1,

φpxhq “ φpxqφphq “ χphqpφ1pxq ` φ2pxqq.

It follows thatp1 ´ χphqqφ1pxq “ χphqφ2pxq ´ φ2pxq

for all x P G. Now, the function x Ñ χphqφ2pxq ´ φ2pxhq vanishes at infinity, andhence so does φ1 since χphq ‰ 1. Therefore φ “ φ1 `φ2 vanishes at infinity, whenceχ “ φ|H vanishes at infinity on H. Since χ is a character, this forces H to becompact, which is a contradiction. �

Lemma 7.4.11. Let G be a locally compact group and let N be a noncompactclosed normal subgroup of G such that G{N is compact and second countable. Sup-pose that


If H is an abelian closed subgroup of G and H is extending, then H must be compact.

Proof. Let χ be a character of H and let φ P P pGq be an extension of χ.Write φ “ φ1 ` φ2, where φ1 P BpG{Nq ˝ q and φ2 P C0pGq. As in the proof ofLemma 7.4.10,

φ1pxhq ´ χphqφ1pxq “ χphqφ2pxq ´ φ2pxhq

for all x P G and h P H. Since the function x Ñ φ1pxhq ´χphqφ1pxq is constant onN , whereas x Ñ χphqφ2pxq´φ2pxhq vanishes at infinity, and sinceN is noncompact,it follows that

φ1pxhq “ χphqφ1pxq and φ2pxhq “ χphqφ2pxq

for all x P G and h P H.Now assume that H is noncompact. Since the function h Ñ φ2pxhq vanishes

at infinity on H and its absolute value equals |φ2pxq|, we conclude that φ2pxq “ 0for every x P G. Thus φ “ φ1 and so φ P P pG{Nq ˝ q. Let L “ HN . Then L{Nis abelian since HN{N is abelian and dense in L{N . By what we have shown, for

each χ P pH there exists φ P P 1pL{Nq ˝ q such that φ|H “ χ. By Lemma 7.1.4, wecan assume that φ P expP 1pLqq and hence, since L{N is abelian,

φ P expP 1pL{Nqq ˝ q “ zL{N ˝ q.


Thus we have seen that pH Ď pzL{N ˝ qq|H . But zL{N is countable since L{N is

compact and second countable. Consequently, pH is countable. This contradicts thenoncompactness of H. �

Example 7.4.12. We find out all the extending subgroups of the ax` b-groupG “ R � Rˆ

`. We identify R with the normal subgroup t1u ˆ R and Rˆ` with the

subgroup Rˆ` ˆ t0u. Lemma 7.4.10 applies to G with N “ R. It is known that the

normal subgroup R is not extending. Let H be a nontrivial extending subgroupof G and assume first that H is connected. Since rH,Hs Ď R, by Lemma 7.4.10,HXR “ rH,Hs. Thus, since rH,Hs is connected, either HXR “ R or HXR “ t0u.In the first case, H “ R or H “ G as H is connected. In the second case, withq : G Ñ Rˆ

` denoting the quotient homomorphism, qpHq “ Rˆ` since qpHq is

connected. Thus G “ R�H, whence H is extending.Now, let H be an arbitrary extending subgroup of G. Since H0 is open in H,

H0 is extending as well. By what we already know, either H0 “ t0u, H0 “ G or Gis the semidirect product of R with H0. In the last case, H0 “ H. Indeed, givenany a P Rˆ

`, there is a unique xa P R such that pa, xaq P H0. Now

p1, yqpa, xaqp1, yq´1

“ pa, xa ` y ´ ayq,

so that y “ ay for all a P Rˆ` and hence y “ 0 whenever p1, yq normalizes H0. Thus,

it remains to consider the case that H is discrete. Then H X N is extending andhence H X N Ď rH X N,H X N s “ t0u by Lemma 7.4.10. Consequently, if H isany proper extending subgroup of G, then H X R “ t0u.

We claim that conversely each closed subgroupH withHXR “ t0u is extending.To that end, we observe that HN is closed in G. In fact, let ptn, anq P H, n P N,be such that an Ñ a P Rˆ

`, and fix some ps, bq P H with b ‰ 1. Then, since H isabelian,

ps ` btn, banq “ ps, bqptn, anq “ ptn, anqps, bq “ ptn ` ans, anbq

and hence pb ´ 1qtn “ pan ´ 1qs. It follows that

ptn, anq Ñ

ˆ

a ´ 1

b ´ 1s, a

˙

P H.

Since HN is closed and H XN “ t0u, H is topologically isomorphic to HN{N . ByLemma 7.1.3, this implies that H is extending since G{N is abelian.

Summarizing, we have seen that a proper closed subgroup H of G is extendingif and only if H X R “ t0u.

Example 7.4.13. Let G “ SOp2q � R2 where SOp2q acts on R2 by rotation,and identify R2 with the normal subgroup N “ tpE, xq : x P R2u, where E is theunit matrix. We claim that a proper closed subgroup of G is extending only if it iscompact. For that, notice first that no non-trivial subgroup of R2 can be extendingsince BpGq Ď BpG{Nq ˝ q ` C0pGq.

Now let H be a proper extending subgroup of G. Since H0 is open in H, H0 isextending as well. By Lemma 7.4.10, H0 X R2 “ rH0, H0s which is connected. SoH0 X R2 is either equal to t0u or R2 or Rv for some v P R2, v ‰ 0. If H Ě R2 then,since H is proper, H0 “ R2 which fails to be extending. Next, let H0 X R2 “ Rv.Then H0 “ Rv since E and ´E are the only elements of SOp2q mapping Rv toitself. However, Rv is not extending. Finally, let H0 X R2 “ t0u. Then eitherH0R2 “ G or H0 is trivial. In the first case, H0 is compact and from H0 XR2 “ t0u

7.5. THE SEPARATION PROPERTY: BASIC FACTS AND EXAMPLES 257

and H0R2 “ G it is easily deduced that H0 is its own normalizer. In particular,H0 “ H. In the second case H is discrete and hence H X R2 is extending. ThusH X R2 “ t0u and hence H is abelian. Now Lemma 7.4.11 shows that H is finite.This finishes the proof of the above claim. Thus a proper closed subgroup of G isextending if and only if it either finite or conjugate to SOp2q.

7.5. The separation property: Basic facts and examples

In the following three sections we study the separation property of positivedefinite functions exposed at the outset of this chapter. To start with, we introducethe basic definitions.

Definition 7.5.1. Let G be a locally compact group. For any closed subgroupH of G, let

PHpGq “ tφ P P pGq : φphq “ 1 for all h P Hu.

(1) We say that G has the H-separation property or H is separating in G iffor every x P GzH, there exists φ P PHpGq such that φpxq ‰ 1. When G has theH-separation property for every closed subgroup H of G, we refer to G as a groupwith the separation property.

(2) We say that G has the separation property for cyclic subgroups if everyclosed subgroup of G which is isomorphic to Z is separating.

Remark 7.5.2. (1) If H is either compact, or open, or normal, then G has theH-separation property.

(2) The separation property for cyclic subgroups, which is apparently weakerthan the separation property, is of interest because for special classes of locally com-pact groups, such as compactly generated nilpotent groups and almost connectedgroups, it will be seen to actually enforce the separation property (see Theorems7.7.10 and 7.6.7). Note, however, that the separation property for cyclic subgroupsis a local property in the sense that a locally compact group has this property if(and only if) every compactly generated open subgroup does so.

(3) Recall that for any φ P P pGq we have

|φpxyq ´ φpxqφpyq|2

ď p1 ´ |φpxq|2qp1 ´ |φpyq|

2q

for all x, y P G (Proposition 1.4.16(iv))). In particular, if H is a closed subgroup ofG and φ P PHpGq, then

φph1xh2q “ φpxq

for all x P G and h1, h2 P H. This simple property will be used frequently.

We now determine the separating subgroups for several classical groups. Someof these examples will be used later when dealing with solvable connected Liegroups.

Example 7.5.3. Let G be the 3-dimensional Heisenberg group. Thus G “ R3

with multiplication

px1, x2, x3qpy1, y2, y3q “ px1 ` y1, x2 ` y2, x3 ` y3 ` x1y2q,

xi, yi P R, i “ 1, 2, 3. Let H be a closed subgroup of G. Then G has the H-separation property (if and) only if H is normal in G. To see this, consider elementsa “ px1, x2, x3q P H and b “ py1, y2, y3q P G. It is straightforward to check that

ra, bs “ aba´1b´1“ p0, 0, x1y2 ´ x2y1q.


For any n P N, let

an “ pnx1, nx2, nx3q and bn “ py1{n, y2{n, y3{nq.

Then an P H and ran, bns “ ra, bs for all n. Now, let ϕ P expPHpGqq Ď expP 1pGqq.Then ϕpxzq “ ϕpxqϕpzq for all x P G and z P ZpGq, and hence

ϕpbnq “ ϕpanbba´1n q “ ϕpran, bnsbnq

“ ϕpran, bnsqϕpbnq “ ϕpra, bsqϕpbnq.

Since bn Ñ e as n Ñ 8, ϕpbnq Ñ 1, and this implies that ϕpra, bsq “ 1. Since thisholds for every ϕ P expPHpGqq, we conclude that φpra, bsq “ 1 for all φ P PHpGq.Finally, because G has the H-separation property, it follows that ra, bs P H, andsince a P H and b P G were arbitrary, it follows that H is normal in G.

The next lemma might be viewed as an analogue of Lemmas 7.4.10 and 7.4.11.

Lemma 7.5.4. Let G be a locally compact group and suppose that G has aclosed normal subgroup N such that, with q : G Ñ G{N denoting the quotienthomomorphism,


If H is a closed subgroup of G and G has the H-separation property, then eitherN Ď H or H is compact.

Proof. Assume that H does not contain N and choose a P N with a R H.There exists φ P PHpGq such that φpaq ‰ 1. By hypothesis, φ “ φ1 ` φ2, whereφ2 P C0pGq and φ1 “ ϕ ˝ q for some ϕ P BpG{Nq. For all h P H we have

φ1phaq “ φphaq ´ φ2phaq “ φpaq ´ φ2phaq

and also, since a P N ,

φ1phaq “ φ1phq “ φphq ´ φ2phq “ 1 ´ φ2phq.

Combining these two equations, we get

φ2phq ´ φ2phaq “ 1 ´ φpaq ‰ 0.

Since the function h Ñ φ2phq ´ φ2phaq vanishes at infinity on H, we conclude thatH must be compact. �

Example 7.5.5. Let G be the ax ` b-group. Then the closed subgroups H ofG such that G has the H-separation property are precisely the normal ones. To seethis, we only have to observe that if H is nontrivial and G has the H-separationproperty, then H contains the normal subgroup Rˆ t1u. This follows from Lemma7.5.4 since BpGq “ BpRˆ

`q ˝ q ` ApGq, where qpb, aq “ a for pb, aq P G.

Example 7.5.6. Let G be the motion group of Rd (d ě 2), that is, G “

Rd �SOpdq where the special orthogonal group SOpdq acts on Rd by rotation. LetH be a closed subgroup of G such that G has the H-separation property. We claimthat H is either compact (and hence conjugate to some closed subgroup of SOpdq)or H contains Rd (and hence H “ Rd �K for some closed subgroup K of SOpdq).It is clear that conversely, for every such subgroup H, G has the H-separationproperty.

Let E denote the d ˆ d-unit matrix. We shall use the following fact. Given0 ă r ă s and v P Rd such that }v} ě sd1{2, there exists D P SOpdq such that}E ´ D} ď r{s and }pE ´ Dqv} “ r. This can be seen as follows. There exists an


orthonormal basis te1, . . . , edu of Rd such that, with v “řd

j“1 αjej , |α1| ě s. Forϕ P R, let

w “ pα1 cos ϕ ´ α2 sin ϕqe1 ` pα1 sin ϕ ` α2 cos ϕqe2 `

dÿ

j“3

αjej .

Then w “ Dϕv, where Dϕ P SOpdq leaves e3, . . . , ed fixed and is the rotationassociated to ϕ in the pe1, e2q-plane. It follows that }E ´ Dϕ} “ |1 ´ eiϕ| and

}pE ´ Dϕqv} “ }v ´ w} “ |1 ´ eiϕ|pα21 ` α2

2q1{2.

Since the range of the continuous function ϕ Ñ |1 ´ eiϕ|pα21 ` α2

2q1{2 contains theinterval r0, 2ss, there exists ϕ such that }pE ´ Dϕqv}r. For this ϕ, it follows that

r “ |1 ´ eiϕ|pα21 ` α2

2q1{2

ě |1 ´ eiϕ|s,

and therefore }E ´ Dϕ} ď r{s.Now, let H be a noncompact subgroup of G and suppose that G has the H-

separation property. We first show that given r ą 0, there exists y P Rd such that}y} “ r and py, Eq P H. Since H is noncompact, there exists a sequence pxn, Anqn

in H with }xn} ě nd1{2 for all n. Then, for C P SOpdq and φ P PHpGq,

φpp0, Cqq “ φppxn, Anq´1

p0, Cqpxn, Anqq

“ φppA´1n pCxn ´ xnq, A´1

n CAnqq.

By what we seen above, for each n large enough, there exists Cn P SOpdq such that}E ´ Cn} ď r{n and }Cnxn ´ xn} “ r. Passing to a subsequence if necessary, wecan assume that A´1

n pCnxn ´ xnq Ñ y for some y P Rd. Then }y} “ r and, forevery φ P PHpGq,

1 “ limnÑ8

φpp0, Cnqq “ limnÑ8

φppA´1n pCxn ´ xnq, A´1

n CAnqq

“ φppy, Eqq.

Since G has the H-separation property, it follows that py, Eq P H. In particular,as r ą 0 is arbitrary, we conclude that H X Rd is a nondiscrete subgroup of Rd.Let V denote the connected component of H X Rd. Then V is a vector space ofpositive dimension. Towards a contradiction, suppose that dimV ă d. Choose alinear subspace W of Rd containing V such that dimpW {V q “ 1, and define closedsubgroups SpW q and SV pW q of SOpdq by

SpW q “ tA P SOpdq : ApW q “ W u

and

SV pW q “ tA P SpW q : ApV q “ V u.

Then, for any A P SpW qzSV pW q, we have

W “ tx ` Ay : x, y P V u.

Now, for x, y P V , A P SOpdq and φ P PHpGq,

φpp0, Aqq “ φppx,Eqp0, Aqpy, Eqq “ φppx ` Ay,Aqq.

There exists a sequence pAnqn in SpW qzSV pW q such that An Ñ E in SOpdq. Itfollows that φpp0, Anqq “ φppw,Anqq and hence

1 “ limnÑ8

φpp0, Anqq “ limnÑ8

φppw,Anqq “ φppw,Eqq


for every φ P PHpGq and w P W . Thus W ˆ tEu Ď H. This contradiction provesthat dimV “ d, whence H Ě Rd, as was to be shown.

Example 7.5.6 could alternatively be treated by appealing to Lemma 7.5.4 andusing that for the motion group G “ Rd � SOpdq one has BpGq Ď BpG{Rdq `

C0pGq. This latter fact can either be deduced from [8, Corollary of Theorem 4]or from a result of Chou [31, Theorem 3.3] stating that W pGq “ AP pGq ` C0pGq,where AP pGq and W pGq denotes the space of almost periodic and weakly almostperiodic functions of G, respectively. However, the proof of [31, Theorem 3.3] isfairly intricate and involves Grothendieck’s criterion for weak almost periodicity ofcontinuous functions.

We now introduce a concept which turns out to be of considerable relevance asit describes precisely the separating subgroups of locally compact groups having analmost connected open normal subgroup (see Theorem 7.5.9).

Definition 7.5.7. Let H be a closed subgroup of G. Then H is called neutralin G if given any neighbourhood U of the identity, there exists a neighbourhood Vof the identity such that V H Ď HU .

Such subgroups were first considered in [239] in the context of invariant mea-sures on homogeneous spaces and have later been studied extensively in [190] and[244].

Remark 7.5.8. (1) A subgroup H of G, which is either open, or compact, orclosed and normal, is obviously neutral. More generally, if G P rSINsH then H isneutral in G. In fact, given U as above, there exists a neighbourhood V of e suchthat V Ď U and hV h´1 “ V for all h P H, and this implies that HV “ V H Ď UH.

(2) If H is a neutral subgroup of G, then there exists a neighbourhood basisV of the identity such that V H “ HV for all V P V . This can be seen as follows.Replacing U by U X U´1, we can assume that U is symmetric. Then choose asymmetric neighbourhood W of the identity with HW Ď UH and let V “ U X

HWH. It is easily verified that V satisfies pV Hq´1 Ď V H and hence pV Hq´1 “

V H. Consequently, HV “ pV Hq´1 “ V H.

Theorem 7.5.9. Let G be a locally compact group and H a closed neutralsubgroup of G. Then, given any compact subset C of G with C X H “ H, thereexists u P PHpGq such that upxq “ 0 for all x P C. In particular, G has theH-separation property.

Proof. Note first that since H is neutral in G, there exists a quasi-invariantmeasure μ on the left coset space G{H (equivalently, the modular functions of Gand H agree on H). So Weil’s formula

ż

G

fpxqdx “

ż

G{H

ˆż

H

fpxhqdh

˙

dμpxHq

holds for all f P L1pGq (Section 1.3). As before, let TH denote the homomorphismfrom L1pGq onto L1pG{H,μq.

We choose a symmetric neighbourhood U of the identity in G such that UCU X

H “ H. Since H is neutral in G, there exists a compact symmetric neighbourhoodV of the identity such that V Ď U and HV “ V H. Let q : G Ñ G{H denote thequotient map and let v be a non-negative function in L1pGq such that THv is equal


to μpqpV qq´1{2 on qpV q and THv vanishes on G{HzqpV q. Then THv has norm onein L2pG{H,μq.

Now define a function u on G by

upxq “

ż

G

vpyqTHvpx´1yHqdy, x P G.

Using the choice of U , V and v, it is then easily verified that upxq “ 0 for all x P C.We claim that uphq “ 1 for all h P H. To see this, observe that if y P V , then thereexist k P H and z P V such that hy “ zk. Thus

ż

H

vphytqdt “

ż

H

vpzsqds “ μpqpV qq´1{2.

Since μ is H-invariant, it follows that

uphq “

ż

G{H

ˆ

THvph´1yHq

ż

H

vpytqdt

˙

dμpyHq

“

ż

qpV q

ˆ

THvpyHq

ż

H

vphytqdt

˙

dμpyHq

“ 1.

Finally, denoting by π the unitary representation of G induced from the trivialone-dimensional representation of H, the formula defining u can be rewritten as

upxq “

ż

G{H

THvpx´1yHqTHvpyHqdμpyHq “ xπpxqTHv, THvy.

This shows that u is positive definite and completes the proof. �

Theorem 7.5.9 especially shows that every SIN-group has the separation prop-erty. It also raises the question of whether conversely a closed subgroup H hasto be neutral whenever G has the H-separation property. This will turn out tobe true for locally compact groups G containing an almost connected open normalsubgroup (Theorem 7.7.2). The proof of Theorem 7.7.2 will build on the followingsequence of lemmas.

Lemma 7.5.10. Let H be a closed subgroup of G and let U be a neighbourhoodbasis of the identity of G. If G has the H-separation property, then

H “Ş

tHUH : U P Uu.

Proof. Let x PŞ

tHUH : U P Uu and, towards a contradiction, assume thatx R H. Then, since G has the H-separation property, there exists φ P PHpGq suchthat φpxq ‰ 1. Let ε “ |1´φpxq| ą 0. There exists U P U such that |φpzq´φpyq| ă

ε2

for all z, y P G with yz´1 P U .Now, since x P HUH , there exist a, b P H and u P U such that aub P Ux. Since

φpaubq “ φpuq, it follows that

|1 ´ φpxq| ď |1 ´ φpuq| ` |φpaubq ´ φpxq| ă ε.

This contradiction shows that x P H. �Lemma 7.5.11. Let H be a closed subgroup of G such that, for some neigh-

bourhood basis U of e in G, H “Ş

tHUH : U P Uu. Let K be a compact normalsubgroup of G and q : G Ñ G{K the quotient homomorphism. Then

qpHq “Ş

tqpHqV qpHq : V P Vu

for some neighbourhood basis V of qpeq in G{K.


Proof. Note first that qpHq is closed in G{K since q´1pqpHqq “ HK is closedin G. Let x P G such that qpxq R qpHq, that is, xK X H “ H. By hypothesis,for each k P K there exists Uk P U such that xk R HUkH . This implies thatxVk X HUkH “ H for some neighbourhood Vk of k in G. Since K is compact,K Ď

Ťnj“1 Vkj

for certain k1, . . . , kn P K. Let U “ Xnj“1Ukj

. Then

xK X HUH Ď x´

Ťnj“1 Vkj

¯

X HUH ĎŤn

j“1

`

xVkjX HUkj

H˘

“ H,

and hence qpxq R qpHUHq. Now, since qpHUHq is closed in G{K,

qpHqqpUqqpHq Ď qpHUHq Ď qpHUHq,

and therefore qpxq R qpHqqpUqqpHq. Taking for V the set of all finite intersectionsof sets qpUq, U P U , the conclusion of the lemma follows. �

Lemma 7.5.12. Let G be a locally connected locally compact group, and letH be a closed subgroup of G such that H “

Ş

tHUH : U P Uu, where U is aneighbourhood basis of e. Then H is neutral in G.

Proof. Let q : G Ñ G{H denote the map x Ñ xH, and equip G{H withthe quotient topology. Let U be the collection of all open neighbourhoods of e inG. It suffices to show that if U P U is relatively compact, then there exists V P Usuch that HV Ď UH. To that end, observe first that there exists V P U such thatqpHV q X qpUq Ď qpUq. In fact, qpUq is compact, qpUq is an open neighbourhood ofqpeq in G{H and

Ş

tqpHVHq X qpUq : V P Uu “ tqpequ,

and this implies that qpHVHq X qpUq Ď qpUq for some V since the sets qpHVHq,V P U , have the finite intersection property.

Since G is locally connected, we can assume that V is connected. Then qpHV q

is connected since, for each h P H, qphV q is connected and qpeq “ qphq P qphV q.

Now qpHV q X qpUq Ď qpUq means that qpHV q X qpUq is open and closed in qpHV q,and this in turn implies that qpHV q Ď qpUq since qpHV q is connected and qpHV qX

qpUq ‰ H. It follows that HV Ď UH. �

Lemma 7.5.13. Let G be a locally compact group and C a compact normalsubgroup of G such that G{C is locally connected. If G has the separation propertyfor cyclic subgroups, then G{C has the corresponding property.

Proof. Let B be a closed subgroup of G{C which is isomorphic to Z andlet q : G Ñ G{C denote the quotient homomorphism. Choose a P G such thatqpaq generates B and let A be the closed subgroup of G generated by a. Then Acannot be compact and hence is isomorphic to Z [125, Theorem (9.1)]. Moreover,qpAq “ B. Let U be a neighbourhood basis of the identity in G. Since A isseparating in G, A “ XtAUA : U P Uu (Lemma 7.5.10). By Lemma 7.5.11, itfollows that

B “Ş

tBV B : V P Vu

for some neighbourhood basis V of tCu in G{C. Since G{C is locally connected,this implies that B is neutral in G{C (Lemma 7.5.11). Finally, B being neutral, itis separating in G{C by Theorem 7.5.9. �


Lemma 7.5.14. Let G be a projective limit of groups G{Kα, α P A, and let Hbe a closed subgroup of G. If HKα{Kα is neutral in G{Kα for all α, then H isneutral in G.

Proof. Let W be a neighbourhood of e in G, and choose K “ Kα such thatK Ď W . Since K is compact, there exists a neighbourhood U of e such thatUK Ď W . Denote by q : G Ñ G{K the quotient homomorphism. By hypothesis,there exists a neighbourhood V of tKu in G{K such that qpHqV Ď qpUqqpHq.Then

Hq´1pV q Ď q´1

pqpUqqpHqq “ UKHK “ UKH Ď WH.

As W was an arbitrary neighbourhood of e, it follows that H is neutral in G. �

Corollary 7.5.15. Let G be a Lie-projective locally compact group and H aclosed subgroup of G. If H “

Ş

tHUH : U P Uu, where U is a neighbourhood basisof the identity, then H is neutral in G.

Proof. Since every Lie group is locally connected, the statement of the corol-lary follows from Lemma 7.5.11 and Lemma 7.5.13. �

We now present an example of a 2-step solvable, totally disconnected groupwhich fails to be an SIN-group and nevertheless has the separation property.

Example 7.5.16. Let p be a prime number and let N be the additive group ofthe p-adic number fields Ωp. Let K be the multiplicative group of p-adic numbersof valuation one and let G “ N �K, where K acts on N by multiplication. We aregoing to show that G has the separation property.

For that we first observe that if H is a noncompact separating subgroup of G,then H Ě N . Indeed, this follows from Lemma 7.5.4 since BpGq “ BpKq˝q`ApGq,where q : G Ñ K is the quotient homomorphism.

We now show that an arbitrary closed subgroup of G either is compact orcontains N . Let H be a closed subgroup of G such that N � H. We have to showthat H is compact. Then H X N is either trivial or a proper closed subgroup of Nand as such is compact. In fact, every nontrivial proper closed subgroup of N is ofthe form pkΔp, where Δp is the subring of p-adic integers and k P Z. Moreover,H{H X N is abelian since the homomorphism H{H X N Ñ G{N “ K is injective.We claim that qpHq is closed in K. To see this, let pxn, anq P H, n P N, such thatan Ñ a for some a P K. Assume that qpHq ‰ t1u and fix py, bq P H with b ‰ 1.Since H{H X N is abelian, there exists a sequence pznqn in H X N such that

pxn ` any, anbq “ pxn, anqpy, bq “ pzn, 1qpy, bqpxn, anq

“ pzn ` y ` bxn, banq,

and consequently, for all n,

xn ` any “ zn ` y ` bxn.

Moreover, since H XN is compact, we can assume that zn Ñ z for some z P H XN .It follows that

p1 ´ bqxn “ zn ` p1 ´ anqy Ñ z ` p1 ´ aqy.

Thus pxnqn converges to x “ p1 ´ bq´1pz ` p1 ´ aqyq P N . Hence pxn, anq Ñ px, aq

in H, whence a P qpHq, as required.


Finally, since H is σ-compact and qpHq is closed in K, the continuous isomor-phism from H{H XN onto HN{N “ qpHq is a homeomorphism. So H is compactsince both H X N and H{H X N are compact.

We conclude this section by briefly pointing out the connection of the separationproperty with the so-called Mautner phenomenon (see also Section 7.8).

Definition 7.5.17. Let G be a locally compact group and p rH,Hq a pair of

closed subgroups of G with rH Ě H. Then the Mautner phenomenon holds for

p rH,Hq if for every unitary representation π of G, each unit vector ξ P Hpπq which

is invariant under H is also invariant under rH , that is, if πphqξ “ ξ for all h P H,

then πpxqξ “ ξ for all x P rH .

Remark 7.5.18. (1) Let H be a closed subgroup of G and ϕ P P pGq. Thereexist a unitary representation π of G and ξ P Hpπq such that ϕpxq “ xπpxqξ, ξy

for all x P G. Then it is easily verified that ϕ P PHpGq if and only if }ξ} “ 1 andφphqξ “ ξ for all h P H. Thus G has the H-separation property precisely when H is

the only closed subgroup rH of G containing H such that the Mautner phenomenon

holds for p rH,Hq.(2) Let ϕ P P 1pGq. Then ϕ´1p1q is a closed subgroup of G. For any closed

subgroup H of G, define Hsep by

Hsep “Ş

tϕ´1p1q : ϕ P PHpGqu.

Then Hsep is the smallest closed subgroup of G containing H for which G has theseparation property. In particular, G has the H-separation property if and only ifHsep “ H. Furthermore, the Mautner phenomenon holds for pHsep, Hq, and Hsepis the largest subgroup of G with this property.

7.6. The separation property: Nilpotent Groups

In the remainder of this chapter our goal is twofold. On the one hand we aimat a criterion for a given closed subgroup to be separating. On the other hand wewant to derive a structure theorem for groups having the separation property. Inthis section we treat nilpotent groups, whereas the next section is devoted to groupswhich possess an almost connected open normal subgroup.

Lemma 7.6.1. Let G be a locally compact group and H a closed subgroup of G.Let

N “ tx P G : |ϕpxq| “ 1 for all ϕ P expPHpGqqu.

Then N is a closed subgroup of G containing the centre of G, and G has the N-separation property.

Proof. Clearly, N is a closed subgroup of G. To see thatN contains the centreofG, let ϕ P expPHpGqq, and let π denote the Gelfand-Naimark-Segal representationof G associated with ϕ. Since expPHpGqq Ď expP 1pGqq, π is irreducible, and hence

π|ZpGq is a multiple of some character χ P zZpGq. Let ξ P Hpπq such that ϕpxq “

xπpxqξ, ξy for all x P G. Then

|ϕpzq| “ |χpzq| ¨ }ξ}2

“ 1

for all z P ZpGq. Now, let

F “ t|ϕ|2 : ϕ P expPHpGqqu Ď PN pGq.

7.6. THE SEPARATION PROPERTY: NILPOTENT GROUPS 265

If x P GzN , then |ϕpxq| ă 1 for some ϕ P expPHpGqq and hence ψpxq ‰ 1 for someψ P F . This shows that G has the N -separation property. �

Lemma 7.6.2. Let G, H and N be as in Lemma 7.6.1. Suppose that G has theH-separation property and that N is normalized by G0, the connected componentof the identity. Then G0 normalizes H.

Proof. Let ϕ P expPHpGqq and put Vϕ “ tx P G : ϕpxq ‰ 0u. Then Vϕ is anopen set containing N . For x P G0 and h P H we have rh, xs P N since H Ď N andG0 normalizes N . So, since ϕ is of absolute value one on N ,

ϕpxq “ ϕprh, xsxq “ ϕprh, xsqϕpxq.

Hence, if x P Vϕ X G0,

ϕpxhx´1q “ ϕprh, xsq “ 1

for all h P H. Now, let x P Vϕ X G0, y P G0 and h P H. Then, since zHz´1 Ď Nfor each z P G0,

ϕpxqϕpyhy´1q “ ϕpxyhy´1

q “ ϕppxyqphpxyq´1xq

“ ϕppxyqhpxyq´1

qϕpxq.

Thus ϕpxyHpxyq´1q “ ϕpyHy´1q since ϕpxq ‰ 0. This equation shows that the setof all elements y in G0 such that ϕpyHy´1q “ t1u contains the subgroup generatedby Vϕ X G0. Since G0 is connected, it follows that ϕpxHx´1q “ t1u for all x P G0

and all ϕ P expPHpGqq. Define a closed subgroup K of G by

K “ ty P G : ϕpyq “ 1 for all ϕ P expPHpGqqu.

We claim that K Ď H. For that, notice first that if C is a closed subgroup of Gand pφαqα is a net in PCpGq converging to some φ P P pGq in the w˚-topology, thenφpcxq “ φpxq for all x P G and c P C. Indeed, for any c P C, Lcφα Ñ Lcφ in thew˚-topology and Lcφα “ φα, whence φ “ Lcφ. It follows that r0, 1sPCpGq “ tλϕ :λ P r0, 1s, ϕ P PCpGqu is w˚-closed in L8pGq. By the Krein-Milman theorem, thisimplies that

r0, 1sPHpGq “ copexpr0, 1sPHpGqqq “ copt0u Y expPHpGqqq,

where the closures are taken in the w˚-topology. Now, by definition ofK, expPHpGqq

Ď PKpGq. Thus r0, 1sPHpGq Ď r0, 1sPKpGq and hence PHpGq Ď PKpGq. Since Ghas the H-separation property, we conclude that K Ď H. We have seen above thatxHx´1 Ď K for all x P G0. This proves that G0 normalizes H. �

Lemma 7.6.3. Let G be a locally compact group such that G0 Ď ZmpGq for somem P N. Let H be a closed subgroup of G and suppose that G has the H-separationproperty. Then G0 normalizes H.

Proof. Notice first that if M is a closed normal subgroup of G which is con-tained in G0, then pG{Mq0 “ G0{M . To prove the statement of the lemma, weapply the preceding two lemmas and induction on m. Let M “ G0 XZ1pGq, and letN be as in Lemma 7.6.1. Then G has the N -separation property and hence G{Mhas the N{M -separation property. Since pG{Mq0 “ G0{M Ď Zm´1pG{Mq, by theinductive hypothesis G0{M normalizes N{M and hence G0 normalizes N . Then,by Lemma 7.6.2, G0 normalizes H since G has the H-separation property. �


The following theorem is the first main result of this section. It in particularshows that if G is a connected nilpotent group and H is a closed subgroup of G,then G has the H-separation property (if and) only if H is normal in G. Moreover,for such a group G, the neutral subgroups are precisely the normal subgroups.

Theorem 7.6.4. Let G be a locally compact group such that G0 is open in Gand G0 Ď ZmpGq for some m P N. For a closed subgroup H of G, the followingconditions are equivalent.

(i) G has the H-separation property.(ii) G0 normalizes H.(iii) H is neutral in G.

Proof. (i) ñ (ii) is a consequence of Lemma 7.6.3. The implication (ii) ñ

(iii) is trivial since G0 is open in G, and (iii) ñ (i) holds for any locally compactgroup by Theorem 7.5.9. �

We now investigate the problem of when a compactly generated nilpotent hasthe separation property.

Corollary 7.6.5. Let G be a connected nilpotent Lie group which has theseparation property for cyclic subgroups. Then G is abelian.

Proof. Let a be an arbitrary element of G and let A be the closed subgroupgenerated by a. Then either A is compact or isomorphic to Z. In any case, A isseparating in G, and hence normal in G by Theorem 7.6.4. If now A “ Z, thenrG, as “ teu because rG, as is connected and contained in A. If A is compact, thenconsider the action of G on the discrete dual group of A. Since G is connected,this action is trivial and hence so is the action of G on A itself. In either case a iscontained in the centre of G. �

Lemma 7.6.6. Let G be a nilpotent Lie group and suppose that G has theseparation property for cyclic subgroups. Then G0 is contained in the centre ofG.

Proof. Since G0 has the separation property for cyclic subgroups, G0 “ V ˆKwhere V is a vector group andK is a compact connected Lie group (Corollary 7.6.5).Moreover, because K is nilpotent, K “ Td for some d P N0.

Let a be an arbitrary element of G and let A denote the closed subgroup ofG generated by a. Since A is either compact or isomorphic to Z and G has theseparation property for cyclic subgroups, it follows from Lemma 7.6.3 that G0

normalizes A. We now distinguish the two cases that a has finite order or infiniteorder modulo the open normal subgroup G0.

Assume first that aq P G0 for some q P N. Using the facts that G0 and A areabelian and that G0 normalizes A, it is easily verified by induction that, for ally P G0 and n P N,

ryn, as “ ry, asn

“ ry, ans.

Let now x be an arbitrary element of G0. Since both V and Td are divisible, thereexists y P G0 such that yq “ x. Then

rx, as “ ryq, as “ ry, aqs “ e.

This shows that rG0, as “ teu. Now let a have infinite order modulo G0. ThenA X G0 “ teu and therefore rG0, as Ď A X G0 “ teu. Summing up we have seenthat G0 is contained in the centre of G. �

7.6. THE SEPARATION PROPERTY: NILPOTENT GROUPS 267

We have seen in Section 7.5 that every SIN-group has the separation property.Now we are able to prove the converse for compactly generated nilpotent groups.Actually, the implication (ii) ñ (iv) of the following theorem constitutes an evenstronger result.

Theorem 7.6.7. Let G be a compactly generated nilpotent locally compactgroup. Then the following conditions are equivalent.

(i) G has the separation property.(ii) G has the separation property for cyclic subgroups.(iii) G is a projective limit of groups each of which has an open centre.(iv) G is an SIN-group.

Proof. Since a projective limit of SIN-groups is again an SIN-group and everySIN-group has the separation property, it only remains to prove the implication (ii)ñ (iii). As G is a compactly generated nilpotent group, it is a projective limit ofLie groups G{Cα [127, 9, Theorem]. By Lemma 7.5.13, the separation property forcyclic subgroups passes to the quotient groups G{Cα. Since G{Cα is a Lie group,the connected component pG{Cαq0 is open in G{Cα. Lemma 7.6.6 now shows thatpG{Cαq0 is contained in the centre of G{Cα. This proves (iii). �

We remind the reader that a locally compact group G is called locally nilpotentif every compactly generated (open) subgroup of G is nilpotent. Examples of suchgroups can be constructed as follows. Let A be an infinite index set, and for eachα P A let Gα be a locally compact nilpotent group of nilpotency length �pGαq.Suppose that there is a finite subset B of A such that for all α P AzB, Gα possessesa compact open subgroup Kα such that supt�pKαq : α P AzBu ă 8. Form therestricted direct product

G “ tpxαqαPA Pś

αPA Gα : xα P Kα for almost all α P Au .

Then, equipped with the product topology, G is a locally compact group whichis locally nilpotent, but not nilpotent whenever the set of numbers �pGαq is un-bounded.

Remark 7.6.8. In the situation of Theorem 7.6.7 condition (iii) is equivalent toG0 being contained in the centre of G. In fact, if G is the projective limit of groupsG{Cα and each G{Cα has an open centre, then G0Cα{Cα “ pG{Cαq0 Ď ZpG{Cαq

for all α, and this implies that G0 Ď ZpGq. Conversely, if G0 Ď ZpGq and G is aprojective limit of Lie groups G{Cα, then pG{Cαq0 “ G0Cα{Cα Ď ZpG{Cαq andpG{Cαq0 is open in G{Cα for every α, so that (iii) holds.

Corollary 7.6.9. Let G be a locally nilpotent locally compact group. Then Ghas the separation property for cyclic subgroups if and only if G0 is contained inthe centre of G.

Proof. If G0 is contained in the centre of G, then by Theorem 7.6.7 andRemark 7.6.8, every compactly generated open subgroup of G has the separationproperty for cyclic subgroups. Hence the same is true of G.

Conversely, suppose thatG has the separation property for cyclic subgroups andlet a be an arbitrary element of G. Choose a compactly generated open subgroupH of G containing a. Then H is nilpotent and has the separation property for cyclicsubgroups. Therefore, Theorem 7.6.7 applies to H and, using Remark 7.6.8 again,


we conclude that G0 “ H0 is contained in the centre of H. Thus rG0, as “ teu, andsince a P G was arbitrary, it follows that G0 is contained in the centre of G. �

7.7. The separation property: Almost connected groups

In this section we consider a class of locally compact groups for which theresults are considerably more complete than for nilpotent groups. The first mainachievement is Theorem 7.7.2 below which, for a locally compact group havingan almost connected open normal subgroup, identifies the separating subgroups asprecisely the neutral subgroups.

Proposition 7.7.1. Let G be a locally compact group containing an almostconnected open normal subgroup N . Let H be a closed subgroup of G such thatG “ HN and G has the H-separation property. Then there exists a compact normalsubgroup C of G such that C Ď H and G{C is a projective limit of Lie groups.

Proof. Note first that N , being almost connected, possesses a maximal com-pact normal subgroup K. Then N{K is a Lie group, and since K is characteristicin N and N is normal in G, K is normal in G.

For any τ P pK, let χτ denote the normalized trace of τ , that is, the functionχτ pxq “ d´1

τ trpτ pxqq, x P G. Here dτ denotes the dimension of Hpτ q and, for anyoperator T in a Hilbert space, trpT q denotes the trace of T . Let EpKq “ tχτ : τ P

pKu Ď L1pKq. If π is a unitary representation of G, then Hpπq “ Hpπ|Kq admitsan orthogonal decomposition

Hpπq “ ‘ tπpχqHpπq : χ P EpKqu.

Consider ξ P Hpπq satisfying πphqξ “ ξ for all h P H. Then, for each χ P EpKq,

πphqπpχqξ “ πphqπpχqπph´1qξ “

ż

K

χpxqπphxh´1qξ dx

“

ż

K

χph´1xhqπpxqξ dx “ πph ¨ χqξ

and hence }πpχqξ} “ }πph ¨ χqξ} for all h P H. This implies that ξ belongs to theclosed linear span of those subspaces πpχqHpπq for which the H-orbit H ¨ χ of χ isfinite. Now, let

C “Ş

xPG xpH X Kqx´1 “ K X`Ş

xPG xHx´1˘

,

and consider any y P KzC. Then there exists x P G such that z “ xyx´1 R H.Since G has the H-separation property, there exist a unitary representation π of Gand ξ P Hpπq such that πpzqξ ‰ ξ and πphqξ “ ξ for all h P H. By what we haveobserved above about such ξ, we can replace ξ by

ř

χ1PH¨χ πpχ1qξ, where χ P EpKq

is such that H ¨ χ is finite. Notice next that if a connected group acts on K, then

the orbits in the discrete dual space pK are trivial. Since N is almost connected, it

follo ws that the N -orbits in pK and hence the N -orbits in EpKq are all finite. Now,by hypothesis, G “ HN “ NH. This implies that the G-orbit G ¨ χ of χ is finite.

Let τ P pK such that χ “ χτ , and define a closed subgroup L of K by

L “ tk P K : x ¨ τ pkq “ I for all x P Gu.

Then L is normal inG, and L is the kernel of σ, the direct sum of the representationsin G ¨ τ . Since G ¨ τ is finite, σ defines a faithful finite dimensional representation

7.7. THE SEPARATION PROPERTY: ALMOST CONNECTED GROUPS 269

of K{L. In particular, K{L is a Lie group, and since N{K is a Lie group and N isopen in G, it follows that G{L is a Lie group.

Summarizing, for each y P KzC we have found a closed normal subgroup Ly

of G such that y R Ly, Ly Ď K and G{Ly is a Lie group. Since then Ly X Vy “ H

for some neighbourhood Vy of y in K, it follows that given any compact subsetY of KzC, there exist y1, . . . , yn P KzC such that Y X p

Şnj“1 Lyj

q “ H. Since

G{ Xnj“1 Lyj

is a Lie group, we conclude that given any neighbourhood V of e inG, there exists a closed normal subgroup LV of G such that C Ď LV Ď V C andG{LV is a Lie group. In other words, G{C is a projective limit of Lie groups. Thisfinishes the proof of the proposition. �

Apart from results obtained in Section 6.5, the preceding proposition is themain tool needed to establish the following theorem.

Theorem 7.7.2. Let G be a locally compact group which has an open almostconnected normal subgroup. Then, for a closed subgroup H of G, the following areequivalent.

(i) G has the H-separation property.(ii) H is neutral in G.

Proof. (ii) ñ (i) follows from Theorem 7.5.9Conversely, suppose that (i) holds and let N be an open almost connected

normal subgroup of G and put rG “ HN . If H is shown to be neutral in rG, then it

is neutral in G because rG is open in G. Therefore we can assume that G “ HN .Then, by Proposition 7.7.1, there exists a compact normal subgroup C of G suchthat C Ď H and G{C is a projective limit of Lie groups. Of course, it sufficesto show that H{C is neutral in G{C. Consequently, we can further assume thatC “ teu, so that G is a projective limit of Lie groups. Using once more thatG has the H-separation property, we have H “

Ş

tHUH : U P Uu, where U isany neighbourhood basis of e in G (Lemma 7.5.10). Since G is locally connected,Corollary 7.5.15 now shows that H is neutral in G. �

In view of Theorem 7.7.2 it is surprising that it requires much more effortto derive a criterion for an almost connected locally compact group to have theseparation property. A first step towards such a criterion is the treatment of solvableconnected Lie groups (Lemma 7.7.4) through reducing to small dimensional groups.

Example 7.7.3. With the notation of [14, p. 180-182], let G “ G3,4pαq,α P R. Then G is a semidirect product of R2 with R and can be realized as R3 withmultiplication given by

px1, y1, t1qpx2, y2, t2q “ px1 ` eαt1px2 cos t1 ´ y2 sin t1q,

y1 ` eαt1px2 sin t1 ` y2 cos t1q, t1 ` t2q,

xj , yj , tj P R, j “ 1, 2. We now give an example of an infinite cyclic subgroup H ofG which is not separating.

First, let α “ 0. Then, for all x, y, t P R,

px, 0, 0qp0, 0, tqpy, 0, 0q “ px ` y cos t, y sin t, tq.

Let H “ tpm, 0, 0q : m P Zu and choose tn P p0, π{2q and xn, yn P R, n P N, suchthat sin tn “ 1{n, yn “ n and xn “ ´tn cos tnu, where for any t P R, ttu denotes the


largest integer ď t. Then, for any φ P PHpGq,

φp0, 0, tnq “ φpn cos tn ´ tn cos tnu, 1, tnq.

Since tn Ñ 0 as n Ñ 8 and some subsequence of the sequence n Ñ n cos tn ´

tn cos tnu converges to some s P r0, 1s, it follows that 1 “ φps, 1, 0q. Thus H is notseparating.

Now let α ą 0, the case n ă 0 being treated similarly. Let H “ tp0, 0, 2πnq :n P Zu and φ P PHpGq. Since, for all x P R and n P Z,

p0, 0, 2πnqpx, 0, 0qp0, 0, 2πnq “ pe2παnx, 0, 0q,

taking xn “ e´2παn, we get

1 “ limnÑ8

φpxn, 0, 0q “ limnÑ8

φpp0, 0, 2πnqpxn, 0, 0qp0, 0,´2πnqq “ φp1, 0, 0q.

This shows that G does not have the H-separation property.

Lemma 7.7.4. Let G be a solvable simply connected Lie group, and suppose thatG has the separation property for cyclic subgroups. Then G is abelian.

Proof. Towards a contradiction, assume that there exists a nonabelian, sim-ply connected, solvable Lie group which has the separation property for cyclicsubgroups, and let G be such a group of minimal dimension. We claim that Gcontains a nonabelian, simply connected subgroup H of dimension 2 or 3.

To that end, let V be a nontrivial normal vector subgroup of G of minimaldimension, and let q : G Ñ G{V denote the quotient homomorphism. It is well-known that V is of dimension 1 or 2 since G is solvable. Since G{V has theseparation property, by the minimality of G, G{V must be abelian and hence avector group. Now, if dim V “ 1, then choose noncommuting elements x and y inG and let H “ q´1pRqpxq ` Rqpyqq. If dimV “ 2, due to the minimality of V , Vcannot be contained in the centre of G. Then choose x P V and y P G such thatrx, ys ‰ e and let H “ q´1pRqpyqq.

Thus it suffices to show that no nonabelian, simply connected, solvable Liegroup G of dimension ď 3 does have the separation property for cyclic subgroups.Now, the noncommutative, solvable, real Lie algebras g of dimension ď 3 are clas-sified in [14, p. 180-182]. Retaining the notation of [14], any such g, except for theHeisenberg Lie algebra and the algebras g3,4pαq, α P R, contain the Lie algebra ofthe ax`b-group as a subalgebra. Since the separation property for cyclic subgroupsis inherited by closed subgroups, it therefore remains to observe that none of theHeisenberg group, the ax ` b-group and G3,4pαq, the simply connected Lie groupcorresponding to g3,4pαq, does have the separation property for cyclic subgroups.However, in the first two cases the only separating subgroups are the normal ones(Examples 7.5.3 and 7.5.6), whereas for G3,4pαq we have exhibited an example of anonseparating subgroup in the preceding example. �

The next step in our approach to Theorem 7.7.10 below consists of generalizingLemma 7.7.4 to the extent that an arbitrary connected Lie group, which has theseparation property for cyclic subgroups, has to be the direct product of a vectorgroup and a compact group (Proposition 7.7.9).

Lemma 7.7.5. Let G be a connected Lie group and let R be the radical of G. IfG has the separation property for cyclic subgroups, then G{R is compact.

7.7. THE SEPARATION PROPERTY: ALMOST CONNECTED GROUPS 271

Proof. Suppose that the semisimple Lie group G{R is noncompact. ThenG{R has an Iwasawa decomposition G “ KAN , where K is a compact group, Ais a nontrivial vector group and N is nontrivial connected nilpotent group. Thegroup A normalizes, but does not centralize N . Then, with q : G Ñ G{R thequotient homomorphism, q´1pANq is a connected solvable Lie group which hasthe separation property for cyclic subgroups. It follows from Lemma 7.7.4 thatq´1pANq is abelian. In particular, AN is abelian, which is a contradiction. ThusG{R is compact. �

Lemma 7.7.6. Let G be a semidirect product G “ V � A, where V is a vectorgroup and A is a connected and locally connected group. If G has the separationproperty for cyclic subgroups, then G is the direct product of the subgroups V andA.

Proof. Let a Ñ αa denote the homomorphism from A into GLpV q definingthis semidirect product, and let e and eA denote the identity of G and A, respec-tively.

We have to show that αapyq “ y for all y P V and a P A. Suppose that thereexist y P V and a P A such that αapyq ‰ y, and let H “ tpny, eAq : n P Zu. Since Ghas the H-separation property, Lemma 7.5.10 implies that H “ X tHUH : U P Uu,where U is any neighbourhood basis of e in G. This in turn implies that H isneutral in G because G is locally connected (Lemma 7.5.11). Since H is discrete andG is locally connected, we find a symmetric, compact, connected neighbourhoodW of e such that HW “ WH and W 2 X H “ teu. The latter condition andW “ W´1 imply that the sets Wh, h P H, are pairwise disjoint. Now the compactset py, eAqW is a finite union of disjoint closed sets of the form py, eAqW X Wh,h P H. Since W is connected, it follows that py, eAqW Ď W py, eAqm for somem P Z. But py, eAq “ vpy, eAqm for some v P W and m P Z with m ‰ 1 impliesthat H XW ‰ teu. This contradiction shows that py, eAqW Ď W py, eAq. The sameargument gives W py, eAq Ď py, eAqW . We now have, for all x P V and a P A suchthat p0, aq P W ,

px ´ αapxq, aq “ px, eAqp0, aqpx, eAq´1

P W.

There exists a P A such that p0, aq P W and αapyq ‰ y. Indeed, otherwise we haveαapyq “ y for all a P A since A is connected. Since every continuous automorphismof a vector group is linear, we conclude that

pnpy ´ αapyqq, aq “ pny ´ αapnyq, aq P W

for all n P N. This is impossible since W is compact. Consequently αapyq “ y forall a P A and y P V and hence G is the direct product of V and A. �

Corollary 7.7.7. Let G be a locally compact group containing a closed normalvector subgroup V such that G{V is a compact connected Lie group. If G has theseparation property for cyclic subgroups, then there exists a compact subgroup K ofG such that G is the direct product of V and K.

Proof. Since V is a vector group and G{V is compact, there exists a compactsubgroup K of G such that G is the semidirect product of V and K [128, TheoremVIII]. Now apply Lemma 7.7.6. �


In the proof of the next proposition we shall use the statement of the followinglemma. This appears to be folklore. However, being unaware of a reference, weinclude the proof for the readers convenience.

Lemma 7.7.8. Let N be a compact and solvable normal subgroup of a connectedlocally compact group G. Then N is contained in the centre ZpGq of G.

Proof. Assume first that N is abelian, and let px, χq Ñ x ¨ χ denote the

continuous action of G on the dual group pN of N given by x ¨ χpsq “ χpx´1sxq for

χ P pN , x P G and y P N . Then, since G is connected and pN is discrete, x ¨ χ “ χfor all χ and x. Since the characters of N separate the points of N , it follows thatN Ď ZpGq.

The statement of the lemma now follows by induction on the length �pNq of thedescending commutator series of N . Suppose that every solvable compact normalsubgroup H of G with �pHq ă �pNq is contained in the centre of G. Then takeH “ rN,N s, the closed commutator subgroup of N . Then rN,N s Ď ZpGq by whatwe have already seen. To conclude that N Ď ZpGq, assume that there exists somex P NzZpGq and let M denote the closed subgroup of G generated by x and N .Then M is abelian, and also normal in G because N{rN,N s is an abelian compactnormal subgroup of the connected group G{rN,N s. Applying the first paragraphagain, we get M Ď ZpGq. This contradicts x R ZpGq and completes the proof. �

Proposition 7.7.9. Let G be a connected Lie group which has the separationproperty for cyclic subgroups. Then G is the direct product of a vector group and acompact group.

Proof. To start with, suppose that G is solvable and let C be the maximalcompact connected normal subgroup of G. Then G{C is simply connected andhas the separation property for cyclic subgroups by Lemma 7.5.13. Lemma 7.7.8implies that G{C is abelian and hence a vector group. Moreover, since C is acompact normal solvable subgroup of the connected Lie group G, C is containedin the centre of G and C “ Td for some d P N0. Now Corollary 7.6.5 applies andyields that G is a direct product G “ W ˆ Td, where W is a vector group.

Now let G be an arbitrary connected Lie group having the separation propertyfor cyclic subgroups, and let R be the radical of G. Then, by the first paragraph,R “ W ˆ Td as above. By Lemma 7.7.5, G{R is compact.

Finally, consider the action of G on R through inner automorphisms. Since Td

is characteristic in R, it is normal in G. Since G{R is connected, the induced actionof G on the discrete dual group of Td is trivial. Thus Td is contained in the centreof G. Since G{R is compact, the stable decomposition theorem [110, Theorem 1.1]asserts the existence of a topological automorphism γ of R such that V “ γpW q

is normal in G. Consequently, V is a normal vector subgroup of G such that thequotient group G{V , which is an extension of G{R by Td, is compact and connected.An application of Corollary 7.7.7 gives that G “ V ˆ K, where K is a compactgroup. This completes the proof of the proposition. �

Combining the preceding proposition with results obtained in Section 7.5, weare now ready to deduce the second main result of this section.

Theorem 7.7.10. Let G be an almost connected locally compact group. Thenthe following four conditions are equivalent.


(i) G has the separation property.(ii) G has the separation property for cyclic subgroups.(iii) G contains an open normal subgroup N of finite index such that N is a

direct product of a compact group and a vector group.(iv) G is an SIN-group.

Proof. The implication (i) ñ (ii) is trivial and (iv) ñ (i) was shown in Theo-rem 7.5.9. Notice next that if G is as in (iii) (and not necessarily almost connected),then G is a projective limit of SIN-groups and hence an SIN-group itself. It there-fore only remains to prove that (ii) implies (iii). Since G is almost connected, itis a projective limit of Lie groups Gα “ G{Cα [214, Theorem 4.6]. Then each Gα

has the separation property for cyclic subgroups because Gα is locally connected(Lemma 7.5.13). By Proposition 7.7.8, the connected component of the identity ofGα, which has finite index in Gα, is a direct product of a compact group and avector group. In particular, each Gα is an SIN-group, and hence so is G. Thus,by Theorem 2.13 of [110], G has an open normal subgroup N such that N is thedirect product of a vector group and a compact group. Finally, since G is almostconnected, N must have finite index in G. Consequently, G is an SIN-group. �

We have seen in Theorem 7.7.2 that if G is a locally compact group havingan almost connected open normal subgroup, then the separating subgroups of Gare precisely the neutral subgroups. It is therefore tempting to try to establishTheorem 7.7.10 by showing that an almost connected locally compact group G hasto be an SIN-group provided that every closed cyclic subgroup of G is neutral inG. However, there appears to be no more direct and simpler proof than the onepresented above.

Example 7.7.11. Let G be a simple Lie group with finite centre. Veech [279,Theorem 1.4] has shown that W pGq “ C ` C0pGq, where W pGq denotes the spaceof all weakly almost periodic functions on G. Therefore, Lemma 7.5.4 applies withN “ G and shows that the compact subgroups are the only separating subgroups ofG. We remark that, alternatively, the fact that BpGq Ď C`C0pGq could be derivedfrom vanishing at infinity theorems for matrix coefficients of unitary representationsof G (see [294, Sections 2.2 and 2.4]).


Let G be a locally compact and let H be a closed subgroup of G. As observed inSection 7.1, it is easy to see thatH is extending in G ifH is either open, or compact,or contained in the centre of G. The last two cases are covered by Theorem 7.3.2,which states thatH is extending whenever G has small H-invariant neighbourhoodsand which was independently shown, using induced representations, by Henrichs[118] and Cowling and Rodway [39]. Of course, this result was raising the questionof whether conversely a locally compact group, which has the extension property,must be an SIN-group. That this is true for connected groups was shown in [39] andcan also be deduced directly from [120, Theorem 2]. Our proof that a connectedLie group which has the extension property, has to be a direct product of a vectorgroup and a compact group (Theorem 7.3.6), is taken from [39]. That the answerto the above question is also affirmative for compactly generated nilpotent groups(Theorem 7.4.6). However, according to Example 7.3.9, there exist locally compactgroups which have the extension property, but nevertheless fail to be SIN-groups.


In connection with Theorem 7.4.6, it is worth pointing out that Example 7.3.9,which was given in [119], is a 2-step solvable and compactly generated group.

The problem of when a single given closed subgroup is extending was studiedby several authors (see [28], [39], [118], [119], [156], [157], [194] and [212]) andnaturally is a more difficult question. The first substantial result in this directionwas Douady’s observation that if N is an abelian closed normal subgroup of G andχ is a character of N , then χ extends to a continuous positive definite function onG only if the stabilizer of χ is open in G (Proposition 7.2.1). Theorem 7.2.3, whichextends Proposition 7.2.1 considerably and which, for a normal subgroup N of Gand ϕ P P 1pNq, gives a necessary and sufficient condition for ϕ to be extendibleto some element of P pGq, was established by Cowling and Rodway [39, Theorem1]. The exposition in Section 7.2 follows closely that in [39]. Since, as indicatedin Section 7.1, the extension property can be formulated in terms of representationtheory, it is not surprising that the extension of ϕ constructed in the proof is realizedas a coordinate function of the representation induced by πϕ. We feel that the fullstrength of [39, Theorem 1] has not yet been exploited. Theorem 7.2.4 saying thatthe connected component of the identity is always extending, is due to Liukkonenand Mislove [194], who were not primarily interested in the extension property, butused Theorem 7.2.4 in their study of symmetry of the Fourier-Stieltjes algebra. Thevarious examples presented in Section 7.4 as well as the useful Lemmas 7.4.10 and7.4.11 are taken from [157]. The same applies to several other results accumulatedin Section 7.1 to 7.4.

The separation property for positive definite functions first appeared in [177]in connection with the invariant complementation problem for w˚-closed invariantsubalgebras of V NpGq. The first general fact that a closed subgroup H of G isseparating whenever G P rSINsH was obtained in [81]. In [81] as well as in [154]the interest in the separation property arose from applications in the ideal theoryof the Fourier algebra. Theorem 7.5.9, which covers the [SIN]H -group case andwhich was the first to indicate the importance of the neutral subgroup notion, isan adaptation of the [SIN]H -group result.

It was expected for some time that a locally compact group having the sepa-ration property, must be an SIN-group. This turns out to be true for compactlygenerated nilpotent groups on the one hand and almost connected groups on theother hand. Moreover, for both classes of locally compact groups the separationproperty for cyclic subgroups already ensures the small invariant neighbourhoodproperty. These results, Theorem 7.6.7 and Theorem 7.7.10, were established in[154] and [156], respectively. The proofs of both theorems involve structure theoryof the groups in question, some Lie group theory and the treatment of several spe-cial cases. For instance, Examples 7.5.3, 7.5.5 and 7.7.3 are used to prove Theorem7.7.10 for solvable connected Lie groups.

However, there exists an example of a 2-step solvable, totally disconnected andcompactly generated group which has the separation property, but nevertheless failsto be an SIN-group. The example was given in [157]. This group, which is oftenreferred to as the Fell group, has served as a counterexample to other conjecturesas well. It has, for instance, been the first example of a noncompact group whichhas a countable dual space.

Of course, the reader will have observed that there are many similarities be-tween the extension and the separation properties for the classes of nilpotent locally


compact groups and almost connected groups. Yet, there are marked differencesas soon as single subgroups or locally compact groups which are neither nilpotentnor almost connected are considered. A striking example is provided by Douady’sobservation: A normal subgroup is always separating, whereas an abelian normalsubgroup of a connected group is extending only if it is contained in the centre.

In [156] several partial results have been obtained towards when a given closedsubgroup H of G is separating in G (see Theorems 1.3 and 2.4 of [156]). Inparticular, a normal subgroup of a connected nilpotent group is separating only ifit is normal. The most satisfying result in this direction, however, is due to Losert[202, Theorem 2]. The proof presented here is a streamlined version of the onegiven in [202].

There is one aspect of the separation property which the authors of [154], [156]and [157] have not been aware of and which was brought to their attention through[202]. This is the relation with the so-called Mautner phenomenon explained inDefinition 7.5.17 and Remark 7.5.18. The Mautner phenomenon describes justthe opposite behaviour to the separation property. The classical case, done byMautner [208], was the ax ` b-group. A very deep study, culminating in [216], ofthe Mautner phenomenon for one-parameter subgroups of connected Lie groups wascarried out by C.C. Moore. Subsequently, S.P. Wang investigated the same problemfor cyclic subgroups of connected Lie groups [283] and for p-adic groups [284],with applications to Kazhdan’s property (T). Moore and Wang established, usingMackey’s theory of induced unitary representations, fairly explicit descriptions ofthe group Hsep (compare Definition 7.5.17) in terms of compactness conditions ofthe group of inner automorphisms defined by the elements of H.

As pointed out above, Theorem 7.7.2 was shown in [202, Theorem 2]. In ad-dition, using the results of Moore and Wang, Losert has also shown that in thesituation of 7.7.2, the neutral subgroups H of G can be built up from normal sub-groups and [SIN]H -groups. More precisely (see (iii) of [202, Theorem 2]), H isneutral in G if and only if H contains a closed subgroup N which is normal in G0Hsuch that G{N is a [SIN]H-group. We felt that this characterization, the proof ofwhich is rather complicated, is far beyond the scope of this monograph. The meth-ods developed in [202] also allow to handle the examples presented in Sections 7.5and 7.7 and to give an alternative, though definitely not shorter, approach to, e.g.,Theorems 7.7.10 and 7.6.4. Moreover, [202] also contains, for closed subgroups Hof G, a comparison of neutrality, local neutrality and the [SIN]H -property. Finally,very recently, Losert [203] has presented a description of the subgroup Hsep in thesituation when G contains an open, almost connected, H-invariant subgroup.

We have seen in Example 7.5.16 that in general the separation property doesnot force a group to be an SIN-group. Heavily exploiting constructions due toOl’shanskii [222], Losert has produced a very striking example demonstrating thisphenomenon; he constructed a locally compact group which is not an SIN-group,but has a compact open normal subgroup (thus, in particular, is a so-called IN-group) and such that every proper closed subgroup is compact. Note that such agroup also has the extension property.

We conclude these comments and historical remarks by pointing out some openproblems which arise naturally from the results and discussion in this chapter.

1. Determine all the extending subgroups of the motion groups Rd � SOpdq,d ě 2 (compare Example 7.4.12 for d “ 2).


2. Let N be a closed normal subgroup of a connected nilpotent group G.Find necessary and sufficient conditions for N to be extending in G. A necessarycondition is that every irreducible representation of N is fixed under the actionof G on the dual space of N . Moreover, it follows by induction from Douady’sobservation that the descending central series of N is obtained by intersecting themembers of the descending central series of G with N .

3. In view of Losert’s example [203], how restrictive for a locally compactgroup G is the hypothesis that G satisfies both, the extension and the separationproperties?

4. Nothing is known concerning both properties for nondiscrete, totally discon-nected, nonabelian, noncompact groups.

5. Is, for general locally compact groups, the separation property (like the onefor cyclic groups) a local property in the sense of Remark 7.5.2(2)?

6. Let G be a simple Lie group with finite centre. Then the proper separatingsubgroups are exactly the compact subgroups (Example 7.7.11). In contrast, itdoes not seem to be known which subgroups are extending.

7. If G is the 3-dimensional Heisenberg group, then a proper closed subgroupof G is extending if and only if it is either cyclic or isomorphic to R. However, forgeneral 2-step nilpotent simply connected Lie groups the question is open.

8. When does a not necessarily compactly generated, nilpotent group have theextension or the separation property? Does there exist a group theoretic charac-terization of extending or separating subgroups?

9. Let G be any locally compact group and H a closed neutral subgroup ofG. Must H be separating? Of course, by Theorem 7.7.2 the answer is affirmativewhen G contains an open, almost connected, normal subgroup.

APPENDIX A

A.1. The closed coset ring

For any (discrete) group H, let RpHq denote the Boolean ring generated by theleft cosets of subgroups of H. RpHq is called the coset ring of H. If G is a locallycompact group and Gd denotes the group G with the discrete topology, then theclosed coset ring RcpGq is defined by

RcpGq “ tE P RpGdq : E is closed inGu.

Gilbert [98] and Schreiber [258] have independently determined explicitly the struc-ture of sets in RcpGq when G is a locally compact abelian group. In [80], Forrestgave the description of sets in RcpGq for arbitrary locally compact groups sayingthat Gilbert’s long and complicated proof can be carried over line by line. However,a very elegant proof for abelian groups was found by Saeki [255]. This proof, whichis based on an idea of Cohen [34], can easily be adapted to work for general locallycompact groups and in what follows we present the details.

The first lemma is due to Cohen [34] for abelian groups. The generalization toarbitrary groups can be found in [81].

Lemma A.1.1. A group G cannot be the union of a finite number of left cosetsof subgroups, each of which has infinite index in G.

Proof. Suppose that there exist subgroups H1, . . . , Hn of G and finite subsetsF1, . . . , Fn of G such that G “

Ťni“1 FiHi, but none of the Hi has finite index in G.

Of course, we can assume that the family tH1, . . . , Hnu is minimal in this respect.Then n ą 1, and because Hn has infinite index, we find some x P G such thatxHn X FnHn “ H. Then xHn Ď

Ťn´1i“1 FiHi and hence

FnHn “ pFnx´1

qxHn ĎŤn´1

i“1 pFnx´1FiqHi.

This implies that

G “Ťn

i“1 FiHi “Ťn´1

i“1 pFi Y Fnx´1FiqHi,

which is impossible by the minimality of the set tH1, . . . , Hnu. �

For any family J of subsets of a group G, let RpJ q denote the smallest Booleanalgebra that contains J and is invariant under left translation. In the followingproposition [34, Lemma p. 223], which is an essential step towards proving TheoremA.1.5 below, fpE, ¨q will denote the characteristic function of a set E.

Proposition A.1.2. Let Hi be finitely many subgroups of a group G, and foreach i, let Kij be finitely many left cosets of Hi. Let cij P C and

fpxq “

ÿ

i,j

cijfpKij , xq, x P G.

277

278 A. APPENDIX

Let Bk denote the pairwise disjoint subsets of G on which f attains its finite numberof values. Then there are finitely many subgroups Li of G such that

RptBkukq “ RptLiuiq.

Proof. As in the proof of Lemma A.1.1, we can assume that for all i and j,the index rHi : pHi XHjqs is either one or infinite. Under this assumption, we shallprove that if the function

fipxq “

ÿ

j

cijfpKij , xq

attains its finitely many values on sets Ck, each of which is of course a finite unionof left cosets Hi, then each Ck is a finite union of left cosets of some subgroup Li,which in turn lies in the Boolean algebra generated by the sets Bk and their lefttranslates. Moreover, Hi has finite index in Li.

We now proceed by induction on the number of distinct subgroups Hi whichoccur. Suppose first that there is only one subgroup Hi “ H. It suffices to provethat the statement of the proposition in the case where the cij are all one. Let Lbe the subgroup of all elements a P G such that fpxaq “ fpxq for all x P G. ThenL Ě H and

fpxq “

rÿ

j“1

fpajL, xq

for certain distinct elements a1, . . . , ar of G. By translating f from the left, wecan assume that a1 “ e. We prove by induction on r that L is in the Booleanalgebra generated by the support of f and its left translates. The case r “ 1 beingtrivial, assume that r ě 2. The union of cosets ajL does not form a group, becauseotherwise this group would have to coincide with L, contradicting r ě 2. Therefore,for some j, the function fpa´1

j xq ´ fpxq is not identically zero. But

fpa´1j xq ´ fpxq “ g1pxq ´ g2pxq,

where g1 and g2 are characteristi c functions of unions of less than r left cosetsof L. By the induction hypothesis, it follows that the group L1, which leaves thesupport of g1 invariant under right translation is in the wanted Boolean algebra.Also, by considering f ´ g1, the subgroup L2 leaving it invariant is also in thatBoolean algebra. Since L is the intersection of L1 and L2, the statement follows inthis case.

Now assume that there are more than just one Hi. Then at least one of them,H1 say, has the property that it is not contained in Hi for any i ‰ 1. ThenrH1 : pH1 X Hiqs “ 8 for all i ‰ 1 and hence H1 is not contained in the union offinitely many left cosets of Hi (Lemma A.1.1). Therefore, we find a1, a2 P H1 suchthat the three cosets K2j , a1K2j and a2K2j are all distinct. Now the two functions

fpxq ´ fpa´11 xq and fpxq ´ fpa´1

2 xq

both don’t involve H1 and so the functionsÿ

j

c2jpfpK2j , xq ´ fpa1K2j , xq andÿ

j

c2jpfpK2j , xq ´ fpaaK2j , xqq

take their different values on unions of left cosets of subgroups M2 and N2, respec-tively, such that H2 Ď M2, N2 and M2 and N2 lie in the Boolean algebra generatedby the sets Bk and their left translates. Since the cosets K2j , a1K2j and a2K2j are

A.1. THE CLOSED COSET RING 279

distinct, it follows th at the function f2 “ř

j c2jfpK2j , ¨q takes its different valueson unions of left cosets of L2 “ M2 X N2. Similarly, for all i ‰ 1. This finishes theproof of the proposition. �

Let C and S denote the family of all closed subsets and all subgroups of G,respectively.

Lemma A.1.3. Let E P RpCq be a coset in G. Then E is closed in G.

Proof. Let E “ xH, where H is a subgroup of G. Then H P RpCq and itsuffices to show that H “ H . Therefore, replacing G by H , we can assume that His dense in G. As H P RpCq, there exist C1, . . . , Cn P C and open subsets U1, . . . , Un

of G such that

H “Ťn

j“1pCj X Ujq.

Since H is dense in G, at least one of the sets Cj X Uj has nonempty interior. Fix

such j and let U ‰ H be open and contained in Cj X Uj . Then U X pCj XUjq ‰ H

and hence

H ‰ U X Uj Ď Cj X Uj Ď H.

It follows that H is open and hence closed. �

Corollary A.1.4. RpC X Sq “ RpCq X RpGdq.

Proof. It is clear that RpC X Sq Ď RpCq X RpGdq. For the reverse inclu-sion, it suffices to show that every closed set E in RpGdq belongs to RpC X Sq.By Lemma A.1.1, there are finitely many subgroups H1, . . . , Hn of G such thatRptEuq Ď RptH1, . . . , Hnuq. Since E P RpCq, it follows that Hi P RpCq, 1 ď i ď n.Now, by Lemma A.1.3, every Hi is closed in G and hence E P RptH1, . . . , Hnuq Ď

RpC X Sq. �

Theorem A.1.5. Let G be any locally compact group and E a subset of G.Then E belongs to RcpGq if and only if E is of the form

(A.1) E “Ťn

i“1

´

EizŤmi

j“1 Eij

¯

,

where Ei, 1 ď i ď n is a left coset of some closed subgroup Hi of G and, for each1 ď i ď n and 1 ď j ď mi, Eij is either empty or a left coset of some open subgroupof Hi.

Proof. It is clear that each such set belongs to RcpGq. Conversely, let E P

RcpGdq; then E P RpCq X RpGdq by Corollary A.1.4 and hence

E “Ťn

i“1

´

EizŤmi

j“1Eij

¯

, n,mi P N, 1 ď i ď n,

where the Ei and Eij are closed cosets in G and Eij Ď Ei (or Eij “ H). Since Eis closed,

E “Ťn

i“1 Ci zŤmi

j“1Cij .

We claim that if C is any closed coset and C1, . . . , Cm are closed cosets containedin C, then

C zŤm

j“1 Cj “ C zŤ

tCj : 1 ď j ď m,Cj open in Gu.

280 A. APPENDIX

To see this, let I be the set of all indices j such that Cj is open in C (equivalently,Cj has nonempty interior relative C). Then

Ť

kRI Ck has empty interior in C and

hence C zŤ

kRI Ck “ C. Since C zŤ

jPI Cj is open and closed in C, it follows that

C zŤ

jPI Cj “

´

C zŤ

jPI Cj

¯

X

´

C zŤ

kRI Ck

¯

“

´

C zŤ

jPI Cj

¯

X

´

CzŤ

kRI Ck

¯

“ C zŤ

jPI Cj .

This completes the proof of the theorem. �Example A.1.6. (1) Every coset of the integer group Z is of the form nZ`m,

n,m P Z. Thus, if n1, . . . , nr,m1, . . . ,mr P Z and F is any finite subset of Z, thenE “

Ťni“1pniZ`miqzF P RpZq. Conversely, every E P RpZq is of this form. Indeed,

this follows from the description of sets in RpGq and the fact that if H and K aresubgroups of Z with H ‰ K Ď H, then K “ mH for some m P N and henceHzK “

Ťmk“1pk ` Kq.

(2) Every nonempty set E P RcpRq is of the form

E “Ťn

i“1

´

ai `

´

HizŤmi

j“1pbij ` Kijq

¯¯

, ai, bij P R,

where Hi and Kij are closed subgroups of G and Kij is open in Hi. To see this,note first that if one of the subgroups Hi equals R, then E “ R. Since every properclosed subgroup of R is of the form αZ, α P R, using the description of RpZq in (1),we conclude that the sets E P RcpRq are precisely the sets of the form

E “Ťn

i“1pαiZ ` βiqzF, αi, βi P R,

where F is a finite (possibly empty) subset of R.(3) Since every proper closed subgroup of the circle group T is finite, RcpTq

consists of T and all the finite subsets of T.

A.2. Amenability and weak amenability of Banach algebras

If A is a Banach algebra, then the diagonal operator is defined through

ΔA : A pbA Ñ A, a b b Ñ ab.

A pbA becomes a Banach A-bimodule through

a ¨ pb b cq “ ab b c and pb b cq ¨ a “ b b ca, a, b, c P A.

Definition A.2.1. A bounded net puαqα in ΔA : A pbA is called an approxi-mate diagonal for A if, for all a P A,

a ¨ uα ´ uα ¨ a Ñ 0 and aΔApuαq Ñ a.

Let A be a Banach algebra, and let E be a Banach A-bimodule. A derivationD : A Ñ E is a bounded linear map satisfying

Dpabq “ a ¨ Dpbq ` Dpaq ¨ b, a, b P A.

A derivation is called inner if there exists x P E such that

Dpaq “ a ¨ x ´ x ¨ a

for all a P A. The dual space E˚ is a Banach A-bimodule through

xx, a ¨ ϕy “ xx ¨ a, ϕy and xx, ϕ ¨ ay “ xa ¨ x, ϕy

A.2. AMENABILITY AND WEAK AMENABILITY OF BANACH ALGEBRAS 281

for a P A, x P E and ϕ P E˚. Modules of this kind are called dual BanachA-bimodules.

Definition A.2.2. A is called amenable if every derivation from A into a dualBanach A-bimodule is inner.

Theorem A.2.3. Let G be a locally compact group. Then L1pGq is amenableif and only if G is amenable.

Lemma A.2.4. Let A be a Banach algebra with bounded approximate identity.and let E be a Banach A-bimodule such that A ¨ E “ t0u. Then every boundedderivation D : A Ñ E˚ is inner.

Proof. Note that E˚ ¨ A “ t0u since xϕ ¨ a, uy “ xϕ, a ¨ uy for ϕ P E˚, a P Aand u P E. Let peαqα be a bounded approximate identity for A, and let ϕ P E˚ bea w˚-cluster point of the net pDpeαqqα. Passing to a subnet if necessary, we mayassume that ϕ “ w˚-limα Dpeαq. Then, for any a P A,

Dpaq “ limα

Dpaeαq “ limαa ¨ Dpeαq “ a ¨ ϕ “ a ¨ ϕ ´ ϕ ¨ a.

Hence D is an inner derivation. �

Definition A.2.5. Let A be a Banach algebra. A Banach A-bimodule E iscalled pseudo-unital if E “ ta ¨ u ¨ b : a, b P A, u P Eu.

Lemma A.2.6. Let A be a Banach algebra with bounded approximate identity.If for each pseudo-unital Banach A-bimodule E every bounded derivation from Ainto E˚ is inner, then A is amenable.

Proof. Let E be an arbitrary Banach A-bimodule and D : A Ñ E˚ a boundedderivation. Let

E0 “ ta ¨ u ¨ b : a, b P A, u P Eu.

By Cohen-Hewitt factorization theorem, E0 is a closed A-submodule of E. Let r :E˚ Ñ E˚

0 denote the restriction map ϕ Ñ ϕ|E0. Then r is a module homomorphism

and hence r ˝ D is a bounded derivation of A into E˚0 . By hypothesis, there exists

ϕ0 P E˚0 such that pr ˝Dqpaq “ a ¨ϕ0 ´ϕ0 ¨ a for all a P A. It is easily checked that

then rD “ D ´ adϕ0 defines a bounded derivation from A into EK0 Ď E˚. �

Proposition A.2.7. Let A be an amenable Banach algebra, B a Banach al-gebra and φ : A Ñ B a continuous homomorphism with dense range. Then B isamenable. In particular, if I is a closed ideal of A, then A{I is amenable.

Theorem A.2.8. Let A be an amenable Banach algebra and I a closed ideal ofA. Then the following are equivalent.

(i) I is amenable.(ii) I has a bounded approximate identity.(iii) I is weakly complemented.

Proposition A.2.9. Let I be a closed ideal of the Banach algebra A. If both Iand A{I are amenable, then A is amenable.

Proposition A.2.10. Let A and B be commutative Banach algebras and φ :A Ñ B a continuous homomorphism with dense range. If A is weakly amenable,then so is B.

282 A. APPENDIX

If ϕ P A˚, the map Δϕ : a ÞÑ a ¨ ϕ ´ ϕ ¨ a is a derivation. Derivations Δϕ arecalled inner. A Banach algebra A is weakly amenable if every continuous derivationfrom A to A˚ is inner. As shown by B. Johnson [142], the group algebra L1pGq ofa locally compact graph is always weakly amenable (see [58] for a beautiful shortproof).

Proposition A.2.11. Let A be a weakly amenable Banach algebra and I aclosed ideal of A. Then I is weakly amenable if and only if I equals the closedlinear span of the set tab : a, b P Iu.

The following theorem was shown in [108, Theorem 3.2].

Theorem A.2.12. Let A be a commutative Banach algebra and let m : Ae pbAe

Ñ Ae be the multiplication map. Then the following are equivalent.

(i) A is weakly amenable.

(ii) kerm “ spanpkermq2.

If A has a bounded approximate identity, then these are equivalent to

(iii) kerpm|A pb Aq “ spanpkerpm|A pb Aq2q.

Theorem A.2.13. (i) Let G be a locally compact group. Then L1pGq is weaklyamenable.

(ii) Every C˚-algebra is weakly amenable.

A.3. Operator spaces

Let H be a Hilbert space and BpHq the C˚-algebra of all bounded linear op-erators on H. A (concrete) operator space is a subspace of BpHq. For n P N, let Ebe a linear subspace of BpHq. We define

MnpEq “ trxijs “ pxijqni,j“1 : xij P Eu,

the space of n ˆ n-matrices with entries in E. Let �n2 pHq denote the Hilbert spacedirect sum of n copies of H. Then there is a natural identification of MnpBpHqq

with Bp�n2 pHqq given by rTijs Ñ T , where T is defined to be˜

T

˜

nÿ

j“1

ξj

¸¸

i

“

nÿ

j“1

Tijξj , 1 ď i ď n,

for ξ1, . . . , ξn P H. Equip MnpBpHqq and its subspace MnpEq with the norminduced from Bp�n2 pHqq, denoted } ¨ }n.

Let H and K be Hilbert spaces and E and F be closed subspaces of BpHq

and BpKq, respectively. Given a linear map φ : E Ñ F , the nth amplification,φn, n P N, of φ is the map φn : MnpEq Ñ MnpF q defined by φnrxijs “ rφpxijqs,rxijs P MepEq. Then φ is said to be completely contractive if }φ} ď 1. Moreover,a linear map φ : V Ñ W is called completely positive if φn ě 0 for all n P N.Let }φn} denote the norm of φn. Then the map φ is called completely boundedif sup t}φn} : n P Nu ă 8. In this case the completely bounded norm }φ}cb of φis defined to be }φ}cb “ sup t}φn} : n P Nu. Let CBpE,F q denote the set of allcompletely bounded maps from E into F . Then pCBpE,F q, } ¨ }cbq is a normedlinear space, and CBpE,F q is complete whenever F is a Banach space.

Theorem A.3.1. Let H be a Hilbert space and let X be a subspace of BpHq

and n,m P N.

A.3. OPERATOR SPACES 283

(i) For each A P MnpXq and B P MmpXq,›

›

›

›

„

A 00 B

j›

›

›

›

n`m

“ maxt}A}n, }B}mu.

(ii) For A P MmpXq, raijs P Mn,mpCq and rbijs P Mm,npCq,

}raijsA rbijs}n ď }raijs} ¨ }A}m ¨ }rbijs}.

Let E be an arbitrary vector space over C. A matricial norm on E is a sequencep} ¨ }nqnPN, where } ¨ }n is a norm on MnpEq such that conditions (i) and (ii) ofTheorem A.3.1 hold for all n,m P N. A linear space equipped with a matricialnorm is called an (abstract) operator space. The following theorem, known asRuan’s representation theorem, shows that every abstract operator space is actuallyisomorphic in the appropriate sense to a concrete operator space.

Theorem A.3.2. Let V be an abstract operator space. Then there exist aHilbert space H, a concrete operator space E Ď BpHq and a complete isometry φof V onto E, that is, for each n P N, the map φn : MnpV q Ñ MnpEq, defined byφnrxijs “ rφpxijqs, is an isometry.

Lemma A.3.3. Let E be an operator space and F a closed subspace of E. ThenF and E{F are also operator spaces.

Proposition A.3.4. Let E be an operator space and F : E Ñ MnpCq, n P N, abounded linear map. Then F is completely bounded and }F }cb “ }F }. In particular,every f P E˚ is completely bounded and }f}cb “ }f}.

The following theorem, which is one of the fundamental results in the theory ofoperator spaces, is usually referred to as the Arveson-Wittstock extension theorem.

Theorem A.3.5. Let V be a subspace of an operator space W , and let H be aHilbert space. Then every complete contraction Φ : V Ñ BpHq admits a completely

contractive extension rΦ : W Ñ BpHq.

Given an operator space E, we can now define the operator space dual E˚ ofE. Each f “ rfijs P MnpE˚q determines a linear map E Ñ MnpCq, x Ñ rxfij , xys.This gives a linear isomorphism MnpE˚q – CBpE,MnpCq, which allows us totransfer the completely bounded norm on CBpE,MnpCqq to MnpE˚q. Thus, usingProposition A.3.4, for any f P MnpE˚q,

}f}n “ }f}cb “ }fn}

“ sup t|xfn, xy| : x P MnpEq, }x} ď 1u.

We call pE˚, } ¨ }n as defined above the operator space dual of E.Let V and W be operator spaces. An element u of MnpV b W q, the space of

nˆn-matrices with entries in the tensor product of V and W , has a representationu “ αpv b wqβ, where v P MppV q, w P MqpW q, α P Mn,pˆq and β P Mpˆq,n,p, q P N. We define

}u}n “ inf t}α} ¨ }v} ¨ }w} ¨ }β} : u “ αpv b wqβu,

where the infimum is taken over all such representations of u.The assignment } ¨ } : n Ñ } ¨ }n defines an operator space matrix norm on

V b W , and the completion of V b W with respect to } ¨ } is called the operatorspace projective tensor product of V and W and denoted V pbop W .

284 A. APPENDIX

Proposition A.3.6. For any operator space E, the canonical embedding E Ñ

E˚˚ is a complete isometry.

More generally, if E and F are operator spaces, then CBpE,F q carries theoperator space structure given by the identification

MnpCBpE,F qq – CBpE,MnpF qq, rφijs Ñ px Ñ rφijpxqs,

φij P CBpE,F q, 1 ď i, j ď n, x P E.

Proposition A.3.7. Given operator spaces E and F and a completely boundedlinear map φ : E Ñ F , the dual map φ˚ : F˚ Ñ E˚ is also completely bounded andsatisfies }pφ˚qn} “ }φn} for all n P N. In particular, }φ˚}cb “ }φ}cb.

We write E – F if there exists a linear complete isometry from E onto F .The following proposition was shown in [27, Lemma 1.5].

Proposition A.3.8. Let M and N be von Neumann algebras, and let φ : M Ñ

M be a σ-weakly continuous, completely bounded map. Then there exists a (unique)

σ-weakly continuous map rφ : MbN Ñ MbN such that

rφpx b yq “ φpxq b y, x P M, y P N ,

where MbN is the usual von Neumann tensor product of M and N .

Moreover, }rφ} ď }φ}cb.

A.4. Operator amenability

A Banach algebra A is called completely contractive if A is a (complete) op-erator space and the multiplication m : A ˆ A Ñ A is a completely contractivemapping, i.e. m determines a completely contractive bilinear map from A pbA intoA. Equivalently, if a “ raijs P MmpAq and b P rbkls P MnpAq, m,n P N, then

}raij , bkls} ď }a} ¨ }b}.

Let A be a completely contractive Banach algebra and X an A-bimodule. ThenX is called ab operator A-bimodule if X is also an operator space and the moduleoperations

pa, xq Ñ a ¨ a and pa, xq Ñ x ¨ a

are completely bounded. Equivalently, there is a constant c ą 0 such that

}raij ¨ xkl} ď c}a} ¨ }x} and }rxkl ¨ aij} ď c}a} ¨ }x}

for all a “ raijs P MmpAq and x “ rxkls P MnpXq, m,n P N. In this case X˚ is alsoan operator A-bimodule with the module actions given by

xa ¨ f, xy “ xf, x ¨ ay and xf ¨ a, xy “ xf, a ¨ xy

for a P A, x P X and f P X˚. Moreover, the bimodule operations are completelybounded.

A completely bounded Banach algebra A is called operator amenable Banachalgebra if for every operator A-bimodule X, every completely bounded derivationfrom A into X˚ is inner.

Proposition A.4.1. Let A be a completely contractive Banach algebra. If Ais operator amenable, then A has a bounded approximate identity.

A.4. OPERATOR AMENABILITY 285

Proof. We first show that A has a bounded right approximate identity. Recallthat the left A-module action of A on A˚˚ is given by xa ¨ϕ, fy “ xϕ, f äy for a P A,f P A˚ and ϕ P A˚˚. We define a right module operation of A on A˚˚ by ϕ ¨ a “ 0for all a P A and ϕ P A˚˚. Then clearly A˚˚ is a dual operator A-bimodule. Thecanonical embedding D : A Ñ A˚˚ is a completely bounded derivation since

xDpabq, fy “ xf, aby “ xf ¨ a, by “ xDpbq, f ¨ ay “ xa ¨ Dpbq, fy

“ xDpaq ¨ b, fy ` xa ¨ Dpbq, fy

for all a, b P A and f P A˚.Since A is operator amenable, there exists ϕ P A˚˚ such thatDpaq “ a¨ϕ´ϕä “

a ¨ ϕ for all a P A. So ϕ is a right identity for A in A˚˚. By Goldstine’s theoremthere exists a net peαqα in A such that }eα} ď }ϕ} for all α and eα Ñ ϕ in thew˚-topology σpA˚˚, A˚q. In particular, aeα Ñ a in the weak topology σpA,A˚q foreach a P A.

Now let C “ tx P A : }x} ď }ϕ}u and let AA denote the product topologicalvector space pA, } ¨ }qA with the product topology. Then the weak topology on AA

is the product of the weak topologies σpA,A˚q. Define φ : C Ñ AA by

φpxqpaq “ ax ´ a, x P C, a P A.

Then φpeαq Ñ 0 in the weak topology of AA. Since on bounded convex sets the weaktopology coincides with the norm topology, replacing eα by a convex combinationif necessary, we may assume that }aeα ´ a} Ñ 0 for all a P A. This shows that Ahas a bounded right approximate identity.

Similarly, by considering the usual right A-module operation of A on A˚˚ andthe zero left A-module operation on A˚˚, one can show that A has a bounded leftapproximate identity, pfβqβ say. Finally, let

upα,βq “ eα ` fβ ´ eαfβ .

Then the upα,βq form a bounded net, and for each a P A,

}aupα,βq ´ a} “ }paeα ´ aq ` pa ´ aeαqfβ}

ď }aeα ´ a}p1 ` }fβ}q Ñ 0

and }upα,βqaá} ď }fβaá}`}eα} ¨}a´fβa} Ñ 0. Thus pupα,βqqpα,βq is a boundedapproximate identity for A. �

Let A be a Banach algebra.(1) A bounded net puαqα in A pbA is called a bounded approximate diagonal for

A if

}a ¨ uα ´ uα ¨ a} Ñ 0 and }mpuαqa ´ a} Ñ 0

for each a P A.(2) An element M of pA qbAq˚ is called a virtual diagonal if

a ¨ M ´ M ¨ a and a ¨ m˚˚pMq “ a

for all a P A.

Proposition A.4.2. Let A be a completely contractive Banach algebra. If Ahas a bounded approximate diagonal, then A is operator amenable.

286 A. APPENDIX

Proof. Let puαqα be a bounded approximate diagonal in A pbA. Let X bean operator A-bimodule and D : A Ñ X˚ a completely bounded derivation. Thenpmpuαqqα is a bounded approximate identity for A.

For x P X, we define Fx P pA pbAq˚ by

xFx, a b by “ xx, a ¨ Dpbqy, a, b P A.

Then }Fx} ď }x} ¨ }D}cb.Let M be a w˚-cluster point of the net pmpuαqqα in A˚˚. Then a ¨ M “ M ¨ a

for all a P A. In fact, for all f P A˚,

xa ¨ M, fy “ xM, f ¨ ay “ limα

xmpuαq, f ¨ ay

“ limα

xampuαq, fy “ limα

xmpauαq, fy

“ limα

xmpuαaq, fy “ xM,a ¨ fy “ xM ¨ a, fy.

Now define f P X˚ by

xf, xy “ xM, , Fxy, x P X.

We are going to show that D “ Df . To see this, let a, b P A and x P X. Then

xb b c, Fx ¨ a ´ a ¨ Fxy “ xa ¨ pb b cq ´ pb b cq ¨ a, Fxy

“ xab b c ´ b b ca, Fxy

“ xx, pabq ¨ Dpcq ´ b ¨ Dpcaqy

“ xx, pabq ¨ Dpcq ´ b ¨ Dpcq ¨ ay ´ xx, pbcq ¨ Dpaqy

“ xx ¨ a ´ a ¨ x, b ¨ Dpcqy ´ xx, pbcq ¨ Dpaqy

“ xb b c, Fxäáäy ´ xx, pbcq ¨ Dpaqy.

This implies that, for each α,

xuα, Fx ¨ a ´ a ¨ Fxy “ xuα, Fxäá¨xy ´ xx,mpuαq ¨ Dpaqy

and therefore, since a ¨ M “ M ¨ a,

xx, a ¨ f ´ f ¨ ay “ xx ¨ a ´ a ¨ x, fy “ xFxäá¨x,My

“ xFx ¨ a ´ a ¨ Fx,My ` limα

xx,mpuαq ¨ Dpaqy

“ xFx, a ¨ M ´ M ¨ ay ` limα

xx,mpuαq ¨ Dpaqy

“ limα

xx,mpuαq ¨ Dpaqy

for all x P X and a P A. Now x is of the form x “ b ¨ y ¨ c for certain y P X andb, c P A. Since pmpuαqqα is an approximate identity for A, it follows that

xx, a ¨ f ´ f ¨ ay “ limα

xb ¨ y ¨ pcmpuαqq, Dpaqy

“ xb ¨ y ¨ c,Dpaqy “ xx,Dpaqy.

This shows that Dpaq “ a ¨ f ´ f ¨ a for all a P A, as required. �

Theorem A.4.3. For a completely contractive Banach algebra A, the followingare equivalent.

(i) A is operator amenable.(ii) A has a virtual diagonal.(iii) A has a bounded approximate diagonal.

A.5. OPERATOR WEAK AMENABILITY 287

Theorem A.4.4. Let A be an operator amenable completely contractive Banachalgebra and let I be a closed ideal of A. Then I is operator amenable if and only ifI has a bounded approximate identity.

Proof. Since I is completely contractive and since we already know fromProposition A.4.1 that every operator amenable completely contractive Banachalgebra has a bounded approximate identity, we only need to show that if converselyI has bounded approximate identity, then I is operator amenable. �

Theorem A.4.5. Let A be a Banach algebra which is an operator space, andlet I be a closed ideal of A such that both I and A{I are operator amenable. ThenA is operator amenable.

A.5. Operator weak amenability

Let A be a completely contractive Banach algebra. Then A is said to be operatorweakly amenable if every completely bounded derivation from A into A˚ is an innerderivation.

Lemma A.5.1. Let A be an operator weakly amenable Banach algebra. ThenA2 is dense in A.

Proof. Towards a contradiction, assume that A2 is not dense in A. Thenthere exists f P A˚ such that f ‰ 0 and fpA2q “ t0u. Define D : A Ñ A˚

by Dpaq “ fpaqf . Then D is a derivation since xf, xyy “ 0 for all x, y P A.By hypothesis, there exists g P A˚ such that Dpaq “ a ¨ g ´ g ¨ a for all a P A.Choose b P A with xf, by ‰ 0. Then xDpbq, by “ xf, by ‰ 0, but on the other handxDpbq, by “ xb ¨ g ´ g ¨ b, by “ 0. This contradiction shows that A2 must be dense inA. �

Proposition A.5.2. Let A be a commutative completely contractive Banachalgebra. Then A is operator weakly amenable if and only if every completely boundedderivation from A into a symmetric operator A-bimodule is trivial.

Proof. Since A is commutative, A˚ is symmetric. Hence the ”if” direction istrivial.

Conversely, suppose that A is operator weakly amenable and let X be a sym-metric operator A-bimodule and D : A Ñ X a completely bounded derivation. ByLemma A.5.1, A2 is dense in A. Assume that D is nonzero and choose a P A andf P X˚ such that Dpa2q ‰ 0 and xf,Dpa2qy ‰ 0. For x P X, define fx P A˚ byxfx, by “ xf, b ¨ xy. Then the map F : x Ñ fx is completely bounded. Indeed, foreach n P N,

}Fn} “ supt}Fnprxijsq} : rxijs P MnpXq, }rxijs} ď 1u

“ supt}rfpxijqs} : rxijs P MnpXq, }rxijs} ď 1u

“ supt}rxfxij, ays} : a P A, }a} ď 1, rxijs P MnpXq, }rxijs} ď 1u

“ supt}xFn, ra ¨ xijsy} : a P A, }a} ď 1, rxijs P MnpXq, }rxijs} ď 1u

ď }F }cb “ }F }.

Now define rD : A Ñ A˚ by rDpaq “ F pDpaqq. It is straightforward to check that rDis a completely bounded derivation. Finally, since X is symmetric,

x rDpaq, ay “ xfDpaq, ay “ xf, a ¨ Dpaqy “1

2xf,Dpa2qy ‰ 0.

288 A. APPENDIX

Since A is operator weakly amenable, rDpaq is inner. So rDpaq “ a ¨ g´ g ¨a for some

g P A˚ and hence x rDpaq, ay “ 0. This contradiction finishes the proof. �

Corollary A.5.3. Let A and B be commutative completely contractive Banachalgebras and let φ : A Ñ B be a completely bounded homomorphism with denserange. If A is operator weakly amenable, then so is B.

Proof. We may assume that }φ}cb “ 1. Observe that B˚ is an operatorA-bimodule with the action given by

pa, fq Ñ φpaq ¨ f “ f ¨ φpaq, a P A, f P B˚.

Let D : B Ñ B˚ be a completely bounded derivation and define rD : A Ñ B˚

by rDpaq “ Dpφpaqq. Then rD is a completely bounded derivation and hence, by

Proposition A.5.2, rD “ 0 since A is operator weakly amenable. Since φ has denserange, it follows that D “ 0. �

Theorem A.5.4. Let A be a commutative operator weakly amenable Banachalgebra and let I be a closed ideal of A. Then I is operator weakly amenable if andonly if I2 is dense in I.

Proof. If I is operator weakly amenable, then I2 is dense in I by Lemma A.5.1.Conversely, suppose that I2 is dense in I and let D : I Ñ I˚ be a completely

bounded derivation. Let HomcbpI, I˚q denote the set of all completely bounded leftI-module homomorphisms, i..e. of all completely bounded linear maps T : I Ñ I˚

such that T pa ¨ fq “ a ¨ T pfq for all a P A and f P I˚.For a P A, we now define Da : A Ñ I˚ by

Dapbq “ Dpabq ´ b ¨ Dpaq, b P A.

Then, since A is commutative and D is a derivation, for all x, y P I and a, b P A,

Dxypabq “ Dpxyabq ´ pabq ¨ Dpxyq

“ paxq ¨ Dpybq ` pybq ¨ Dpaxq ´ pabq ¨ Dpxyq

“ a ¨ rx ¨ Dpybq ´ pbxq ¨ Dpyqs ` b ¨ rpaxq ¨ Dpyq ` y ¨ Dpaxq ´ a ¨ Dpxyqs

“ a ¨ rDpxybq ´ b ¨ Dpxyqs ` b ¨ rDpxyaq ´ a ¨ Dpxyqs

“ a ¨ Dxypbq ` b ¨ Dxypaq.

This shows that, for every z P I2, a Ñ Dzpaq is a derivation from A into I˚.Being the composition of completely bounded maps, Dz is completely bounded.Since A is an operator space, so is I and hence I˚. Moreover, I˚ is a symmetricA-bimodule since A is commutative. As A is operator weakly amenable, it followsfrom Proposition A.5.2 that Dz “ 0 for all z P I2. Now, for x, a P I,

Dxpaq “ Dpxaq ´ a ¨ Dpxq “ x ¨ Dpaq ` Dpxq ¨ a ´ a ¨ Dpxq “ x ¨ Dpaq.

So, if x, y, a P I, then pxyq ¨ Dpaq “ Dxypaq “ 0 and hence xI2y ¨ DpIq “ t0u. AsxI2y is dense in I, x ¨DpIq “ DpIq ¨x “ t0u for all x P I and so DpI2q “ t0u. Again,

using that xI2y “ I, we conclude that DpIq “ t0u. �

Lemma A.5.5. If S P CBpA,X˚q – pAe ` Xq˚ is a derivation, then S annihi-lates rK;As.

A.5. OPERATOR WEAK AMENABILITY 289

Proof. First note that

K “ tu ´ e b mlpuq : u P Ae pbXu “ spanta b c ´ e b a ¨ x : a P Ae, x P Xu.

Since S is a derivation, for any a, b P Ae and x P X, we have

xb b x ´ e b b ¨ xy, S ¨ a ´ a ¨ Sy “ xx, Spabq ´ a ¨ Spbqy

´xb ¨ x, Spaq ´ a ¨ Speqy

“ xx, Spabq ´ a ¨ Spbqy ´ xx, Spaq ¨ by

“ xx, Spabq ´ a ¨ Spbq ´ Spaq ¨ by “ 0.

Thus S ¨ a ´ a ¨ S P KK for all a P K. This implies

xa ¨ u ´ u ¨ a, Sy “ xu, S ¨ a ´ a ¨ Sy “ 0

for all u P K and a P Ae, and hence S annihilates rK;As. �Proposition A.5.6. Let A be a commutative completely contractive Banach

algebra and m : Ae pbAe Ñ Ae the multiplication map. Let K1 “ kerm and

K0 “ K1 X pA pbAq. If A2 “ A and K20 “ pA pbAqK1, then Ae and hence A is

operator weakly amenable.

Proof. Let D : Ae Ñ A˚e be a completely bounded derivation and view D as

an element of pAe b Aeq˚. Then D vanishes on K21 . Indeed, with uj “ aj b bj ´

e b ajbj , aj , bj P Ae, j “ 1, 2, we have

Dpu1u2q “ xa1a2 b bbb2, Dy ´ xa1 b a2b1b2, Dy

´xa2 b a1b1b2, Dy ` xe b a1a2b1b2, Dy

“ xb1b2, Dpa1a2qy ´ xa2b1b2, Dpa1qy

´xa2b1b2, Dpa2qy ´ xa1a2b1b2, .Dpeqy “ 0.

Since K1 is the closed linear span of elements of the form ab b´ebab, a, b P Ae, it

follows that D vanishes on K21 . Since K0 Ď K1 and K2

0 “ pA pb AqK1, D vanishes

on K1pA pbAq. In particular, D vanishes on elements of the form

pe b cq ´ c b eqpa b bq “ a b bc ´ ac b b, a, b, c P A.

D being a derivation, we conclude that

xab,Dpcqy “ xb,Dpacqy ´ xbc,Dpaqy “ xac b b ´ a b bc,Dy “ 0

for all a, b, c P A. Because A2 is dense in A, it follows that D vanishes on A andhence also on Ae since Dpeq “ 0. Thus Ae is operator weakly amenable and henceso is the ideal A by Theorem A.5.4. �

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Index

C˚-algebra, 1

H-separation property, 257

L1-algebra, 14

rSINsH -group, 246˚-algebra, 1

cb-multiplier norm, 180

n-step nilpotent group, 8

n-step solvable, 8

p-adic integers, 11

p-adic number field, 10

abstract opeartor space, 283

affine group, 9

affine map, 141

algebra

˚-, 1C˚-, 1

L1pGq, 14

Banach, 1

Banach-˚, 1

Figa-Talamanca-Herz, 88Fourier, 52

Fourier-Stieltjes, 44

group C˚-, 32

measure, 15

multiplier, 5normed, 1

normed ˚, 1

of almost periodic functions, 11

reduced group C˚-, 32

regular, 3Tauberian, 3

unital, 1

almost connected group, 7

almost periodic function, 11

amenable Banach algebra, 281amenable group, 33

amplification, 282

antidiagonal, 129

approximate diagonal, 280

approximate identity, 2

bounded, 2multiplier bounded, 191

sequential, 159

ascending central series, 8

Banach ˚-algebra, 1

Banach algebra, 1

amenable, 281

completely contractive, 284

operator amenable, 284

semisimple, 5

bimodule

operator, 284

pseudo-unital, 281

Bochner’s theorem, 27

Bohr compactification, 11

bounded approximate diagonal, 285

bounded approximate identity, 2

character, 24

closed coset ring, 277

of R, 280of T, 280of Z, 280

coefficient function, 29

Cohen-Hewitt factorization theorem, 2

commutator series, 8

compact-free group, 250

compactly generated group, 6

completely bounded map, 282

completely bounded multiplier, 179

completely contractive Banach algebra, 284

completely contractive map, 282

completely positive map, 282

concrete operator space, 282

conjugate representation, 20

convolution

of measures, 14

convolution of functions, 13

coset ring, 277

coset space, 6

cyclic representation, 19

Day’s fixed point theorem, 34

derivation, 280

descending central series, 8

diagonal

303

304 INDEX

bounded approximate, 285

virtual, 285diagonal operator, 280

direct sum of representations, 268disjoint representations, 81Ditkin set, 205

Douady’s observation, 242dual

reduced, 32dual group, 24

of R, 25of T, 25of Z, 25of Ωp, 25of direct product, 25

dual space topology, 32

Euclidian motion group, 9extending subgroup, 238

extension property, 238

Fell group, 85Fell topology, 30

Figa-Talamanca-Herz algebra, 88flip map, 129Folner’s condition, 34

formulainversion, 27

Weil, 15Fourier algebra, 52

Fourier transform, 26Fourier-Stieltjes algebra, 44

Fourier-Stieltjes transform, 26, 45full host algebra, 90function

almost periodic, 11coefficient, 29

modular, 13negative definite, 23, 196

positive definite, 22uniformly continuous, 11

functionalpositive, 28

functions

convolution of, 13

Gelfand homomorphism, 3Gelfand representation, 3

Gelfand space, 3Gelfand transform, 3

Gelfand-Mazur theorem, 2Gelfand-Naimark theorem, 2Gelfand-Raikov theorem, 19

GNS-construction, 29GNS-representation, 29

grouprSINsH , 246

ax ` b, 9n-step nilpotent, 8

n-step solvable, 8affine, 9

almost connected, 7amenable, 33compact-free, 250compactly generated, 6

dual, 24Euclidian motion, 9Fell, 85Heisenberg, 10integer Heisenberg, 10

locally compact, 6locally finite, 35maximally almost periodic, 11nilpotent, 8semidirect product, 9

SIN, 246solvable, 8unimodular, 13

group C˚-algebra, 32

Haar measureon semi-direct product, 17

Heisenberg group, 10host algebra, 90

Host’s idempotent theorem, 91hull-kernel topology, 31

ideal

primitive, 31IN-group, 247induced representation, 21induction in stages, 21inner derivation, 280

integer Heisenberg group, 10intertwining operator, 18inverse Fourier transform, 27inverse Fourier-Stieltjes transform, 27inversion formula, 27

inversion theorem, 27involution, 1involution on L1pGq, 14irreducible representation, 19

Jacobson topology, 31

Kakutani-Kodaira theorem, 11

lattice, 193left Haar measure, 12left invariant mean, 33left invariant measure, 12

left regular representation, 18local Ditkin set, 206local spectral set, 206local synthesis, 206locally compact group, 6

locally finite group, 35

Malliavin’s theorem, x, 205

INDEX 305

map

affine, 141

completely bounded, 282

completely contractive, 282

completely positive, 282

flip, 129

piecewise affine, 142

matricial norm, 283

Mautner phenomenon, 264

maximally almost periodic group, 11

mean, 33

left invariant, 33right invariant, 33

measure

Radon, 12

measure algebra, 15

modular function, 13

multiplier, 5

completely bounded, 179

multiplier algebra, 5

multiplier bounded approximate identity,191

negative definite function, 23, 196

neutral subgroup, 260

nilpotent group, 8

normed ˚-algebra, 1

normed algebra, 1

operator

intertwining, 18

operator A-bimodule, 284operator amenable Banach algebra, 284

operator space projective tensor product,283

Parseval identity, 27

piecewise affine map, 142

Plancherel theorem, 26

Plancherel transform, 27

Pontryagin duality theorem, 24

positive definite function, 22

positive linear functional, 28

primitive ideal, 31

primitive ideal space, 31

product group, 16

pseudo-unital bimodule, 281

quasi-equivalent representations, 82

radical of G, 7

Radon measure, 12

reduced dual, 32reduced group C˚-algebra, 32

regular algebra, 3

Reiter’s condition (P1), 35

representation

conjugate, 20

cyclic, 19

GNS-, 29irreducible, 19left regular, 18nondegenerate, 28right regular, 18support of, 30uniformly bounded, 186

representationsdirect sum, 19, 268quasi-equivalent, 82similar, 186tensor product, 20weakly equivalent, 30

right invariant mean, 33right regular representation, 18Ruan’s representation theorem, 283

Schoenberg’s theorem, 23semidirect product group, 9semisimple Banach algebra, 5separating subgroup, 257separation property, 257

for cyclic subgroups, 257sequential approximate identity, 159series

ascending central, 8commutator, 8descending central, 8

setDitkin, 205local Ditkin, 206local spectral, 206of synthesis, 206spectral, 206

set of synthesis, 206similar representations, 186SIN-group, 246small H-invariant neighbourhoods, 246solvable group, 8space

abstract operator, 283concrete operator, 282

spectral set, 206spectrum, 3strong convergence to invariance, 34subgroup

extending, 238neutral, 260

separating, 257torsion, 250

support of a representation, 30support of an operator, 62

Tauberian algebra, 3tensor product

operator space projective, 283tensor product of representations, 20theorem

Bochner, 27

306 INDEX

Cohen-Hewitt factorization, 2Day’s fixed point, 34Gelfand-Mazur, 2Gelfand-Naimark, 2Gelfand-Raikov, 19Host’s idempotent, 91induction in stages, 21

inversion, 27Kakutani-Kodaira, 11Malliavin, x, 205Plancherel, 26Pontryagin duality, 24Ruan’s representation, 283Schoenberg, 23Wendel, 18

topologydual space, 32Fell, 30hull-kernel, 31Jacobson, 31

torsion subgroup, 250transform

Fourier, 26Fourier-Stieltjes, 26Gelfand, 3inverse Fourier, 27inverse Fourier-Stieltjes, 27Plancherel, 27

uniformly bounded representation, 186uniformly continuous function, 11unimodular group, 13unital algebra, 1

virtual diagonal, 285

weak containment, 30weakly equivalent representations, 30Weil’s formula, 15Wendel’s theorem, 18

word length, 195

Selected Published Titles in This Series

231 Eberhard Kaniuth and Anthony To-Ming Lau, Fourier and Fourier-StieltjesAlgebras on Locally Compact Groups, 2018

228 Zhenbo Qin, Hilbert Schemes of Points and Infinite Dimensional Lie Algebras, 2018

227 Roberto Frigerio, Bounded Cohomology of Discrete Groups, 2017

226 Marcelo Aguiar and Swapneel Mahajan, Topics in Hyperplane Arrangements, 2017

225 Mario Bonk and Daniel Meyer, Expanding Thurston Maps, 2017

224 Ruy Exel, Partial Dynamical Systems, Fell Bundles and Applications, 2017

223 Guillaume Aubrun and Stanis�law J. Szarek, Alice and Bob Meet Banach, 2017

222 Alexandru Buium, Foundations of Arithmetic Differential Geometry, 2017

221 Dennis Gaitsgory and Nick Rozenblyum, A Study in Derived Algebraic Geometry,2017

220 A. Shen, V. A. Uspensky, and N. Vereshchagin, Kolmogorov Complexity andAlgorithmic Randomness, 2017

219 Richard Evan Schwartz, The Projective Heat Map, 2017

218 Tushar Das, David Simmons, and Mariusz Urbanski, Geometry and Dynamics inGromov Hyperbolic Metric Spaces, 2017

217 Benoit Fresse, Homotopy of Operads and Grothendieck–Teichmuller Groups, 2017

216 Frederick W. Gehring, Gaven J. Martin, and Bruce P. Palka, An Introduction tothe Theory of Higher-Dimensional Quasiconformal Mappings, 2017

215 Robert Bieri and Ralph Strebel, On Groups of PL-homeomorphisms of the Real Line,2016

214 Jared Speck, Shock Formation in Small-Data Solutions to 3D Quasilinear WaveEquations, 2016

213 Harold G. Diamond and Wen-Bin Zhang (Cheung Man Ping), BeurlingGeneralized Numbers, 2016

212 Pandelis Dodos and Vassilis Kanellopoulos, Ramsey Theory for Product Spaces,2016

211 Charlotte Hardouin, Jacques Sauloy, and Michael F. Singer, Galois Theories ofLinear Difference Equations: An Introduction, 2016

210 Jason P. Bell, Dragos Ghioca, and Thomas J. Tucker, The DynamicalMordell–Lang Conjecture, 2016

209 Steve Y. Oudot, Persistence Theory: From Quiver Representations to Data Analysis,2015

208 Peter S. Ozsvath, Andras I. Stipsicz, and Zoltan Szabo, Grid Homology for Knots

and Links, 2015

207 Vladimir I. Bogachev, Nicolai V. Krylov, Michael Rockner, and Stanislav V.Shaposhnikov, Fokker–Planck–Kolmogorov Equations, 2015

206 Bennett Chow, Sun-Chin Chu, David Glickenstein, Christine Guenther, JamesIsenberg, Tom Ivey, Dan Knopf, Peng Lu, Feng Luo, and Lei Ni, The Ricci Flow:Techniques and Applications: Part IV: Long-Time Solutions and Related Topics, 2015

205 Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik, TensorCategories, 2015

204 Victor M. Buchstaber and Taras E. Panov, Toric Topology, 2015

203 Donald Yau and Mark W. Johnson, A Foundation for PROPs, Algebras, andModules, 2015

202 Shiri Artstein-Avidan, Apostolos Giannopoulos, and Vitali D. Milman,Asymptotic Geometric Analysis, Part I, 2015

For a complete list of titles in this series, visit theAMS Bookstore at www.ams.org/bookstore/survseries/.

For additional information and updates on this book, visit

www.ams.org/bookpages/surv-231

The theory of the Fourier algebra lies at the crossroads of several areas of analysis. Its roots are in locally compact groups and group representations, but it requires a considerable amount of functional analysis, mainly Banach algebras. In recent years it has made a major connection to the subject of operator spaces, to the enrichment of both. In this book two leading experts provide a road map to roughly 50 years of research detailing the role that the Fourier and Fourier-Stieltjes algebras have played in not only helping to better understand the nature of locally compact groups, but also in building bridges between abstract harmonic analysis, Banach algebras, and operator algebras. All of the important topics have been included, which makes this book a comprehensive survey of the field as it currently exists.

Since the book is, in part, aimed at graduate students, the authors offer complete and readable proofs of all results. The book will be well received by the community in abstract harmonic analysis and will be particularly useful for doctoral and postdoctoral mathematicians conducting research in this important and vibrant area.

www.ams.orgSURV/231

Fourier and Fourier-Stieltjes algebras on locally compact groups

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