A note on noncommutative unique ergodicity and weighted means

Linear Algebra and its Applications 430 (2009) 782ndash790

Contents lists available at ScienceDirect

Linear Algebra and its Applications

j ourna l homepage wwwe lsev ie r com loca te laa

A note on noncommutative unique ergodicity

and weighted means

Luigi Accardi a Farrukh Mukhamedov blowast1

a Centro Interdisciplinare Vito Volterra II Universitagrave di Roma ldquoTor Vergatardquo Via Columbia 2 00133 Roma Italyb Department of Computational and Theoretical Sciences Faculty of Sciences International Islamic University Malaysia

PO Box 141 25710 Kuantan Pahang Malaysia

A R T I C L E I N F O A B S T R A C T

Article history

Received 27 February 2008

Accepted 17 September 2008

Submitted by H Schneider

AMS classification

47A35

46L35

46L55

Keywords

Uniquely ergodic

Markov operator

Riesz means

In this paper we study unique ergodicity of Clowast-dynamical system

(A T) consisting of a unital Clowast-algebra A and a Markov operator

T A rarr A relative to its fixed point subspace in terms of Riesz

summation which is weaker than Cesaro one Namely it is proven

that (A T) is uniquely ergodic relative to its fixed point subspace if

and only if its Riesz means

1

p1 + middot middot middot + pn

nsumk=1

pkTkx

converge to ET (x) in A for any x isin A as n rarr infin here ET is an pro-

jection of A to the fixed point subspace of T It is also constructed

a uniquely ergodic entangled Markov operator relative to its fixed

point subspace which is not ergodic

copy 2008 Elsevier Inc All rights reserved

1 Introduction

It is known [1622] that one of the important notions in ergodic theory is unique ergodicity of a

homeomorphism T of a compact Hausdorff space Recall that T is uniquely ergodic if there is a unique

T-invariant Borel probabilitymeasureμ on Thewell knownKrylovndashBogolyubov theorem [16] states

that T is uniquely ergodic if and only if for every f isin C() the averages

lowast Corresponding author

E-mail addresses accardivolterrauniroma2it (L Accardi) far75myandexru farrukhminusmiiuedumy (F Mukhamedov)1 The author (FM) is on leave from National University of Uzbekistan Department of Mechanics and Mathematics Vuzgor-

odok 100174 Tashkent Uzbekistan

0024-3795$ - see front matter copy 2008 Elsevier Inc All rights reserved

doi101016jlaa200809029

L Accardi F Mukhamedov Linear Algebra and its Applications 430 (2009) 782ndash790 783

1

n

nminus1sumk=0

f (Tkx)

converge uniformly to the constantintf dμ as n rarr infin

The study of ergodic theorems in recent years showed that the ordinary Cesaro means have been

replaced by weighted averages

nminus1sumk=0

akf (Tkx) (11)

Therefore it is natural to ask is there aweaker summation thanCesaro ensuring theunique ergodicity

In [15] it has been established that unique ergodicity implies uniform convergence of (11) when

ak is Riesz weight (see also [14] for similar results) In [4] similar problems were considered for

transformations of Hilbert spaces

On the other hand since the theory of quantum dynamical systems provides a convenient math-

ematical description of irreversible dynamics of an open quantum system (see [15]) investigation of

ergodic properties of such dynamical systems have had a considerable growth In a quantum setting

the matter is more complicated than in the classical case Some differences between classical and

quantum situations are pointed out in [119] Thismotivates an interest to study dynamics of quantum

systems (see [8912]) Therefore it is then natural to address the study of the possible generalizations

to quantum case of various ergodic properties known for classical dynamical systems In [1718] a non-

commutative notion of unique ergodicity was defined and certain properties were studied Recently

in [2] a general notion of unique ergodicity for automorphisms of a Clowast-algebra with respect to its fixed

point subalgebra has been introduced The present paper is devoted to a generalization of such a notion

for positive mappings of Clowast-algebras and its characterization in term of Riesz means

The paper is organized as follows Section 2 is devoted to preliminaries where we recall some facts

about Clowast-dynamical systems and the Riesz summation of a sequence on Clowast-algebras Here we define

a notion of unique ergodicity of Clowast-dynamical system relative to its fixed point subspace In Section 3

we prove that a Clowast-dynamical system (A T) is uniquely ergodic relative to its fixed point subspace if

and only if its Riesz means (see below)

1


nsumk=1

pkTkx

converge to ET (x) in A for any x isin A here ET is a projection of A onto the fixed point subspace of

T Note however that if T is completely positive then ET is a conditional expectation (see [620]) On

the other hand it is known [18] that unique ergodicity implies ergodicity Therefore one can ask can

a Clowast-dynamical system which is uniquely ergodic relative to its fixed point subspace be ergodic It

turns out that this question has a negative answer More precisely in Section 4we construct entangled

Markov operatorwhich is uniquely ergodic relative to its fixedpoint subspace butwhich is not ergodic

2 Preliminaries

In this section we recall some preliminaries concerning Clowast-dynamical systems

Let A be a Clowast-algebra with unit An element x isin A is called positive if there is an element y isin Asuch that x = ylowasty The set of all positive elementswill bedenotedbyA+ ByAlowast

wedenote the conjugate

space to A A linear functional ϕ isin Alowastis called Hermitian if ϕ(xlowast) = ϕ(x) for every x isin A A Hermitian

functional ϕ is called state if ϕ(xlowastx) 0 for every x isin A and ϕ( ) = 1 By SA (resp Alowasth) we denote the

set of all states (resp Hermitian functionals) on A By Mn(A) we denote the set of all n times n-matrices

a = (aij)with entries aij in A

Definition 21 A linear operator T A rarr A is called

(i) positive if Tx 0 whenever x 0

784 L Accardi F Mukhamedov Linear Algebra and its Applications 430 (2009) 782ndash790

(ii) n-positive if the linear mapping Tn Mn(A) rarr Mn(A) given by Tn(aij) = (T(aij)) is positive

(iii) completely positive if it is n-positive for all n isin N

A positive mapping T with T = is calledMarkov operator A pair (A T) consisting of a Clowast-algebraAandaMarkovoperatorT A rarr A is calledaClowast-dynamical system TheClowast-dynamical system (Aϕ T)

is calleduniquely ergodic if there is a unique invariant stateϕ (ieϕ(Tx) = ϕ(x) for all x isin A)with respect

to T Denote

AT = x isin A Tx = x (21)

It is clear thatATis a closed linear subspace ofA but in general it is not a subalgebra ofA (see Section

3) We say that (A T) is uniquely ergodic relative to ATif every state of AT

has a unique T-invariant

state extension to A In the case when ATconsists only of scalar multiples of the identity element

this reduces to the usual notion of unique ergodicity Note that for an automorphism such a notion has

been introduced in [2]

Now suppose we are given a sequence of numbers pn such that p1 gt 0 pk 0withsuminfin

k=1 pk = infin

We say that a sequence sn sub A is Riesz convergent to an element s isin A if the sequence

1sumnk=1pk

nsumk=1

pksk

converges to s in A and it is denoted by sn rarr s(R pn) The numbers pn are called weights If sn rarr s

implies sn rarr s(R pn) then Riesz-converges is said to be regular The regularity condition (see [13

Theorem 14]) is equivalent to

pn

p1 + p2 + middot middot middot + pnrarr 0 as n rarr infin (22)

Basics about (R pn) convergence can be found in [13]

Recall the following lemmawhich shows that Riesz convergence isweaker thanCesaro convergence

(see [1315])

Lemma 22 [13 Theorem 16] Assume that pn+1 pn and

npn

p1 + middot middot middot + pn C foralln isin N (23)

for some constant C gt 0 Then Cesaro convergence implies (R pn) convergence

3 Unique ergodicity

In this section we are going to characterize unique ergodicity relative to ATof Clowast-dynamical sys-

tems To do it we need the following

Lemma 31 (cf [182]) Let (A T) be uniquely ergodic relative to AT If h isin Alowast

is invariant with respect to

T and hAT = 0 then h = 0

Proof Let us first assume that h is Hermitian Then there is a unique Jordan decomposition [21] of h

such that

h = h+ minus hminus h1 = h+1 + hminus1 (31)

where middot 1 is the dual norm on Alowast The invariance of h implies that

h T = h+ T minus hminus T = h+ minus hminus

Using h+ T1 = h+( ) = h+1 similarly h+ T1 = h+1 from uniqueness of the decomposition

we find h+ T = h+ and hminus T = hminus From hAT = 0 one gets h( ) = 0 which implies that h+1 =


hminus1 On the other hand we also have h+h+1 = hminus

hminus1 on AT So according to the unique ergodicity

relative toATwe obtain h+ = hminus onA Consequently h = 0 Now let h be an arbitrary bounded linear

functional Then it can bewritten as h = h1 + ih2 where h1 and h2 are Hermitian Again invariance of h

implies that hi T = hi i = 1 2 From hAT = 0 one gets hkAT = 0 k = 1 2 Consequently according

to the above argument we obtain h = 0

Now we are ready to formulate a criterion for unique ergodicity of Clowast-dynamical system in terms

of (R pn) convergence In the proof we will follow some ideas used in [21518]

Theorem 32 Let (A T) be a Clowast-dynamical system Assume that the weight pn satisfiesP(n) = p1 + |p2 minus p1| + middot middot middot + |pn minus pnminus1| + pn


Then the following conditions are equivalent

(i) (A T) is uniquely ergodic relative to AT

(ii) The set AT + a minus T(a) a isin A is dense in A

(iii) For all x isin A

Tnx rarr ET (x) (R pn)

where ET (x) is a positive norm one projection onto ATsuch that ETT = TET = ET Moreover the

following estimation holds∥∥∥∥∥∥1sumn

k=1pk

nsumk=1

pkTk(x)minus ET (x)

∥∥∥∥∥∥ P(n)x n isin N (33)

for every x isin A

(iv) For every x isin A and ψ isin SA

ψ(Tk(x)) rarr ψ(ET (x))(R pn)

Proof Consider the implication (i)rArr(ii) Assume that AT + a minus T(a) a isin A = A then there is an

element x0 isin A such that x0 isin AT + a minus T(a) a isin A Then according to the HahnndashBanach theorem

there is a functional h isin Alowastsuch that h(x0) = 1 and hAT + a minus T(a) a isin A = 0 The last condi-

tion implies that hAT = 0 and h T = h Hence Lemma 31 yields that h = 0 which contradicts to

h(x0) = 1

(ii)rArr(iii) It is clear that for every element of the form y = x minus T(x) x isin A by (32) we have

1sumnk=1 pk

∥∥∥∥∥∥nsum

k=1

pkTk(y)

∥∥∥∥∥∥ = 1sumnk=1 pk


k=1

pk(Tk+1(x)minus Tkx)

∥∥∥∥∥∥= 1sumn

k=1 pkp1Tx + (p2 minus p1)T

2x + middot middot middot

+(pn minus pnminus1)Tnx minus pnT

n+1xP(n)x rarr 0 as n rarr infin (34)

Now let x isin AT then

limnrarrinfin

1sumnk=1 pk

nsumk=1

pkTk(x) = x (35)

Hence for every x isin AT + a minus T(a) a isin A the limit


limnrarrinfin

1sumnk=1 pk

nsumk=1

pkTkx

exists which is denoted by ET (x) It is clear that ET is a positive linear operator from AT + a minus T(a) a isin A onto AT

Positivity and ET = imply that ET is bounded From (34) one obviously gets that

ETT = TET = ET According to (ii) the operator ET can be uniquely extended to A this extension is

denoted by the same symbol ET It is evident that ET is a positive projection with ET = 1

Now take an arbitrary x isin A Then again using (ii) for any ε gt 0 we can find xε isin AT + a minus T(a) a isin A such that x minus xε ε By means of (34) (35) we conclude that∥∥∥∥∥∥

1sumnk=1 pk

nsumk=1

pkTk(xε)minus ET (xε)

∥∥∥∥∥∥ P(n)xε

Hence one has∥∥∥∥∥∥1sumn

k=1 pk

nsumk=1

pkTk(x)minus ET (x)

∥∥∥∥∥∥ ∥∥∥∥∥∥

1sumnk=1 pk

nsumk=1

pkTk(x minus xε)

∥∥∥∥∥∥+

∥∥∥∥∥∥1sumn

k=1 pk

nsumk=1


∥∥∥∥∥∥+ET (x minus xε)

2x minus xε + P(n)xεP(n)x + (2 + P(n))ε

which with the arbitrariness of ε implies (33)

Consequently

limnrarrinfin

1sumnk=1 pk

nsumk=1

pkTkx = ET (x)

is valid for every x isin A

Themapping ET is a unique T-invariant positive projection Indeed if E A rarr ATis any T-invariant

positive projection onto AT then

E(x) = 1sumnk=1 pk

nsumk=1

pkE(Tk(x)) = E

⎛⎝ 1sumn

k=1 pk

nsumk=1

pkTk(x)

⎞⎠

Taking the limit as n rarr infin gives

E(x) = E(ET (x)) = ET (x)

The implication (iii)rArr(iv) is obvious Let us consider (iv)rArr(i) Let ψ be any state onAT then ψ ET

is a T-invariant extension of ψ to A Assume that φ is any T-invariant linear extension of ψ Then

φ(x) = 1sumnk=1 pk

nsumk=1

pkφ(Tk(x)) = φ

⎛⎝ 1sumn

k=1 pk

nsumk=1

pkTk(x)

⎞⎠

Now taking the limit from both sides of the last equality as n rarr infin one gives

φ(x) = φ(ET (x)) = ψ(ET (x))

so φ = ψ ET

Remark 33 If we choose pn = 1 for all n isin N then it is clear that the condition (32) is satisfied hence

we infer that unique ergodicity relative to ATis equivalent to the norm convergence of the mean

averages ie


1

n

nsumk=1

Tk(x)

which recovers the result of [2]

Remark 34 If the condition (23) is satisfied then condition (32) is valid as well This means that

uniqueergodicitywould remain true if Cesaro summation is replacedbyaweaker Theorem32extends

a result of Mukhamedov and Temir [18]

Example If we define pn = nα with α gt 0 then one can see that pn is an increasing sequence and

condition (32) is also satisfied This provides a concrete example of weights

Remark 35 Note that some nontrivial examples of uniquely ergodic quantum dynamical systems

based on automorphisms has been given in [2] Namely itwas proved that free shifts based on reduced

Clowast-algebras of RD-groups (including the free group on infinitely many generators) and amalgamated

free product Clowast-algebras are uniquely ergodic relative to the fixed-point subalgebra In [11] it has been

proved that such shifts possess a stronger property called F-strict weak mixing (see also [18])

Observation We note that in general the projection ET is not a conditional expectation but when T

is an automorphism then it is so Now we are going to provide an example of Markov operator which

is uniquely ergodic relative to its fixed point subspace for which the projector ET is not a conditional

expectation

Consider the algebra Md(C) ndash d times d matrices over C For a matrix x = (xij) by xt we denote its

transpose matrix ie xt = (xji) Define a mapping φ Md(C) rarr Md(C) by φ(x) = xt Then it is known

[20] that such a mapping is positive but not completely positive One can see that φ is a Markov

operator Due to the equality

x = x + xt

2+ x minus xt

2condition (ii) of Theorem 32 is satisfied so φ is uniquely ergodic with respect to Md(C)

φ Hence the

corresponding projection Eφ is given by Eφ(x) = (x + xt)2which is not completely positiveMoreover

Md(C)φ is the set of all symmetric matrices which do not form an algebra So Eφ is not a conditional

expectation

4 A uniquely ergodic entangled Markov operator

In recentdevelopmentsofquantuminformationmanypeoplehavediscussed theproblemoffinding

a satisfactory quantum generalization of classical random walks Motivating this in [310] a new class

of quantumMarkov chainswas constructedwhich are at the same time purely generated and uniquely

determined by a corresponding classical Markov chain Such a class of Markov chains was constructed

bymeans of entangledMarkov operators In onersquos turn theywere associatedwith Schurmultiplication

In that paper ergodicity andweak clustering properties of such chainswere established In this section

we are going to provide entangledMarkov operatorwhich is uniquely ergodic relative to its fixed point

subspace but which is not ergodic

Let us recall some notations To define Schur multiplication we choose an orthonormal basis ejj = 1 d in a d-dimensional Hilbert space Hd which is kept fixed during the analysis In such a way

we have the natural identification Hd with Cd The corresponding system of matrix units eij = ei otimes ej

identifies B(Hd) with Md(C) Then for x = sumdij=1 xijeij y = sumd

ij=1 yijeij elements of Md(C) we define

Schur multiplication in Md(C) as usual

x 13 y =dsum

ij=1

(xijyij)eij (41)

that is componentwise (x 13 y)ij =xijyij


A linear map P Md(C) rarr Md(C) is said to be Schur identity-preserving if its diagonal projection is

the identity ie 13 P( ) = It is called an entangledMarkov operator if in addition P( ) =

The entangled Markov operator (see [3]) associated to a stochastic matrix = (pij)dij=1

and to the

canonical systems of matrix units eijdij=1ofMd(C) is defined by

P(x)ij =dsum

kl=1

radicpikpjlxkl (42)

where as before x = sumdij=1 xijeij

Define a Markov operator Md(C) rarr Md(C) by

(x) = 13 P(x) x isin Md(C) (43)

Given a stochastic matrix = (pij) put

Fix() = ψ isin Cd ψ = ψTo every vector a = (a1 ad) isin Cd

corresponds a diagonal matrix xa inMd(C) defined by

xa =

⎛⎜⎜⎝a1 0 middot middot middot 0

0 a2 middot middot middot 0

middot middot middot middot middot middot middot middot middot middot middot middot0 0 middot middot middot ad

⎞⎟⎟⎠ (44)

Lemma 41 For a Markov operator given by (43) one has

Md(C) = xψ ψ isin Fix()

Proof Let x = (xij) isin Md(C) ie(x) = x From (41) and (43)we conclude that xij = 0 if i = j There-

fore due to (42) one finds

dsumj=1

radicpijpijxjj = xii

which implies that (x11 xdd) isin Fix()

Furthermoreweassumethat thedimensionof Fix() is greaterorequal than2 iedim(Fix()) 2

Hence according to Lemma 41 we conclude that Md(C) is a nontrivial commutative subalgebra of

Md(C)

Theorem 42 Let be a stochastic matrix such that dim(Fix()) 2 Then the corresponding Markov

operator given by (43) is uniquely ergodic wrtMd(C)

Proof To prove the statement it is enough to establish condition (ii) of Theorem 32 Take any x =(xij) isin Md(C) Now we are going to show that it can be represented as follows

x = x1 + x2 (45)

where x1 isin Md(C) and x2 isin y minus (y) y isin Md(C)

Due to Lemma 41 there is a vector ψ isin Fix() such that x1 = xψ and hence from (43) (45) one

finds that

x2 =

⎛⎜⎜⎝ϕ11 x12 middot middot middot x1dx21 ϕ22 middot middot middot x2dmiddot middot middot middot middot middot middot middot middot middot middot middotxd1 xd2 middot middot middot ϕdd

⎞⎟⎟⎠ (46)

where


ϕii = ξi minusdsum

j=1

pijξj minusdsum

kl=1k =j

radicpikpilxkl (47)

The existenceof the vectorsψ = (ψ1 ψd) and (ξ1 ξd) follows immediately from the following

relations

ψi + ξi minusdsum

j=1

pijξj = xii +dsum

kl=1k =j

radicpikpilxkl i = 1 d (48)

since the number of unknowns is greater than the number of equations Note that the equality (48)

comes from (43)ndash(47) Hence one concludes that the equality

Md(C) + x minus (x) x isin Md(C) = Md(C)

which completes the proof

Let us provide a more concrete example

Example Consider on M3(C) the following stochastic matrix 0 defined by

0 =⎛⎝1 0 0

0 0 1

0 u v

⎞⎠ (49)

here u v 0 u + v = 1

One can immediately find that

Fix(0) = (x y y) x y isin C (410)

Then for the corresponding Markov operator 0 given by (43) (42) due to Lemma 41 one has

M3(C)0 =

⎧⎨⎩

⎛⎝x 0 0

0 y 0

0 0 y

⎞⎠ x y isin C

⎫⎬⎭ (411)

SoM3(C)0 is a nontrivial commutative subalgebra of M3(C) having dimension 2

So according to Theorem 42 we see that 0 is uniquely ergodic relative to M3(C)0 But (411)

implies that 0 is not ergodic Note that ergodicity of entangled Markov chains has been studied in

[3]

Acknowledgments

The second named author (FM) thanks Prof L Accardi for kind hospitality at ldquoUniversitagrave di Roma

Tor Vergatardquo during 18ndash22 June of 2006 Finally the authors also would like to thank to the referee for

his useful suggestions which allowed them to improve the text of the paper

References

[1] S Albeverio R Hoslashegh-Krohn Frobenius theory for positive maps of von Neumann algebras Comm Math Phys 64 (1978)

83ndash94

[2] B Abadie K Dykema Unique ergodicity of free shifts and some other automorphisms of Clowast-algebras J Operator Theoryin press lthttpwwwarxivorgmathOA0608227gt

[3] L Accardi F Fidaleo Entangled markov chains Ann Mat Pura Appl 184 (2005) 327ndash346

[4] D Berend M Lin J Rosenblatt A Tempelman Modulated and subsequential ergodic theorems in Hilbert and Banach

spaces Ergodic Theory Dynam Systems 22 (2002) 1653ndash1665


[5] O Bratteli DW Robinson Operator Algebras and Quantum Statistical Mechanics I Springer New York Heidelberg Berlin

1979

[6] M-D Choi A Schwarz inequality for positive linear maps on Clowast-algebras Illinois J Math 18 (1974) 565ndash574

[7] M-D Choi Completely Positive Linear Maps on Complex Matrices Linear Algebra Appl 10 (1975) 285ndash290

[8] F Fagnola R Rebolledo On the existance of stationary states for quantum dyanamical semigroups J Math Phys 42 (2001)

1296ndash1308

[9] F Fagnola R Rebolledo Transience and recurrence of quantum Markov semi-groups Probab Theory Related Fields 126

(2003) 289ndash306

[10] F Fidaleo Infinite dimensional entangled Markov chains Random Oper Stochastic Equations 12 (2004) 4 393ndash404

[11] F Fidaleo F Mukhamedov Strict weakmixing of some Clowast-dynamical systems based on free shifts J Math Anal Appl 336

(2007) 180ndash187

[12] A Frigerio M Verri Long-time asymptotic properties of dynamical semi-groups on Wlowast-algebras Math Z 180 (1982)

275ndash286

[13] GH Hardy Divergent Series Cambridge Univ Press 1949

[14] A Iwanik Unique ergodicity of irreducible Markov operators on C(X) Studia Math 77 (1983) 81ndash86

[15] VV Kozlov Weighted means strict ergodicity and uniform distributions Math Notes 78 (2005) 329ndash337

[16] IP Kornfeld YaG Sinai SV Fomin Ergodic Theory Springer Berlin Heidelberg New York 1982

[17] R Longo C Peligrad Noncommutative topological dynamics and compact actions on Clowast-algebras J Funct Anal 58 (1984)

157ndash174

[18] FMukhamedov S Temir A few remarks onmixing properties of Clowast-dynamical systemsRockyMountain JMath 37 (2007)

1685ndash1703

[19] C Nicolescu A Stroumlh L Zsidoacute Noncommutative extensions of classical and multiple recurrence theorems J Operator

Theory 50 (2003) 3ndash52

[20] V Paulsen Completely Bounded Maps and Operator Algebras Cambridge Univ Press 2002

[21] M Takesaki Theory of Operator Algebras I Springer Berlin Heidelberg New York 1979

[22] P Walters An Introduction to Ergodic Theory Springer Berlin Heidelberg New York 1982

Introduction

Preliminaries

Unique ergodicity

A uniquely ergodic entangled Markov operator

Acknowledgments

References


1

n

nminus1sumk=0

f (Tkx)

converge uniformly to the constantintf dμ as n rarr infin

The study of ergodic theorems in recent years showed that the ordinary Cesaro means have been

replaced by weighted averages

nminus1sumk=0

akf (Tkx) (11)

Therefore it is natural to ask is there aweaker summation thanCesaro ensuring theunique ergodicity

In [15] it has been established that unique ergodicity implies uniform convergence of (11) when

ak is Riesz weight (see also [14] for similar results) In [4] similar problems were considered for

transformations of Hilbert spaces

On the other hand since the theory of quantum dynamical systems provides a convenient math-

ematical description of irreversible dynamics of an open quantum system (see [15]) investigation of

ergodic properties of such dynamical systems have had a considerable growth In a quantum setting

the matter is more complicated than in the classical case Some differences between classical and

quantum situations are pointed out in [119] Thismotivates an interest to study dynamics of quantum

systems (see [8912]) Therefore it is then natural to address the study of the possible generalizations

to quantum case of various ergodic properties known for classical dynamical systems In [1718] a non-

commutative notion of unique ergodicity was defined and certain properties were studied Recently

in [2] a general notion of unique ergodicity for automorphisms of a Clowast-algebra with respect to its fixed

point subalgebra has been introduced The present paper is devoted to a generalization of such a notion

for positive mappings of Clowast-algebras and its characterization in term of Riesz means

The paper is organized as follows Section 2 is devoted to preliminaries where we recall some facts

about Clowast-dynamical systems and the Riesz summation of a sequence on Clowast-algebras Here we define

a notion of unique ergodicity of Clowast-dynamical system relative to its fixed point subspace In Section 3

we prove that a Clowast-dynamical system (A T) is uniquely ergodic relative to its fixed point subspace if

and only if its Riesz means (see below)

1


nsumk=1

pkTkx

converge to ET (x) in A for any x isin A here ET is a projection of A onto the fixed point subspace of

T Note however that if T is completely positive then ET is a conditional expectation (see [620]) On

the other hand it is known [18] that unique ergodicity implies ergodicity Therefore one can ask can

a Clowast-dynamical system which is uniquely ergodic relative to its fixed point subspace be ergodic It

turns out that this question has a negative answer More precisely in Section 4we construct entangled

Markov operatorwhich is uniquely ergodic relative to its fixedpoint subspace butwhich is not ergodic

2 Preliminaries

In this section we recall some preliminaries concerning Clowast-dynamical systems

Let A be a Clowast-algebra with unit An element x isin A is called positive if there is an element y isin Asuch that x = ylowasty The set of all positive elementswill bedenotedbyA+ ByAlowast

wedenote the conjugate

space to A A linear functional ϕ isin Alowastis called Hermitian if ϕ(xlowast) = ϕ(x) for every x isin A A Hermitian

functional ϕ is called state if ϕ(xlowastx) 0 for every x isin A and ϕ( ) = 1 By SA (resp Alowasth) we denote the

set of all states (resp Hermitian functionals) on A By Mn(A) we denote the set of all n times n-matrices

a = (aij)with entries aij in A

Definition 21 A linear operator T A rarr A is called

(i) positive if Tx 0 whenever x 0






to T Denote









k=1 pk = infin


1sumnk=1pk

nsumk=1

pksk




pn




(see [1315])


npn



3 Unique ergodicity







such that
























k=1pk

nsumk=1

pkTk(x)minus ET (x)

∥∥∥∥∥∥ P(n)x n isin N (33)

for every x isin A







h(x0) = 1


1sumnk=1 pk


k=1

pkTk(y)

∥∥∥∥∥∥ = 1sumnk=1 pk


k=1


∥∥∥∥∥∥= 1sumn






limnrarrinfin

1sumnk=1 pk

nsumk=1

pkTk(x) = x (35)



limnrarrinfin

1sumnk=1 pk

nsumk=1

pkTkx






1sumnk=1 pk

nsumk=1


∥∥∥∥∥∥ P(n)xε


k=1 pk

nsumk=1

pkTk(x)minus ET (x)

∥∥∥∥∥∥ ∥∥∥∥∥∥

1sumnk=1 pk

nsumk=1

pkTk(x minus xε)

∥∥∥∥∥∥+

∥∥∥∥∥∥1sumn

k=1 pk

nsumk=1





Consequently

limnrarrinfin

1sumnk=1 pk

nsumk=1

pkTkx = ET (x)




E(x) = 1sumnk=1 pk

nsumk=1

pkE(Tk(x)) = E

⎛⎝ 1sumn

k=1 pk

nsumk=1

pkTk(x)

⎞⎠


E(x) = E(ET (x)) = ET (x)



φ(x) = 1sumnk=1 pk

nsumk=1

pkφ(Tk(x)) = φ

⎛⎝ 1sumn

k=1 pk

nsumk=1

pkTk(x)

⎞⎠


φ(x) = φ(ET (x)) = ψ(ET (x))

so φ = ψ ET



averages ie


1

n

nsumk=1

Tk(x)















expectation





x = x + xt

2+ x minus xt


φ Hence the



expectation















x 13 y =dsum

ij=1

(xijyij)eij (41)






and to the


P(x)ij =dsum

kl=1

radicpikpjlxkl (42)



(x) = 13 P(x) x isin Md(C) (43)




xa =




⎞⎟⎟⎠ (44)





dsumj=1





Md(C)




x = x1 + x2 (45)



finds that

x2 =


⎞⎟⎟⎠ (46)

where



j=1

pijξj minusdsum

kl=1k =j

radicpikpilxkl (47)


relations

ψi + ξi minusdsum

j=1

pijξj = xii +dsum

kl=1k =j








0 =⎛⎝1 0 0

0 0 1

0 u v

⎞⎠ (49)



Fix(0) = (x y y) x y isin C (410)


M3(C)0 =

⎧⎨⎩

⎛⎝x 0 0

0 y 0

0 0 y

⎞⎠ x y isin C

⎫⎬⎭ (411)




[3]

Acknowledgments




References


83ndash94







1979




1296ndash1308


(2003) 289ndash306



(2007) 180ndash187


275ndash286






157ndash174


1685ndash1703






Introduction

Preliminaries

Unique ergodicity


Acknowledgments

References






to T Denote









k=1 pk = infin


1sumnk=1pk

nsumk=1

pksk




pn




(see [1315])


npn



3 Unique ergodicity







such that
























k=1pk

nsumk=1

pkTk(x)minus ET (x)

∥∥∥∥∥∥ P(n)x n isin N (33)

for every x isin A







h(x0) = 1


1sumnk=1 pk


k=1

pkTk(y)

∥∥∥∥∥∥ = 1sumnk=1 pk


k=1


∥∥∥∥∥∥= 1sumn






limnrarrinfin

1sumnk=1 pk

nsumk=1

pkTk(x) = x (35)



limnrarrinfin

1sumnk=1 pk

nsumk=1

pkTkx






1sumnk=1 pk

nsumk=1


∥∥∥∥∥∥ P(n)xε


k=1 pk

nsumk=1

pkTk(x)minus ET (x)

∥∥∥∥∥∥ ∥∥∥∥∥∥

1sumnk=1 pk

nsumk=1

pkTk(x minus xε)

∥∥∥∥∥∥+

∥∥∥∥∥∥1sumn

k=1 pk

nsumk=1





Consequently

limnrarrinfin

1sumnk=1 pk

nsumk=1

pkTkx = ET (x)




E(x) = 1sumnk=1 pk

nsumk=1

pkE(Tk(x)) = E

⎛⎝ 1sumn

k=1 pk

nsumk=1

pkTk(x)

⎞⎠


E(x) = E(ET (x)) = ET (x)



φ(x) = 1sumnk=1 pk

nsumk=1

pkφ(Tk(x)) = φ

⎛⎝ 1sumn

k=1 pk

nsumk=1

pkTk(x)

⎞⎠


φ(x) = φ(ET (x)) = ψ(ET (x))

so φ = ψ ET



averages ie


1

n

nsumk=1

Tk(x)















expectation





x = x + xt

2+ x minus xt


φ Hence the



expectation















x 13 y =dsum

ij=1

(xijyij)eij (41)






and to the


P(x)ij =dsum

kl=1

radicpikpjlxkl (42)



(x) = 13 P(x) x isin Md(C) (43)




xa =




⎞⎟⎟⎠ (44)





dsumj=1





Md(C)




x = x1 + x2 (45)



finds that

x2 =


⎞⎟⎟⎠ (46)

where



j=1

pijξj minusdsum

kl=1k =j

radicpikpilxkl (47)


relations

ψi + ξi minusdsum

j=1

pijξj = xii +dsum

kl=1k =j








0 =⎛⎝1 0 0

0 0 1

0 u v

⎞⎠ (49)



Fix(0) = (x y y) x y isin C (410)


M3(C)0 =

⎧⎨⎩

⎛⎝x 0 0

0 y 0

0 0 y

⎞⎠ x y isin C

⎫⎬⎭ (411)




[3]

Acknowledgments




References


83ndash94







1979




1296ndash1308


(2003) 289ndash306



(2007) 180ndash187


275ndash286






157ndash174


1685ndash1703






Introduction

Preliminaries

Unique ergodicity


Acknowledgments

References



















k=1pk

nsumk=1

pkTk(x)minus ET (x)

∥∥∥∥∥∥ P(n)x n isin N (33)

for every x isin A







h(x0) = 1


1sumnk=1 pk


k=1

pkTk(y)

∥∥∥∥∥∥ = 1sumnk=1 pk


k=1


∥∥∥∥∥∥= 1sumn






limnrarrinfin

1sumnk=1 pk

nsumk=1

pkTk(x) = x (35)



limnrarrinfin

1sumnk=1 pk

nsumk=1

pkTkx






1sumnk=1 pk

nsumk=1


∥∥∥∥∥∥ P(n)xε


k=1 pk

nsumk=1

pkTk(x)minus ET (x)

∥∥∥∥∥∥ ∥∥∥∥∥∥

1sumnk=1 pk

nsumk=1

pkTk(x minus xε)

∥∥∥∥∥∥+

∥∥∥∥∥∥1sumn

k=1 pk

nsumk=1





Consequently

limnrarrinfin

1sumnk=1 pk

nsumk=1

pkTkx = ET (x)




E(x) = 1sumnk=1 pk

nsumk=1

pkE(Tk(x)) = E

⎛⎝ 1sumn

k=1 pk

nsumk=1

pkTk(x)

⎞⎠


E(x) = E(ET (x)) = ET (x)



φ(x) = 1sumnk=1 pk

nsumk=1

pkφ(Tk(x)) = φ

⎛⎝ 1sumn

k=1 pk

nsumk=1

pkTk(x)

⎞⎠


φ(x) = φ(ET (x)) = ψ(ET (x))

so φ = ψ ET



averages ie


1

n

nsumk=1

Tk(x)















expectation





x = x + xt

2+ x minus xt


φ Hence the



expectation















x 13 y =dsum

ij=1

(xijyij)eij (41)






and to the


P(x)ij =dsum

kl=1

radicpikpjlxkl (42)



(x) = 13 P(x) x isin Md(C) (43)




xa =




⎞⎟⎟⎠ (44)





dsumj=1





Md(C)




x = x1 + x2 (45)



finds that

x2 =


⎞⎟⎟⎠ (46)

where



j=1

pijξj minusdsum

kl=1k =j

radicpikpilxkl (47)


relations

ψi + ξi minusdsum

j=1

pijξj = xii +dsum

kl=1k =j








0 =⎛⎝1 0 0

0 0 1

0 u v

⎞⎠ (49)



Fix(0) = (x y y) x y isin C (410)


M3(C)0 =

⎧⎨⎩

⎛⎝x 0 0

0 y 0

0 0 y

⎞⎠ x y isin C

⎫⎬⎭ (411)




[3]

Acknowledgments




References


83ndash94







1979




1296ndash1308


(2003) 289ndash306



(2007) 180ndash187


275ndash286






157ndash174


1685ndash1703






Introduction

Preliminaries

Unique ergodicity


Acknowledgments

References


limnrarrinfin

1sumnk=1 pk

nsumk=1

pkTkx






1sumnk=1 pk

nsumk=1


∥∥∥∥∥∥ P(n)xε


k=1 pk

nsumk=1

pkTk(x)minus ET (x)

∥∥∥∥∥∥ ∥∥∥∥∥∥

1sumnk=1 pk

nsumk=1

pkTk(x minus xε)

∥∥∥∥∥∥+

∥∥∥∥∥∥1sumn

k=1 pk

nsumk=1





Consequently

limnrarrinfin

1sumnk=1 pk

nsumk=1

pkTkx = ET (x)




E(x) = 1sumnk=1 pk

nsumk=1

pkE(Tk(x)) = E

⎛⎝ 1sumn

k=1 pk

nsumk=1

pkTk(x)

⎞⎠


E(x) = E(ET (x)) = ET (x)



φ(x) = 1sumnk=1 pk

nsumk=1

pkφ(Tk(x)) = φ

⎛⎝ 1sumn

k=1 pk

nsumk=1

pkTk(x)

⎞⎠


φ(x) = φ(ET (x)) = ψ(ET (x))

so φ = ψ ET



averages ie


1

n

nsumk=1

Tk(x)















expectation





x = x + xt

2+ x minus xt


φ Hence the



expectation















x 13 y =dsum

ij=1

(xijyij)eij (41)






and to the


P(x)ij =dsum

kl=1

radicpikpjlxkl (42)



(x) = 13 P(x) x isin Md(C) (43)




xa =




⎞⎟⎟⎠ (44)





dsumj=1





Md(C)




x = x1 + x2 (45)



finds that

x2 =


⎞⎟⎟⎠ (46)

where



j=1

pijξj minusdsum

kl=1k =j

radicpikpilxkl (47)


relations

ψi + ξi minusdsum

j=1

pijξj = xii +dsum

kl=1k =j








0 =⎛⎝1 0 0

0 0 1

0 u v

⎞⎠ (49)



Fix(0) = (x y y) x y isin C (410)


M3(C)0 =

⎧⎨⎩

⎛⎝x 0 0

0 y 0

0 0 y

⎞⎠ x y isin C

⎫⎬⎭ (411)




[3]

Acknowledgments




References


83ndash94







1979




1296ndash1308


(2003) 289ndash306



(2007) 180ndash187


275ndash286






157ndash174


1685ndash1703






Introduction

Preliminaries

Unique ergodicity


Acknowledgments

References


1

n

nsumk=1

Tk(x)















expectation





x = x + xt

2+ x minus xt


φ Hence the



expectation















x 13 y =dsum

ij=1

(xijyij)eij (41)






and to the


P(x)ij =dsum

kl=1

radicpikpjlxkl (42)



(x) = 13 P(x) x isin Md(C) (43)




xa =




⎞⎟⎟⎠ (44)





dsumj=1





Md(C)




x = x1 + x2 (45)



finds that

x2 =


⎞⎟⎟⎠ (46)

where



j=1

pijξj minusdsum

kl=1k =j

radicpikpilxkl (47)


relations

ψi + ξi minusdsum

j=1

pijξj = xii +dsum

kl=1k =j








0 =⎛⎝1 0 0

0 0 1

0 u v

⎞⎠ (49)



Fix(0) = (x y y) x y isin C (410)


M3(C)0 =

⎧⎨⎩

⎛⎝x 0 0

0 y 0

0 0 y

⎞⎠ x y isin C

⎫⎬⎭ (411)




[3]

Acknowledgments




References


83ndash94







1979




1296ndash1308


(2003) 289ndash306



(2007) 180ndash187


275ndash286






157ndash174


1685ndash1703






Introduction

Preliminaries

Unique ergodicity


Acknowledgments

References





and to the


P(x)ij =dsum

kl=1

radicpikpjlxkl (42)



(x) = 13 P(x) x isin Md(C) (43)




xa =




⎞⎟⎟⎠ (44)





dsumj=1





Md(C)




x = x1 + x2 (45)



finds that

x2 =


⎞⎟⎟⎠ (46)

where



j=1

pijξj minusdsum

kl=1k =j

radicpikpilxkl (47)


relations

ψi + ξi minusdsum

j=1

pijξj = xii +dsum

kl=1k =j








0 =⎛⎝1 0 0

0 0 1

0 u v

⎞⎠ (49)



Fix(0) = (x y y) x y isin C (410)


M3(C)0 =

⎧⎨⎩

⎛⎝x 0 0

0 y 0

0 0 y

⎞⎠ x y isin C

⎫⎬⎭ (411)




[3]

Acknowledgments




References


83ndash94







1979




1296ndash1308


(2003) 289ndash306



(2007) 180ndash187


275ndash286






157ndash174


1685ndash1703






Introduction

Preliminaries

Unique ergodicity


Acknowledgments

References



j=1

pijξj minusdsum

kl=1k =j

radicpikpilxkl (47)


relations

ψi + ξi minusdsum

j=1

pijξj = xii +dsum

kl=1k =j








0 =⎛⎝1 0 0

0 0 1

0 u v

⎞⎠ (49)



Fix(0) = (x y y) x y isin C (410)


M3(C)0 =

⎧⎨⎩

⎛⎝x 0 0

0 y 0

0 0 y

⎞⎠ x y isin C

⎫⎬⎭ (411)




[3]

Acknowledgments




References


83ndash94







1979




1296ndash1308


(2003) 289ndash306



(2007) 180ndash187


275ndash286






157ndash174


1685ndash1703






Introduction

Preliminaries

Unique ergodicity


Acknowledgments

References



1979




1296ndash1308


(2003) 289ndash306



(2007) 180ndash187


275ndash286






157ndash174


1685ndash1703






Introduction

Preliminaries

Unique ergodicity


Acknowledgments

References

A note on noncommutative unique ergodicity and weighted means

Documents

Transcript of A note on noncommutative unique ergodicity and weighted means