On weak convergence of iterates in quantum Lp spaces (p≥1)

IN QUANTUM

L

-SPACES

(

p

≥

1)

GENADY YA. GRABARNIK, ALEXANDER A. KATZ, AND LAURA SHWARTZ

Received 13 April 2005

Equivalent conditions are obtained for weak convergence of iterates of positive contrac-tions in theL1-spaces for general von Neumann algebra and general JBW algebras, as well

as for Segal-DixmierLp-spaces (1≤p <∞) affiliated to semifinite von Neumann algebras and semifinite JBW algebras without direct summands of typeI2.

1. Introduction and preliminaries

This paper is devoted to a presentation of some results concerning ergodic-type prop-erties of weak convergence of iterates of operators acting in L1-space for general von

Neumann algebras and JBW algebras, as well as Segal-DixmierLp-spaces (1≤p <∞) of operators affiliated with semifinite von Neumann algebras and semifinite JBW algebras.

The first results in the field of noncommutative ergodic theory were obtained inde-pendently by Sina˘i and Ansˇeleviˇc [21] and Lance [15]. Developments of the subject are reflected in the monographs of Jajte [13] and Krengel [14] (see also [8,9,10,18]).

We will use facts and the terminology from the general theory of von Neumann alge-bras (see [5,7,17,19,22]), the general theory of Jordan and real operator algebras (see [2,3,11,16]), and the theory of noncommutative integration (see [20,23,24]).

Let M be a von Neumann algebra, acting on a separable Hilbert space H,M∗is a

predual space ofM, which always exists according to the Sakai theorem [19]. It is well known thatM∗could be identified withL1-space forM.

SpacesL1andL2of the operators affiliated with the semifinite von Neumann algebra

M with semifinite faithful traceτ were introduced by Segal (see [20]). This result was extended toLp-space of operators affiliated with von Neumann algebrasM,τ, and in-tegrated withpth power by Dixmier (see [6]). For an alternative exposition of building Lpbased on Grothendieck’s idea of using rearrangements of functions, see also [24]. The theory ofLp-spaces was extended further to the von Neumann algebras with faithful nor-mal weightρ. However, these spaces lack some of the properties, for example, in general, these spaces do not intersect.

Recall some standard terminology (see [8,9,10,14]).

Copyright©2005 Hindawi Publishing Corporation

p

Definition 1.1. A linear mappingTfromM∗in itself is called acontractionif its norm is

not greater than one.

Definition 1.2. A contractionTis said to bepositiveif

TM∗+⊂M∗+. (1.1)

We will consider the two topologies on the spaceM∗: theweak topology, or theσ(M∗,

M) topology, and thestrong topologyof theM∗-space norm convergence.

Definition 1.3. A matrix (an,i),i,n=1, 2,..., of real numbers is calleduniformly regularif

sup n

∞

i=1

_{an,i≤C <∞; lim}

n→∞sup_i an,i=0; nlim→∞

i an

,i=1. (1.2)

2. Main result: the case of quantumL1-spaces

2.1. The case of noncommutativeL1-spaces. The following theorem is valid.

Theorem2.1. The following conditions for a positive contractionT in the predual space of

a complex von Neumann algebrasMare equivalent.

(i)The sequence{Ti_{}i=1,2,...converges weakly.}

(ii)For each strictly increasing sequence of natural numbers{ki}i=1,2,...,

n−1

i<nT

ki _(2.1)

converges strongly.

(iii)For any uniformly regular matrix(an,i), the sequence{An(T)}n=1,2,...,

An(T)= i an

,iTi, (2.2)

converges strongly.

Proof ofTheorem 2.1. We first prove the following lemma.

Lemma2.2. Let there exist a uniformly regular matrix(an,i)such that for each strictly

in-creasing sequence{ki}i=1,2,...of natural numbers,

Bn=

i an

,iTki (2.3)

converges strongly. Then the sequence{Ti_}

i=1,2,...converges weakly.

Proof. Let (an,i) be a matrix with the aforementioned properties. Then the limitBnis not dependant upon the choice of the sequence{ki}i=1,2,.... In fact, let{ki}i=1,2,...and{li}i=1,2,... be the sequences for which the limitsBnare different. This means that for somex∈M∗,

i an

,iTkix−→x1,

i an

forn→ ∞. For a matrix (an,i), we build increasing sequences{ij}j=1,2,...and{nj}j=1,2,..., such that

lim j→∞

i<ij−1

_an j,i+

i>ij _an

j,i

=0. (2.5)

Let

mi=ki fori∈i2j−1,i2j

, mi=li fori∈i2j,i2j+1

, j=1, 2,... . (2.6)

Then

lim j

i an2j+1

,iTmix−x1

=0; lim

j

i an2j

,iTmix−x2

=0, (2.7)

which contradict (2.3), and thereforex1=x2. Let nowy∈Msuch that

Tn_x−x

1,y

−→0, (2.8)

whenn→ ∞. We choose a subsequence{ki}such that

Tkix₋x 1,y

−→γ=0, (2.9)

whereγis a real number. Then, from the uniform regularity of the matrix (an,i), it follows

that

lim n

i an

,iTkix−x1,y

=γ, (2.10)

which contradicts the choice of the matrix (an,i).

The implication (iii)⇒(ii) is trivial because the matrix (an,i),

an,i=1

n

i<nδj

,ki, (2.11)

is uniformly regular. ApplyingLemma 2.2to the matrix

an,i=_n1, (2.12)

i≤nandan,i=0 fori > n, we get the implication (ii)⇒(i).

To prove the implication (i)⇒(iii), we would need the following lemma.

Lemma2.3. LetQbe a contraction in the Hilbert spaceH. Then the weak convergence of

Qn_{xinH, wherex∈H, implies the strong convergence of}

i an

,iQix (2.13)

p

Proof. If the weak limitQnxexists and is equal tox1, then

Qx1=Q lim

n→∞Q

n_x=x

1, (2.14)

where the limit is considered in the weak topology, that is,x1isQ-invariant. Replacingx

onx−x1(if necessary), we may suppose thatQnxconverges weakly to0, and hence

Qn_{x,x−→0. (2.15)}

We are going to show that

n ai,nQ n_x ·

−−→0, (2.16)

where (ai,n) is uniformly regular matrix. One can see that

i aN

,iQix

2

≤

i

j aN

,iaN,j

Qi_x,Qj_x≤ i

j

_aN,iaN,j

Qi_x,Qj_{x. (2.17)}

We fixε >0. BecauseQis a contraction, the limitQn_{xdoes exist. Now, we can find} K >0, such that fork > Kand j≥0,

_Qk_x−Qk+j_x≤ε2_{, Q}k_{x,x≤ε. (2.18)}

Then,

_Qk_x,x−Qk+j_x,Qj_x

=Qk_x,x−Q∗j_Qk+j_x,x

≤Qk_x−Q∗j_Qk+j_{x· x =Q}k_x−Q∗j_Qk+j21/2_{· x}

=Qk_x2

−2Qk+j_x2_+Q∗j_Qk+j_x21/2_{· x}

≤Qk_x2

−Qk+j_x2_{· x ≤ε· x,}

(2.19)

and therefore

_Qk+j_x,Qj_{x≤ε·1 +x (2.20)}

for allk > Kand j≥0, or for|i−j| ≥k, the inequality

_Qi_x,Qj_{x≤ε·1 +x (2.21)}

is valid. We will fixη >0, and letNbe a natural number such that

max

forn≥N. Then the expression (1) forn≥Ncould be estimated in the following way:

i

j

_aN,iaN,j

Qi_x,Qj_x

=

|i−j|≤k

_an,ian,j

Qi_x,Qj_x+

|i−j|>k

_an,ian,j

Qi_x,Qj_x

≤

i

_{an,i·η· x}2_{·(2k−1) +}

i

j

_{an,ian,j·ε·}

1 +x

≤C·η· x2_{·(2k−1) +C}2_{·ε·1 +x.}

(2.23)

From the arbitrarity of the values of εandη, it follows that the strong convergence is

present and the lemma is proven.

We prove the implication (i)⇒(iii). Letx∈M∗+ and the sequence{Tix}i=1,2,... con-verges weakly. Without the loss of generality, we can considerx ≤1, and let

x=lim n→∞T

n_{x, (2.24)}

where the limit is understood in the weak sense. We consider

y=

∞

n=0

2−nTnx. (2.25)

The series that definesyis convergent in the norm of the spaceM∗. From the positivity

ofxand the properties of the operatorT, it follows that

T y≤2y, (2.26)

and, therefore, for allk=1, 2,...,

sTk_{y≤s(y), (2.27)}

where we denote bys(z) the support of the normal functionalz.

Lemma2.4. Letu∈M∗+ands(u)≤s(y). Thens(u)≤s(x), where

u=lim n→∞T

n_{u. (2.28)}

Proof. In fact, We fixε >0. From the density of the set

Ly=w∈M∗+,w≤λy, for someλ >0 (2.29)

in the set

S=w∈M∗+,s(w)≤s(y)

(2.30)

in the norm of the spaceM∗, it follows that there areλ >0 andw∈Lysuch that

p

Let

w=lim

n→∞T

n_{w. (2.32)}

Then

w1−s(x)=lim n→∞

Tn_(w)(1−s(x)

≤λ·lim n→∞

Tn_y(1−s(x)

≤λ·_nlim

→∞

_∞

k=0

2−k·Tn+kx1−s(x)

=λ·∞ k=0

2−klim n→∞

Tn+k_x1−s(x)=0.

(2.33)

Because the operatorT does not increase the norm of the functionals fromM∗, we get

that

u1−s(x)=lim n→∞

Tn_{u1−s(x)≤lim} n→∞

Tn_{w1−s(x)+ lim} n→∞T

n_{(w−u)≤ε.}

(2.34)

The needed inequality follows from the arbitrarity ofε.

We introduce the following notion. Forµ∈M∗, we will denote byµ·E, whereEis a

projection from the algebraM, the functional

(µ·E)(A)=µ(EAE), (2.35)

whereA∈M.

We fixε >0. We will find a numberN, such that

Tn_{x1−s(x)< ε}2 _(2.36)

forn > N. Then,

_TN_x·s(x)−TN_x

= sup

A∈M

A∞≤1

_TN_{x(1−s(x)A(1−s(x)}

+TNxs(x)A1−s(x)+TNx1−s(x)As(x) ≤ε·ε+ 2x1/2_,

(2.37)

because

_µ(AB)2

≤µA∗_A·µB∗_{B, (2.38)}

Letw∈Lybe such that

w≤λx (2.39)

for someλ >0 and

_TN_{x·s(x)−w≤ε. (2.40)}

Then, forn > N, the following is valid:

_Tn_x−Tn−N_w≤Tn−N_TN_x−TN_x·s(x)+Tn−N_TN_{x·s(x)−w≤4·ε.} (2.41)

By taking the weak limit in the inequality (2.37) and because the unit ball ofM∗ is

closed weakly, we will get

x−w ≤4·ε, (2.42)

where

w=lim

n→∞T

n_{w. (2.43)}

We now consider the algebraMs(x). The functionalxis faithful on the algebraMs(x). We

will consider the representationπxof the algebraMs(x)constructed using the functional

x[7]. Because the functionalxis faithful, we can conclude that the representationπx is faithful on the algebraMs(x), and thereforeπxis an isomorphism of the algebraMs(x)and

some algebraA. The algebraAis a von Neumann algebra, and its preconjugate spaceA∗

is isomorphic to the spaceM∗·s(x) ([19]). We note now that

TM∗·s(x)⊂M∗·s(x). (2.44)

In fact,

TLy⊂Ly, (2.45)

and therefore, by taking the norm closure, we get

TS⊂S; (2.46)

by taking now the linear span, we get

TM∗·s(x)⊂M∗·s(x). (2.47)

We denote byTthe isomorphic image of the operatorT, acting on the spaceA∗. Let

p

for someλ >0. Then there exists the operatorB∈A, where A is a commutant ofA, such that

(ABΩ,Ω)=u(A) (2.49)

for allA∈A. Note, that fromLemma 2.3,

Tu(A)=uT∗A=T∗ABΩ,Ω=AT∗BΩ,Ω. (2.50)

Also, from

TA∗+⊂A∗+, Tu ≤ u, T x=x, (2.51)

it follows that

T∗A+;

T∗1≤1, T∗A_∞≤A_∞, (2.52)

for allA∈A. Based on the lemma, we now conclude that _T∗_B

∞≤ B∞; T∗A+⊂A+; T ∗

1≤1, (2.53)

for allB∈A_.

The spaceAsais a pre-Hilbert space of the selfadjoint operators fromAwith the scalar

product

(B,C)x=(CBΩ,Ω), (2.54)

and using the Kadison inequality [5], we have

T∗BT∗BΩ,Ω≤T∗B2Ω,Ω≤(BΩ,BΩ), (2.55)

that is, the operatorT∗is a contraction in the pre-Hilbert space (Asa, (·,·)x).

We will identifyM∗·s(x) andA∗. Becausew∈L, that is,

w≤λx (2.56)

for someλ >0, then

w≤λx (2.57)

as well. Let

w(A)=(BAΩ,Ω), w(A)=BAΩ,Ω, (2.58)

Let now (an,i) be a uniformly regular matrix. UsingLemma 2.3, we will findk∈Nso that i a

k,iTiw−w

= sup

A∈A

A∞=1

_∞ i=1 a

k,iT∗iB−BAΩ,Ω ≤ _∞ i=1 a

k,i

T∗iB−BΩ,

∞

i=1

a

k,i

T∗iB−BΩ

1/2

· sup A∈A

A∞≤1

(AΩ,AΩ)1/2

≤x(1)1/2·

∞

i=1

a

k,i

T∗i(B−B)

(·,·)x

< ε

(2.59)

fork > K, where by (a_n,i), we will denote a matrix with the elements

a

n,i=

i>Nan

,j −1

an,j+N. (2.60)

It is easy to see that the matrix (a

n,i) will be uniformly regular as well. Then, for a big enoughk > K, we will have

i ak

,iTix−x

≤

i≤N

_ak,iTi_x−x+ i>N

_ak,iTi_x−Ti−N_w

+

i>N _ak,i

1−

i>Nak

,i

−1

Ti−Nw

+

∞

j=1

ak,j+N·

i>Nak

,i

−1

Tj_w−w

+

i≤Nak

,i

·w+ i>Nak

,i w−x

≤

i≤N

2·_Nε + i>N

_ak,i·4ε+ i>N

_{ak,i1−(1 +ε)}−1_·2

+

i≤N

2·_Nε + (1 +ε)·4ε

≤2ε+ (1 +ε)·4ε+ε·2·(1 +ε) +ε+ 2ε+ (1 +ε)·4ε≤25ε.

(2.61)

The arbitrarity ofεproves the needed statement. The proof of the theorem is now

completed.

2.2. The case ofL1-spaces forJBWalgebras. TheL1-spaces for semifinite JBW algebras

p

with predual spaces. A semifinite JBW algebraAis always represented as

A=AspAex, (2.62)

whereAsp is isometrically isomorphic to operator JW algebra, andAex is isometrically

isomorphic to the spaceC(X,M38) of all continuous mappings from a Hyperstonean

com-pact topological spaceXonto the exceptional Jordan algebraM8

3 (see [11]). In this case,

whenAdoes not have direct summands of typeI2, it is going to be a selfadjoint part of a

real von Neumann algebraR(Asp), whose complexification

RAsp

iRAsp

=M, (2.63)

whereMis the enveloping von Neumann algebra ofAsp, and the predual space ofA, and

the space

A∗=Asp

∗

Aex

∗, (2.64)

where (Asp)∗ is the predual space ofAsp, and (Aex)∗ is the predual space of Aex (see,

e.g., [2,11]). The main result for the summandAexfollows immediately from the result

forC(X), and the fact that the algebraM38is finite dimentional. So, without the loss of

generality, we are interested in the operator case only. But in the operator case, the space (Asp)∗is a selfadjoint part ofR∗=(R(Asp))∗, and

M∗=R∗iR∗ (2.65)

(see [2,16] for details). So, the main result forR∗thus follows from the complex case by

restriction of scalars, and we obtain the main result forL1-spaces affiliated to semifinite

JBW algebras without direct typeI2summand.

3. Main result: the case of quantumLp-spaces(1< p <∞)

In the case of a noncommutativeLp-space for a semifinite von Neumann algebra, the

main result is discussed in [25].

We will discuss here the nonassociative case.

In this section,Adenotes a semifinite JBW algebra without direct summands of type I2, with a faithful normal traceτ. ByLp, we denote the space of operators affiliated toA,

and integrated withpth power (p >1, see, e.g., [1,2,12]). SpaceLq(hereq=p/(p−1)) is a dual as Banach space toLp(see [1,12]). The following theorem is valid.

Theorem3.1. The following conditions for a positive contractionTin theLpare equivalent.

(i)The sequence{Ti_{x}i=1,2,...converges inσ(Lp,Lq)topology forx∈Lp.} (ii)For each strictly increasing sequence of natural numbers{ki}i=1,2,...,

n−1

i<nT

kix _(3.1)

(iii)For any uniformly regular matrix(an,i), the sequence{An(T)x}n=1,2,...,

An(T)x=

i an

,iTix, (3.2)

converges in norm ofLpfor allx∈Lp.

For the sake of completeness, we give the following definitions (see, e.g., [25]) and sketch of the proof. Letφbe a gauge function

φ:R+_−→R+_{, (3.3)}

with

φ(0)=0, lim

t→∞φ(t)= ∞. (3.4)

Hahn-Banach theorem implies for strictly convex Banach spacesEwith conjugateEthat there exists a duality map

Φ:E−→E_{, (3.5)}

associated withφsuch that

x,Φ(x)= xΦ(x), Φ(x)=φ(x). (3.6)

Definition 3.2. MapΦis said to satisfy property (S) uniformly if for every>0, there existsδ()>0, such that for anyx,y∈E,

_{x,Φ(y)< δ() (3.7)}

implies that

_{y,Φ(x)<. (3.8)}

Proof. From [12, Section 4], it follows that the duality map defined as

Φ(a)=s|a|p−1_{, (3.9)}

for

a=s|a| ∈A (3.10)

(where a=s|a|is a polar decomposition of elementa) satisfies the property (S)

uni-formly. Hence, the statement of the theorem follows from [25, Theorem 3.1].

Acknowledgments

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Genady Ya. Grabarnik: IBM T. J. Watson Research Center, 19 Skyline Drive, Hawthorne, NY 10532, USA

E-mail address:[email protected]

Alexander A. Katz: Department of Mathematics and Computer Science, St. John’s College of Lib-eral Arts and Sciences, St. John’s University, 300 Howard Avenue, DaSilva Hall 314, Staten Island, NY 10301, USA

E-mail address:[email protected]

Laura Shwartz: Department of Mathematical Sciences, College of Science, Engineering, and Tech-nology, University of South Africa (UNISA), P.O. Box 392, Pretoria 0003, South Africa