Ergodic Theorems for Noncommutative Lorentz Spaces∗

Arxen T Yerkex,Turdebek N Bekjan

(College of Mathematics and System Sciences,Xinjiang University,Urumqi,Xinjiang 830046,China)

0 Introduction

First,werecalltheclassicalLorentzspaces.Givenameasurespace(X,Σ,ν),let0

wheref∗(t)isthedecreasingrearrangementoff.TheclassicalLorentzspacesLp,q(X)isthesetallmeasurablefunctionsfon function(X,Σ,ν)with kfkLp,q(X)<∞.We refer to[1–4]for more information aboutLp,q(X).

Using the operator space theory and theory of interpolation of Banach spaces,Junge and Xu[5]proved a noncommutative analogue of the classical Dunford-Schwartz maximal ergodic inequality for positive contraction onLp(M),and the analogue of Stein’s maximal inequality for symmetric positive contractions onLp(M),1

The purpose of this article is to present the individual ergodic theorems in the noncommutative Lorentz spaceLp,q(M).

The remainder of this article is organized as follows.Section 2 contains some preliminaries and notations on the noncommutativeLp-spaces and noncommutative Lorentz spaces,including the noncommutative Lorentz spaces and their elementary properties.In section 3 we give some theorems,de finitions about uniform equicontinuity of sequences of operators.Finally,using the techniques in[7]we prove the individual ergodic theorems in noncommutative Lorentz spaceLp,q(M).

1 Preliminaries

We use standard notation and notions from theory of noncommutativeLpspaces.Our main references are[8–10](see[9]for more historical references).Let M be a finite von Neumann algebra on a Hilbert space H with a normal finite faithful trace τ and denote the lattice of(orthogonal)projections in M by P(M).The closed densely de fined linear operatorxin H with domainD(x)is said to be affiliated with M if and only ifu∗xu=xfor all unitary operatorsuwhich belong to the commutant M0of M.Ifxis affiliated with M,thenxis said to be τ-measurable if for every ε >0 there exists a projectione∈P(M)such thate(H)⊆D(x)and τ(e⊥)< ε(where for any projectionewe lete⊥=1−e).The set of all τ-measurable operators will be denoted byL0(M;τ)or simply byL0(M).The setL0(M)is a ∗-algebra with sum and product to be the respective closure of the algebraic sum and product.

The measure topologytτinL0(M)is given by the system

ε>0,δ>0,of neighborhoods of zero.Note that if one replaces the condition kxek∞≤ δ above by the generally weaker condition kexek∞≤ δ,then the corresponding family of neighborhoods of zero generates the same topologytτ(see[11]).

The trace τ can be extended to the positive coneL+0(M)ofL0(M):

Given 0

whereThen(M,k·kp)is a normed(or quasi-normed forp<1)space,whose completion is the noncommutativeLp-space associated with(M,τ),satisfying all the expected properties such as duality(see[9,10]),denoted byLp(M,τ)or simply byLp(M).As usual,we setL∞(M,τ)=M and denote by k·k∞(=k·k)the usual operator norm.

Letxbe a τ-measurable operator andt>0.The ”t-th singular number(or generalizeds-number)”ofxµt(x)is de fined by

It is clear that,ifxis a τ-measurable operator,then µt(x)< ∞ for everyt>0.See[8]for more information about generalized s number.

We say that a sequence{xn}⊂L0(M)converges in measure tox∈L0(M),xn→x(m),if,given ε>0,δ>0,there is a numberN=N(ε,δ)such that for anyn≥Nthere exists a projectionen∈P(M)satisfying the conditions τ(e⊥n)< ε and k(xn−x)enk<δ.

Accordingly,we also say that a sequence{xn}⊂L0(M)converges tox∈L0(M)almost uniformly(a.u.)(bilaterally almost uniformly(b.a.u.))if for every ε>0 there exists such ane∈P(M)that τ(e⊥) ≤ ε and k(x−xn)ek∞→ 0(respectively,ke(x−xn)ek∞→0).By Theorem 2.3 and Remark 2.4 of[11],we find that the algebraL0(M)is complete with respect to botha.u.andb.a.u.convergences.

De finition 1Letxbe a τ-measurable operator affiliated with a semi- finite von Neumann algebra M,and 0

The set of allx∈L0(M)with kxkLp,q(M)<∞is denoted byLp,q(M)and is called the noncommutative Lorentz space with indicespandq.

It is easy to check that(Lp,q(M),k·kLp,q(M))is a quasi-Banach space.Moreover,ifp>1,q≥ 1,and equipped with the equivalent norm that

Remark 1(i)Ifp=q,thenLp,p(M)=Lp(M).(ii)Let 0

is dense inLp,q(M)(cf.[4]).Hence M is also dense inLp,q(M).

A positive linear map α :L1(M)→L1(M)will be called an absolute contraction if α(x)≤ 1 and τ(α(x))≤ τ(x)for everyx∈M with 0≤x≤1.Note that,as it is shown in[6],kα(x)kp≤kxkpfor eachx=x∗∈S and all 1≤p≤∞.Besides,there exist unique continuous extensions α:Lp(M)→Lp(M)for every 1≤p≤∞.Its restriction α:M→M is ultra-weakly continuous.Note that since α is positive,it is hermitian i.e.α(x)∗=α(x∗)for allx∈L1(M).

Lemma 1Let 1≤p<∞and 1≤q<∞.If α is the above absolute contraction,then

Proof Letx=x∗∈M,by the formula on page 289 of[8],we have that

Hence

From this lemma follows that if 1≤p<∞,1≤q<∞,then α has unique continuous extension α :Lp,q(M)→Lp,q(M).Forx∈Lp,q(M),we denote

LetXbe any set.Ifai:X→L0(M),i∈I,ande∈P(M)are such that{ai(x)e}i∈I⊂ M(respectively,{eai(x)e}i∈I}⊂M),we denote

De finition 2Let(X,k·k)be a normed space andX0a subset ofXsuch that 0 ∈X0,where 0 is the neutral element ofX.A familyai:X→L0(M),i∈I,of additive maps is calleduniformly equicontinuous in measure(u.e.m.)(bilaterally uniformly equicontinuous in measure(b.u.e.m.))at 0 onX0,if,given ε>0,δ>0,there is such a γ >0 that for everyx∈X0with kxk<γ there existse∈P(M)for which

2 Ergodic theorems for noncommutative Lorentz spaces

Lemma 2Let 1

ProofLet us consider the bilateral case.The “only if”part is obvious.To prove the“if”part, fix ε>0 and δ>0.Then there is such a γ >0 thatx∈Lp,q+(M)and kxkLp,q(M)< γ imply thatfor some

Letx∈Lp,q(M)with kxkLp,q(M)<γ.Then we can writex=x1−x2+i(x3−x4)such thatandSinceandthere existsej∈P(M)such thatand

Thus,the assertion holds.

Proposition 1If 1

ProofDue to Lemma 2,it is sufficient to show that{an}isb.u.e.m.at 0 onSince M+is dense inby Theorem 3.1 of[7],we only show that{an}b.u.e.m.at 0 on

Fix ε >0 and δ>0,there exists such at=t(p,q,δ)thatt·λr≥ λ wheneverwherer=min{p,q}.Let ν>0 and γ>0 be such thatwhereCrsatis fies

Letx∈M+satisfy kxkLp,q(M)≤γ,then

andThen we derive

Therefore,{an}isb.u.e.m.at 0 on(M+,k·kLp,q(M)).

Theorem 1Let 1

Proof By Proposition 1,we know that{an}isb.u.e.m.at 0 on(Lp,q(M),k·kLp,q(M)),and by Theorem 4.2 of[7],for eachx∈M,the averages(1)convergeb.a.u..Applying Theorem 2.1 of[7],we have for eachx∈Lp,q(M)the averages(1)convergeb.a.u..Letis the b.a.u.limit of averages(1),then by Theorem 2.3 of[11],we know that∈L0(M).It implies that the averages(1)converge to∈L0(M)in measure,for eachx∈Lp,q(M).Similarly to the proof of Lemma 1,for eachn∈N+,we can prove that

Applying Lemma 2.5 of[8]and Fatou lemma,we obtain

Therefore∈Lp,q(M).

Proposition 2Let 2

ProofBy Theorem 3.2 of[7]and Lemma 2,it is enough to show that{an}isu.e.m.at 0 on(M+,k·kLp,q(M)).

Fix ε >0 and δ>0.Since,due to Proposition 1,it is sufficient to show that{an}isb.u.e.m.at 0 onthere exists such a γ >0 thatimply thatfor somee∈P(M)with

Therefore,there ise∈P(M)such that τ(e⊥)≤ ε and(x2,e)≤δ2,applying Kadison’s inequality again,we arrive at

Thus,a∗(x,e)≤δ.

Theorem 2If 2≤p<∞,2≤q<∞,then for everyx∈Lp,q(M),the averages(1)converge a.u.to some∈Lp,q(M).

ProofBy Corollary 4.1 of[7],the sequence{an(x)}convergesa.u.for everyx∈M.Since the set M is dense inLp,q(M),applying Theorem 2.1 of[7]and Proposition 2,we infer that the averagesanconvergea.u.for all∈Lp,q(M).The fact that the corresponding limitsbelong toLp,q(M)follows as in Theorem 1.

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