Abstract Harmonic Analysis and Related Operator Algebras...
Transcript of Abstract Harmonic Analysis and Related Operator Algebras...
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Abstract Harmonic Analysis and Related
Operator Algebras on Locally Compact
Quantum Groups
Zhong-Jin Ruan
University of Illinois at Urbana-Champaign
ICM Satellite Conference on Operator Algebras and Applications
August 8-12, 2014
Cheongpung, Korea
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In this talk, we plan to discuss
• Historical Backgroup of Locally Compact Quantum Groups
• Abstract Quantum Harmonic Analysis
• Amenability and Approximation Properties of Quantum Groups
• Exotic Quantum Group C*-algebras
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Historical Background
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Pontryagin Duality (1934)
Let G be a locally compact abelian group. There exists a dual group
G = {γ : G→ T : continuous homomorphisms}.
G has a canonical abelian group structure and we have the homeomor-
phism
ˆG ∼= G.
In particular, we have the Plancherel theorem
L2(G) = L2(G).
How to generalize such duality results to non-abelian groups ?
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Fourier transformation gives injective *-homomorphism
λ : L1(G)→ C0(G) ⊆ L∞(G)
This suggests that we can consider the corresponding group C*-algebras
and group von Neumann algebras as the duality !
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Group C*-algebras and Group von Neumann Algebras
Let G be a locally compact group and µ be the left Haar measure of G.
The left regular representation of G
λ : s ∈ G→ λs ∈ B(L2(G))
is given by
λs(ξ)(t) = ξ(s−1t).
We get
• reduced group C*-algebra C∗λ(G) = {λ(f) : f ∈ L1(G)}−‖·‖ ⊆ B(L2(G))
• the left group von Neumann algebra L(G) = {λs}′′ ⊆ B(L2(G)).
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We get a natural “duality” between
L∞(G) and L(G)
or
C0(G) and C∗λ(G)
since if G is an abelian group with dual group G, we have
L(G) = L∞(G) and C∗λ(G) = C0(G)
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How do we recover the group structure from L∞(G) or L(G) ?
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Examples
For the Dihedral group D4 or the quaternion group Q (of order 8), we
have
`∞(D4) = `∞(8) = `∞(Q)
and
L(D4) = C⊕ C⊕ C⊕ C⊕M2(C) = L(Q).
We can not recover the group structure from their algebraic structure.
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Co-multiplication on L∞(G)
To recover the group structure from L∞(G), we need to consider a
co-multiplication
Γa : f ∈ L∞(G)→ Γa(f) ∈ L∞(G)⊗L∞(G)
given by
Γa(f)(s, t) = f(st).
It is easy to see the Γa is a normal injective *-homomorphism such that
it satisfies the co-associativity condition
(Γa ⊗ ι)Γa = (ι⊗ Γa)Γa
i.e. we have
(Γa ⊗ ι)Γa(f)(s, t, u) = f((st)u) = f(s(tu)) = (ι⊗ Γa)Γa(f)(s, t, u).
We call (L∞(G),Γa) a commutative Hopf von Neumann algebra.
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Co-involution on L∞(G)
To recover the inverse of G, we can consider a co-involution
κa(h) = h
which is a weak* continuous *-anti-automorphsim on L∞(G) such that
Γa ◦ κa = Σ(κa ⊗ κa) ◦ Γa,
where Σ(a⊗ b) = b⊗ a is the flip map.
Moreover, we can use the left Haar measure to obtain a normal faithful
weight
ϕa : h ∈ L∞(G)+ → ϕa(h) =∫Gh(s)ds ∈ [0,+∞]
on L∞(G) which satisfies (ι⊗ ϕa)Γ(h) = ϕa(h)1 since
(ι⊗ ϕa)Γ(h)(s) =∫Gh(st)dt = (
∫Gh(t)dt) = ϕa(h).
So we call ϕa a left Haar weight on L∞(G).
This gives a Kac algebra structure on L∞(G) !
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Kac Algebras
G. Kac introduced Kac algebras K = (M,Γ, κ, ϕ), for unimodular case,in the 60’s.
The theory was completed for general (non-unimodular) case in the70’s by two groups: Kac-Vainerman in Ukraine and Enock-Schwartz inFrance (see Enock-Schwartz’s book 1992).
There exists a perfect Pontryagin duality
ˆK = K.
Examples:
• Commutative Kac algebras: Ka(G) = (L∞(G),Γa, κa, ϕa)
• Co-commutative Kac algebras: Ks(G) = (L(G),ΓG, κG, ϕG)
We have the natural duality
Ka(G) = Ks(G) and Ks(G) = Ka(G).
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Kac Algebra Structure on L(G)
There is a natural co-associative co-multiplication
ΓG : L(G)→ L(G)⊗L(G)
on L(G) given by
ΓG(λ(s)) = λ(s)⊗ λ(s).
This is co-commutative in the sense that
Σ ◦ ΓG = ΓG.
To recover the inverse, we consider a co-involution κG on L(G) givenby
κG(λ(s)) = λ(s−1).
Moreover, we can obtain a normal faithful (Plancherel) weight ϕG onL(G). So (L(G),ΓG, κG, ϕG) is a co-commutative Kac algebra.
Remark If G is a discrete group, then
ϕG(x) = ψG(x) = 〈xδe|δe〉.
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Locally Compact Quantum Groups (LCQG)
In 1987, Woronowicz introduced SUq(2,C), a natural quantum defor-
mation of SU(2,C). He showed that SUq(2,C) does not correspond to
any Kac algebra since the “co-involution” is unbounded on SUq(2,C).
Since then, several different definitions of LCQG have been investigated:
• Baaj and Skandalis 1993: Regular Multiplicative Unitaries
• Woronowicz 1996: Manageable Multiplicative Unitaries
• Kustermans and Vaes 2000: Quantum Groups, C∗-algebra setting
• Kustermans and Vaes: 2003: Quantum Groups, von Neumann algebra
setting.
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Kustermans and Vaes’ Definition
A LCQG is G = (M,Γ, ϕ, ψ) consisting of
(1) a von Neumann algebra M
(2) a co-multiplication Γ : M → M⊗M , i.e. a unital normal *-homomorphism satisfying the co-associativity condition
(id⊗ Γ) ◦ Γ = (Γ⊗ id) ◦ Γ.
(3) a left Haar wight ϕ, i.e. a n.f.s weight ϕ on M satisfying
(ι⊗ ϕ)Γ(x) = ϕ(x)1
(4) a right Haar weight ψ, i.e. n.f.s weight ψ on M satisfying
(ψ ⊗ ι)Γ(x) = ψ(x)1.
It is known that for every locally compact quantum group G = (M,Γ, ϕ, ψ),there exists a dual quantum group G = (M, Γ, ϕ, ψ) such that we mayobtain the perfect Pontryagin duality
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Every Kac algebra K = (M,Γ, κ, ϕ) is a LCQG since we can obtain the
right Haar weight
ψ = ϕ ◦ κ
via ϕ and κ !
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Abstract Quantum Harmonic Analysis
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Quantum Convolution Algebras (L1(G), ?)
Let G = (L∞(G),Γ, ϕ, ψ) be a locally compact quantum group. Here we
use the notion L∞(G) for the quantum group von Neumann algebra M .
The co-multiplication
Γ : L∞(G)→ L∞(G)⊗L∞(G)
induces an associative completely contractive multiplication
? = Γ∗ : f1 ⊗ f2 ∈ L1(G)⊗L1(G)→ f1 ? f2 = (f1 ⊗ f2) ◦ Γ ∈ L1(G)
on L1(G) = L∞(G)∗.
Remark: In general, we do not have an involution on L1(G) since the
antipode S is unbounded on L∞(G). But we have a dense *-subalgebra
L]1(G) = {ω ∈ L1(G) : ∃ ω] ∈ L1(G) such that ω] = ω∗ ◦ S}.
For Kac algebras, we have L1(G) = L]1(G) and the involution is com-
pletely isometric.
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If G = (L∞(G),Γa, ϕa, ψa) is a commutative LCQG, then ? = Γ∗ is just
the multiplication on the convolution algebra L1(G) = L1(G).
If G = (L(G),ΓG, ϕG, ψG) is a co-commutative LCQG, then ? = ΓG∗ is
just the pointwise multiplication on the Fourier algebra
L1(G) = L(G)∗ = A(G).
We note that Fourier algebra was first studied by Eymard in 1964. It
can be defined to be the space
A(G) = {f : G→ C : continuous such that f(s) = 〈λ(s)ξ|η〉}
It was shown by Eymard in 1964 that A(G) is a commutative Banach
algebra with the pointwise multiplication and norm
‖f‖A(G) = inf{‖ξ‖‖η‖}.
We have a natural operator space structure on A(G), regarded as the
operator predual of L(G).
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Representations
A representation of G is a non-degenerate c.c. homomorphism π :L1(G)→ B(H) such that π restricted to L
]1(G) is a *-homomorphism.
Each representation (π,H) is uniquely associated with a unitary operatorUπ ∈M(C0(G)⊗K(H)) ⊆ L∞(G)⊗B(H) such that
π(ω) = (ω ⊗ ι)(Uπ) (ω ∈ L1(G))
In particular, we have the left regular representation
λ : ω ∈ L1(G)→ λ(ω) = (ω ⊗ ι)(W ) ∈ Cλ(G) = C0(G) ⊆ L∞(G) ⊆ B(L2(G))
and the universal representation
πu : ω ∈ L1(G)→ πu(ω) = (ω ⊗ ι)(W) ∈ C∗(G) = Cu(G) ⊆ B(Hu)
Here W is the left fundamental unitary of G and W is Kustermans semi-universal version of W . If G is a finite group, we can explicitly write
W =∑s∈G
δs ⊗ λ(s) and W =∑s∈G
δs ⊗ πu(s).
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Summary
We have the following quantum group algebras:
L∞(G) and L∞(G) = L(G)
C0(G) and C0(G) = C∗λ(G)
Cu(G) and Cu(G) = C∗(G)
L1(G) ⊇ L]1(G) and L]1(G) ⊆ L1(G) = A(G)
Here C0(G) = C∗λ(G) is the reduced quantum group C*-algebra and
C∗u(G) = C∗(G) is the universal/full quantum group C*-algebra.
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Quantum Fourier-Stieltjes Algebra B(G) = Cu(G)∗
Let G be a locally compact quantum group and let G be its quantum
dual. There is a natural co-multiplication
Γu : Cu(G)→M(Cu(G)⊗ Cu(G))
on the universal (or full) quantum group C*-algebra Cu(G) = C∗(G).
Taking the adjoint, we obtain a completely contractive multiplication
on Cu(G)∗ given by
µ ? ν = (µ⊗ ν) ◦ Γu (µ, ν ∈ Cu(G)∗).
In this case, we call B(G) = Cu(G)∗ to be the quantum Fourier-Stieltjes
algebra of G. It contains A(G) = L1(G) as a norm closed two-sided
ideal.
Given µ in B(G), we can define two completely bounded maps
mlµ(f) = µ ? f and mr
µ(f) = f ? µ
on L1(G). In fact these maps are completely bounded left (resp. right)
centralizers of L1(G).
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Completely Bounded/Herz-Schur Multipliers of Groups
A function ϕ : G→ C is a multiplier of G if the multiplication map
mϕ : f ∈ A(G)→ ϕf ∈ A(G)
is well defined on A(G). Such a map is automatically bounded.
A function ϕ : G → C is a completely bounded multiplier of G if mϕ iscompletely bounded on A(G).
Remarks: Completely bounded multipliers ϕ are contained in Cb(G) =M(C0(G)) and can be characterized by the following Gilbert theorem.
Theorem: A continuous function ϕ : G → C is a cb-multiplier with‖mϕ‖cb ≤ C if and only if there exist bounded continuous maps α =[αi], β = [βi] : G→ `2(I) for some Hilbert space `2(I) such that sup{‖α(t)‖} sup{‖β(s)‖} ≤C and
ϕ(st−1) = β∗(s)α(t) =∑i
βi(s)αi(t).
In this case, we also call ϕ a Herz-Schur multiplier.
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Completely Bounded/Herz-Schur Multipliers of Quantum Groups
Let G be a locally compact quantum group.
•We first expect that a Herz-Schur multiplier of G should be an element
in Cb(G) = M(C0(G)) ⊆ L∞(G) .
• Then we expect that it is associated with a completely bounded left
(right, or double) centralizer on A(G) = L1(G).
• Finally, we expect that it could have the corresponding Gilbert theo-
rem.
This was first studied by Kraus and R around 1996, for completely
bounded double multipliers of Kac algebras. Later on, these results
have generalized to all cases for locally compact quantum groups by
Junge-Neufang-R, Hu-Neufang-R (around 2009-11) and Daws (2011-
12).
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Let us consider the left multiplier case and we recall W ∈ L∞(G)⊗L∞(G).
Theorem : [Kraus-R 1996, Junge-Neufang-R 2009] Let S be acompletely bounded map on L1(G) = A(G). TFAE:
1. S is a completely bounded left centralizer , i.e. S(f ? g) = S(f) ? g,on L1(G). Then taking the adjoint, we get a normal cb-map S∗ onL∞(G) such that
Γ(S∗(x)) = (S∗ ⊗ ι)Γ(x) for x ∈ L∞(G).
2. There exists a unique normal completely bounded L∞(G)′-bimodulemap Φ on B(L2(G)) such that
W ∗(1⊗ Φ(x))W = (S∗ ⊗ ι)(W ∗(1⊗ x)W ) for x ∈ B(L2(G)).
In this case, we have Φ|L∞(G) = S∗.
3. There exists a unique operator a ∈ L∞(G) such that
(S∗ ⊗ ι)(W ) = (1⊗ a)W or (Φ⊗ ι)(W ) = (1⊗ a)W
It was shown by Daws that the operator a ∈ L∞(G) is actually containedin M(C0(G)). We call a the Herz-Schur multiplier of G. In this case,we say a ∈M l
cb(L1(G)).
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Remark 1: Since the left regular representation of G is given by
λ(f) = (f ⊗ ι)(W ),
condition 3
(S∗ ⊗ ι)(W ) = (1⊗ a)W
implies that for any f ∈ L1(G),
aλ(f) = (f ⊗ ι)((1⊗ a)W ) = (f ⊗ ι)((S∗ ⊗ ι)(W )) = λ(S(f)).
In this case, we get
λ(f)→ aλ(f) = λ(S(f))
and thus
S(f) = λ−1(aλ(f))
is a completely bounded left multiplier map (or a left centralizer) of
L1(G).
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A Gilbert Characterization for Herz-Schur Multipliers
Theorem : [Daws 2012] An element a ∈ Cb(G) is a Herz-Schur multiplier
if and only if there exist bounded elements [αi], [βi] ∈MI,1(L∞(G)) such
that ∑i
(1⊗ β∗i )Γ(αi) = a⊗ 1.
Daws used a Hilbert module approach. Actually, this result is an imme-
diate consequence of Condition 3 in above theorem.
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Theorem : [Daws 2012] An element a ∈ Cb(G) is a Herz-Schur multiplier
if and only if there exist bounded elements [αi], [βi] ∈MI,1(L∞(G)) such
that ∑i
(1⊗ β∗i )Γ(αi) = a⊗ 1.
Proof: Let a be a completely bounded left multiplier. By condition 3,
there exists a normal completely bounded L∞(G)′-bimodule map Φ on
B(L2(G)) such that (Φ⊗ ι)(W ) = (1⊗ a)W . Since W = ΣW ∗Σ, we get
(ι⊗ Φ)(W ∗) = (a⊗ 1)W ∗ or (ι⊗ Φ)(W ∗)W = a⊗ 1
Now by Haagerup’s result, the normal L∞(G)’-bimodule map Φ has the
expression Φ(x) =∑i β∗xαi for some [αi], [βi] ∈ MI,1(L∞(G)). There-
fore, ∑i
(1⊗ β∗i )Γ(αi) = (ι⊗ Φ)(W ∗)W = a⊗ 1.
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Positive Definite Herz-Schur Multipliers
We note that L1(G) = A(G) is an ideal in B(G) = Cu(G)∗ and the left
regular representation
λ : f ∈ L1(G)→ (f ⊗ ι)(W ) = (ι⊗ f)(W ∗) ∈ C0(G) ⊆ L∞(G)
extends naturally to a representation
λ : µ ∈ B(G)→ (ι⊗ µ)(W∗) ∈ Cb(G) = M(C0(G)).
It is easy to see that for every µ ∈ B(G), a = λ(µ) is a Herz-Schur
multiplier. The converse is not true in general (even for groups).
Theorem: [Daws 2012] A Herz-Schur multiplier a is positive definite
if and only if a = λ(µ) for some positive µ ∈ B(G)+.
A completely bounded/Herz-Schur multiplier a is positive definite if the
corresponding map Φ is completely positive.
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Amenability and Approximation Properties of Quantum Groups
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Amenability
Let G be a locally compact quantum group.
• G is amenable if there exists a state m : L∞(G)→ C such that
(ι⊗m)Γ(x) = m(x)1 (x ∈ L∞(G)).
• G is co-amenable if L1(G) has a bounded approximate identity. Thisis equivalent to Cu(G) = C0(G).
Remark: For general locally compact quantum groups, we have
G is co-amenable ⇒ G is amenable ⇒ L∞(G) is injective.
Theorem: [R 1996] All three statements are equivalent if G is adiscrete Kac algebra.
Theorem: [Tamatsu 2006] For discrete quantum groups, the firsttwo statements are equivalent. It is open question whether the thirdone is equivalent to the first two.
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Note: A quantum group G is a discrete if L1(G) is unital. In this case,
its dual quantum group G is compact, i.e., C0(G) is unital. In this case,
we simply write C(G) = C0(G).
It follows from definition that discrete quantum groups are always co-
amenable since L1(G) is unital.
However, compact quantum groups are not necessarily co-amenable.
•Woronowicz’s twisted SUq(2) is co-amenable. So Cu(SUq(2)) = C(SUq(2))
is nuclear.
• Free orthogonal quantum group O+N (N ≥ 3) and free unitary quantum
groups U+N (N ≥ 2) are not co-amenable.
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Weak Amenability of G
Let G be a locally compact group. For each f ∈ L1(G) = A(G), we candefine a completely bounded map
mf : g ∈ L1(G)→ f g ∈ L1(G).
We say that G is weakly amenable if there exists a net of functions {fi}in L1(G) such that 1) ‖mfi
‖cb ≤ C <∞ for some constant C > 0 and 2)
‖fi ? g − g‖ → 0 for all g ∈ L1(G).
Theorem: [Kraus-R 1999] Let G be a discrete Kac algebra. TFAE:
1. G is weakly amenable,
2. L∞(G) has the weak* CBAP
3. C0(G) = C∗λ(G) has the CBAP.
In this case, we have Λ(G) = Λ(L∞(G)) = Λ(C0(G)).
Remark: For discrete quantum groups, we have 1) ⇒ 2) and 3). Butwe do not know whether 2) or 3) implies 1).
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Free Quantum Groups [van Dale and Wang 1996]:
Let N ≥ 2 and F ∈ GLN(C) such that FF = R1.
The free orthogonal quantum group O+F is the compact quantum group
given by the universal C*-algebra
Cu(O+F ) = C∗({uij}1≤i,j≤N : U = [uij] is unitary and U = FUF−1}.
If, in addition, F ∈ CUN , then O+F is of Kac type, i.e. the Haar state is
tracial. We use the notion O+N for O+
IN.
The free unitary quantum group U+F is the compact quantum group
given by the universal C*-algebra
Cu(U+F ) = C∗({uij}1≤i,j≤N : U = [uij] and FUF−1}.
Similarly, F is a scalar multiple of unitary if and only if U+F is of Kac
type. We use the notion U+N for U+
IN.
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Examples:
For N ≥ 3, Brannan first showed (in 2011) that the free othergonalquantum group C(O+
N ) and free unitary quantum group C(O+N ) have
the metric approximation property.
Theorem: [Brannan 2012, Freslon 2013, De Commer-Freslon-
Yamashita 2013] Let N ≥ 2 and F ∈ GLN(C) such that FF = RIN .
The discrete quantum groups FOF = O+F and FUF = U+
F are weaklyamenable with
Λ(FOF ) = Λ(FUF ) = 1.
• The reduced quantum group C*-algebras C(O+F ) and C(U+
F ) have theCCAP.
• The quantum group von Neumann algebras L∞(O+F ) and L∞(U+
F )have the weak* CCAP.
In this case, quantum group C*-algebras C(O+F ) and C(U+
F ) are exact.
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Haagerup Property
Haagerup has even intensively studied for locally compact groups.
For locally compact quantum groups, we adopt the following (equiva-
lent) definition given by Daws, Fima, Skalski and White in 2013.
Definition: A locally compact quantum group G has the Haagerup
property if the quantum group C*-algebra C0(G) has a bounded ap-
proximate identity {ai} ∈ C0(G) which consists of norm-one positive
definite multipliers/functions.
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Coefficient Functions of Representations
Let π : L1(G)→ B(H) be a representation of G and let Uπ ∈ L∞(G)⊗B(H)
be the associated unitary operator such that π(ω) = (ω ⊗ ι)(Uπ).
For any ξ, η ∈ H, we call
ϕπξ,η = (ι⊗ ωξ,η)(Uπ) ∈ Cb(G) ⊆ L∞(G)
a coefficient function (associated with the representation).
Remark: For any representation π, there is a unique *-homo π from
Cu(G) = C∗u(G) onto C∗π(G) = π(L1(G))−‖·‖
such that
Uπ = (ι⊗ π)(W).
In this case, we can write
ϕπξ,η = (ι⊗ ωξ,η ◦ π)(W)
and regard it as a coefficient function of µ = ωξ,η ◦ π ∈ Cu(G)∗.
A coefficient is positive definite if it has the form ϕπξ,ξ.
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Von Neumann Algebra Haagerup Property
Haagerup property for finite von Neumann algebras (with trace) was
first studied by Connes in 1985. A definition for general von Neumann
algebras is given recently (both in 2013) by two independent groups:
Caspers-Skalski and Okayasu-Tomatsu.
Definition: A von Neumann algebra M with a normal faithful weight
ϕ has the Haagerup property if there exists a net of unital normal cp
maps Φi on M such that
0) ϕ ◦Φi ≤ ϕ1) each Φi extends to a compact operator on L2(M,ϕ)
2) ‖Φi(x)− x‖2 → 0 for every x ∈M (resp. for every x ∈ L2(M,ϕ)).
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Theorem: [Choda 1983] A discrete group G has the Haagerup prop-
erty if and only if its group von Neumann algebra L∞(G) = L(G) (with
the canonical trace) has the von Neumann algebra Haagerup property.
Theorem: [Daws, Fima, Skalski, White 2013] Let G be a discrete
Kac algebra. Then G has the Haagerup property if and only if L∞(G)
(with the canonical trace) has the von Neumann algebra Haagerup prop-
erty.
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Examples:
• Every co-amenable discrete quantum groups has the Haagerup prop-
erty.
Theorem: [Brannan 2012, De Commer-Freslon-Yamashita 2013]
The discrete quantum groups FOF and FUF have the Haagerup property.
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Exotic Quantum Group C*-algebras
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Exotic C*-algebras
Let us recall that a quantum group G is co-amenable if and only if
Cu(G) = C0(G).
So if G is not co-amenable, then we have a proper quotient
Cu(G) 7→ C0(G)
Question: If G is not co-amenable, do we have any (quantum group)
C*-algebra A, sitting in between Cu(G) and C0(G), i.e., do we have any
(quantum group) C*-algebra A such that we get proper quotients
Cu(G) 7→ A 7→ C0(G).
We call such C*-algebra A an exotic C*-algebra.
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Brown-Guentner’s D- C*-algebras C∗D(G)
Let G be a discrete group and D an ideal in `∞(G). A unitary repre-
sentation π : G → B(H) is called an D-representation if there eixsts a
dense subset H0 ⊆ H such that the coefficient functions
πξ,η(s) = 〈π(s)ξ|η〉 ∈ D
for all ξ, η ∈ H0.
• Direct sum of D-representations is a D-representation
• Tensor product of a D-representations is a D-representation.
• If Cc(G) ⊆ D ⊆ `∞(G), then for any f ∈ `1(G),
‖f‖D = {‖π(f)‖ : π is a D-representation}
defines a faithful C*-algebra norm on `1(G). The associated C*-algebra
C∗D(G) is a compact quantum group C*-algebra.
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Quantum Group Approach to D- C*-algebras C∗D(G)
This is a recent joint work with Michael Brannan.
Let us assume that G is a unimodular discrete quantum group. In this
case, we get
`∞(G) =∏α∈I B(Hα) with tracial Haar weight h = ⊕α∈IdαTrα
C0(G) = c0 −⊕α∈I B(Hα)
Cc(G) =⊕α∈I B(Hα).
`1(G) = `1 −⊕α∈I B(Hα)∗
`p(G) = `p −⊕α∈I Lp(Hα)
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Let D a subspace in `∞(G). A representation
π : ω ∈ `1(G)→ π(ω) = (ω ⊗ ι)(Uπ) ∈ B(H)
is an D-representation if there eixsts a dense subset H0 ⊆ H such that
the coefficient functions
ϕπξ,η = (ι⊗ ωξ,η)(Uπ) ∈ D
for all ξ, η ∈ H0.
• Direct sum of D-representations is a D-representation
• Tensor product of a D-representations is a D-representation.
• If Cc(G) ⊆ D ⊆ `∞(G), then for any ω ∈ `1(G),
‖ω‖D = {‖π(ω)‖ : π is a D-representation}
defines a faithful C*-algebra norm on `1(G). The associated C*-algebra
C∗D(G) is a compact quantum group C*-algebra.
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It is clear that if Cc(G) ⊆ D1 ⊆ D2 ⊆ `∞(G), then we have the quotient
C∗D2→ C∗D1
.
It is also easy to see that C∗`∞(G) = C∗(G).
Theorem: [Brown-Guenter 2012, Brannan-R 2014]
1) If Cc(G) ⊆ D ⊆ `2(G), then C∗D(G) = C∗λ(G).
2) G is amenable if and only if there exists an 1 ≤ p <∞ such that
C∗`p(G) = C∗u(G).
3) G has the Haagerup property if and only if C∗(G) = C∗C0(G)(G).
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Theorem: [Brown-Guentner 2012] For d ≥ 2, there exists some
2 < p <∞ such that C∗`p(Fd) is exotic, i.e., the quotents
C∗(Fd)→ C∗`p(Fd)→ C∗λ(Fd)
are proper.
Theorem : [Okayasu 2012] Let 2 < p < q <∞. The quotients
C∗(Fd)→ C∗`q(Fd)→ C∗`p(Fd)→ C∗λ(Fd)
are all proper. So there are uncountablly many of exotic compact quan-
tum group C*-algebras C∗`p(Fd).
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Quantum Group Case
Let N ≥ 3 and F ∈ GLN(C) such that FF ∈ RIN and F ∈ CUN . In this
case, FOF and FUF are unimodular discrete quantum groups.
Theorem : [Brannan-R 2014] Let 2 < p < q <∞. The quotients
C∗(FOF )→ C∗`q(FOF )→ C∗`p(FOF )→ C∗λ(FOF ) = C(O+F )
and
C∗(FUF )→ C∗`q(FUF )→ C∗`p(FUF )→ C∗λ(FUF ) = C(U+F )
are all proper. So we get uncountablly many of exotic compact quantum
group C*-algebras.
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The main ingredient in the proof for the FOF case is that we need
• Haagerup property
• Rapid decay property
More precisely, we consider a family of positive definite functions
ϕr =∏
n∈N0=Irr(O+F )
Sn(rN)
Sn(N)1B(Hn) ∈ C0(FOF ) (0 < 1 < r < 1)
where (Sn) are type 2 Chebyschev polynomials defined by
S0(x) = 1, S1(x) = 1, and Sn(x) = xSn(x)− Sn−1(x) (n ≥ 1).
Since each ϕr is a positive definite function in C0(FOF ), it corresponds
a positive linear functional µr ∈ C∗(FOF )∗ = B(FOF ).
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Sinse 2 < p < q <∞, we have the natural quotients
C∗(FOF )→ C∗`q(FOF )→ C∗`p(FOF )→ C∗λ(FOF ) = C(O+F )
and thus get the inclusions
C∗(FOF )∗ ← C∗`q(FOF )∗ ← C∗`p(FOF )∗ ← C∗λ(FOF )∗ = C(O+F )∗.
Then we can show that if 2 < p < q <∞, then there exists r0 such that
ϕr0 corresponds to a positive linear functional in C∗`q(FOF ), but not in
C∗`p(FOF ). This completes the proof.
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Thank you !
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