INDIVIDUAL ERGODIC THEOREMS IN NONCOMMUTATIVE ORLICZ SPACES 6 1 VLADIMIRCHILIN, SEMYONLITVINOV 0 2 n Abstract. For a noncommutative Orlicz space associated with a semifinite a vonNeumannalgebra,afaithfulnormalsemifinitetraceandanOrliczfunction J satisfying(δ2,∆2)−condition, anindividualergodictheoremisproved. 1 3 ] A 1. Introduction O Development of the theory of noncommutative integration with respect to a . faithful normal semifinite trace τ defined on a semifinite von Neumann algebra h , has given rise to a systematic study of various classes of noncommutative re- t M a arrangement invariant Banach spaces. The noncommutative Lp spaces Lp( ,τ) m − M [20,22,19]and,moregenerally,noncommutativeOrliczspacesLΦ( ,τ)[16,17,13] M [ are important examples of such spaces. Since every LΦ( ,τ) is an exact interpolation space for the Banach couple 1 M v (L1( ,τ), ), for any linear operator T :L1( ,τ)+ L1( ,τ)+ such M M M M→ M M 1 that 8 T(x) x x and T(x) x x L1( ,τ) 2 ∞ ∞ 1 1 k k ≤k k ∀ ∈M k k ≤k k ∀ ∈ M 0 (such operators are called Dunford-Schwartz operators),we have 0 2. T(LΦ)⊂LΦ and kTkLΦ→LΦ ≤1. 0 Thus, it is natural to study noncommutative Dunford-Shwartz ergodic theorem 6 in LΦ( ,τ). The first result in this direction was obtained in [23] for the space 1 M L1( ,τ) (as it is noticed in [3, Proposition 1.1], the class of operators α that was : v M employed in [23] coincides with the class of positive Dunford-Schwartz operators). i X In [10], the result of [23] was extended to the noncommutative Lp spaces with − r 1 < p < . For general noncommutative fully symmetric spaces with non trivial a Boyd ind∞exes, an individual ergodic theorem was established in [3]. Note that the class of Orlicz spaces LΦ( ,τ) is significantly wider than the M class of spaces Lp( ,τ). Besides, there are Orlicz spaces LΦ( ,τ), with the M M Orliczfunctionsatisfyingthe so-called(δ ,∆ ) condition,whichhavetrivialBoyd 2 2 − index pLΦ = 1 (see Remark 2.3 below). Therefore an individual ergodic theorem for positive Dunford-Schwartz operators in Orlicz spaces does not follow from the results mentioned above. Theaimofthisarticleistoestablishanindividualergodictheoremforapositive Dunford-SchwartzoperatorinanoncommutativeOrliczspaceLΦ( ,τ)associated M Date:January31,2016. 2010 Mathematics Subject Classification. 47A35(primary),46L52(secondary). Key words and phrases. Noncommutative Orlicz space, Dunford-Schwartz operator, uniform equicontinuityatzero,individualergodictheorem. 1 2 VLADIMIRCHILIN, SEMYONLITVINOV withanOrliczfunctionΦsatisfying(δ ,∆ ) condition. Ourargumentisessentially 2 2 − based on the notion of uniform equicontinuity in measure at zero of a sequence of linear maps from a normed space into the space of measurable operators affiliated with ( ,τ). This notion was introduced in [2] and then applied in [15] to pro- M vide a simplified proof of noncommutative individual ergodic theorem for positive Dunford-Schwartz operators in Lp( ,τ), 1<p< . M ∞ 2. Preliminaries Assume that is a semifinite von Neumann algebra with a faithful normal M semifinite trace τ, and let ( ) be the complete lattice of projections in . If P M M 1 is the multiplicative identity of and e ( ), we denote e⊥ = 1 e. Let M ∈ P M − L0 = L0( ,τ) be the -algebra of τ-measurable operators. Recall that L0 is a M ∗ metrizable topological -algebra with respect to the measure topology that can be ∗ equivalently (see [1, Theorem 2.2]) defined by either of the families V(ǫ,δ)= x L0 : xe δ for some e ( ) with τ(e⊥) ǫ ∞ { ∈ k k ≤ ∈P M ≤ } or W(ǫ,δ)= x L0 : exe δ for some e ( ) with τ(e⊥) ǫ , ∞ { ∈ k k ≤ ∈P M ≤ } ǫ>0, δ >0, of neighborhoods of zero [18]. For a positive operator x= ∞λde L0 one can define 0 λ ∈ R n ∞ τ(x)=supτ λde = λdτ(e ). λ λ n (cid:18)Z0 (cid:19) Z0 If1 p< ,thenthenoncommutativeLp spaceassociatedwith( ,τ)isdefined ≤ ∞ − M as Lp =(Lp( ,τ), )= x L0 : x =(τ(xp))1/p < , p p M k·k { ∈ k k | | ∞} where x =(x∗x)1/2 is the absolute value of x; naturally, L∞ = . | | M For detailed accounts on noncommutative Lp-spaces, see [19, 22]. Given x L0, let e be the spectral family of projections of x. If t > 0, λ λ≥0 ∈ { } | | the t-th generalized singular number of x [9] is defined as µ (x)=inf λ>0:τ(e⊥) t . t { λ ≤ } A Banach space (E, ) L0 is called fully symmetric if the conditions E k·k ⊂ s s x E, y L0, µ (y)dt µ (x)dt s>0 t t ∈ ∈ Z ≤Z ∀ 0 0 imply that y E and y x . E E ∈ k k ≤k k If L L0, the set of all positive operators in L will be denoted by L . + ⊂ A fully symmetric space (E, ) is said to possess Fatou property if the con- E k·k ditions x E , x x for α β, and sup x < α + α β α E ∈ ≤ ≤ k k ∞ α imply that there exists x=supx E and x =sup x . α E α E ∈ k k k k α α Let m be Lebesgue measure on the interval (0, ), and let L0(0, ) be the ∞ ∞ linear space of all (equivalence classes of) almost everywherefinite complex-valued m measurable functions on (0, ). We identify L∞(0, ) with the commutative − ∞ ∞ von Neumann algebra acting on the Hilbert space L2(0, ) via multiplication by ∞ INDIVIDUAL ERGODIC THEOREMS IN NONCOMMUTATIVE ORLICZ SPACES 3 theelementsfromL∞(0, )withthetracegivenbytheintegrationwithrespectto ∞ Lebesgue measure. A fully symmetric space E L0( ,τ), where =L∞(0, ) ⊂ M M ∞ andτ is givenby the Lebesgue integral,is calledfully symmetric function space on (0, ). ∞ Let E = (E(0, ), ) be a fully symmetric function space. For each s > 0 E ∞ k·k let D :E(0, ) E(0, ) be the bounded linear operator given by s ∞ → ∞ D (f)(t)=f(t/s), t>0. s The Boyd indices p and q are defined as E E logs logs p = lim , q = lim . E E s→∞log Ds E s→+0log Ds E k k k k It is known that 1 p q [14, II, Ch.2, Proposition 2.b.2]. A fully E E ≤ ≤ ≤ ∞ symmetric function space is said to have non-trivial Boyd indices if 1 < p and E q < . For example, the spaces Lp(0, ), 1 < p < , have non-trivial Boyd E ∞ ∞ ∞ indices: p =q =p Lp(0,∞) Lp(0,∞) [14, II, Ch.2, 2.b.1]. If E is a fully symmetric function space on (0, ), define ∞ E( )=E( ,τ)= x L0( ,τ): µ (x) E t M M { ∈ M ∈ } and set x = µ (x) , x E( ). E(M) t E k k k k ∈ M It is shown in [4] that (E( ), ) is a fully symmetric space. E(M) M k·k If 1 p< and E =Lp(0, ), the space (E( ), )coincides with the E(M) ≤ ∞ ∞ M k·k noncommutative Lp-space (Lp( ,τ), ) because p M k·k ∞ 1/p kxkp =Z µpt(x)dt =kxkLp(M,τ). 0  [22, Proposition 2.4]. Since for a fully symmetric function space E on (0, ), ∞ L1(0, ) L∞(0, ) E L1(0, )+L∞(0, ) ∞ ∩ ∞ ⊂ ⊂ ∞ ∞ with continuous embeddings [12, Ch.II, 4, Theorem 4.1], we also have § L1( ,τ) E( ,τ) L1( ,τ)+ , M ∩M⊂ M ⊂ M M with continuous embeddings. Definition 2.1. A convex continuous at 0 function Φ :[0, ) [0, ) such that ∞ → ∞ Φ(0)=0 and Φ(u)>0 if u=0 is called an Orlicz function. 6 Remark 2.1. (1) Since an Orlicz function is convex and continuous at 0, it is necessarily continuous on [0, ). ∞ (2) If Φ is an Orlicz function, then Φ(λu) λΦ(u) for all λ [0,1]. Therefore Φ is ≤ ∈ increasing, that is, Φ(u )<Φ(u ) whenever 0 u <u . 1 2 1 2 ≤ We will need the following lemma. 4 VLADIMIRCHILIN, SEMYONLITVINOV Lemma 2.1. Let Φ be an Orlicz function. Then for any given δ > 0 there exists t>0 satisfying the condition t Φ(u) u whenever u δ. · ≥ ≥ In particular, lim Φ(u)= . u→∞ ∞ Proof. Since Φ(u)>0 as u>0, it is possible to find a>0 such that the equation Φ(u)=au has a solutionu=u >0. Then, as Φ is convex,we haveΦ(u) au for 0 ≥ all u u . 0 ≥ Fix δ >0. If δ u , then we have 0 ≥ 1 Φ(u) u u δ. a · ≥ ∀ ≥ If δ < u , then, since Φ(δ) > 0 and Φ is increasing on the interval [δ,u ], there 0 0 exists such s>1 that s Φ(u) au, or · ≥ s Φ(u) u, u δ. a · ≥ ∀ ≥ (cid:3) Remark 2.2. Since an Orlicz function Φ is continuous, increasing and such that lim Φ(u) = , there exists continuous increasing inverse function Φ−1 from u→∞ ∞ [0, ) onto [0, ). ∞ ∞ If Φ is an Orlicz function, x L0 and x= ∞λde its spectral decomposition, ∞ ∈ + 0 λ one can define Φ(x) = Φ(λ)de . The noncRommutative Orlicz space associated 0 λ with ( ,τ) for an OrliRcz function Φ is the set M x LΦ =LΦ( ,τ)= x L0( ,τ): τ Φ | | < for some a>0 . M (cid:26) ∈ M (cid:18) (cid:18) a (cid:19)(cid:19) ∞ (cid:27) The Luxemburg norm of an operator x LΦ is defined as ∈ x x =inf a>0:τ Φ | | 1 . Φ k k (cid:26) (cid:18) (cid:18) a (cid:19)(cid:19)≤ (cid:27) Theorem 2.1. [13, Proposition 2.5]. (LΦ, ) is a Banach space. Φ k·k Proposition 2.1. If x LΦ, then Φ(x) L0 and µ (Φ(x))=Φ(µ (x)), t>0. t t ∈ ∞ | | ∈ | | In addition, τ(Φ(x))= Φ(µ (x))dt. | | 0 t R Proof. As x LΦ, we have τ Φ |x| < for some a > 0. This implies that ∈ (cid:16) (cid:16) a (cid:17)(cid:17) ∞ Φ |x| L1, so τ Φ |x| >λ < for all λ>0. Since (cid:16) a (cid:17)∈ (cid:16)n (cid:16) a (cid:17) o(cid:17) ∞ x x Φ | | >λ = Φ−1 Φ | | >Φ−1(λ) = x >aΦ−1(λ) , (cid:26) (cid:18) a (cid:19) (cid:27) (cid:26) (cid:18) (cid:18) a (cid:19)(cid:19) (cid:27) {| | } it follows that τ( Φ(x)>µ ) = τ x >Φ−1(µ) < for all µ > 0, thus { | | } {| | } ∞ Φ(x) L0. (cid:0) (cid:1) | | ∈ By [9, Lemma 2.5, Corollary 2.8], given x L0, we have µ (ϕ(x)) =ϕ(µ (x)), t t ∈ | | t > 0, for every continuous increasing function ϕ : [0, ) [0, ) with ϕ(0) = 0 ∞ ∞ → ∞ and, in addition, τ(ϕ(x)) = ϕ(µ (x))dt. Therefore µ (Φ(x)) =Φ(µ (x)) and τ(Φ(x))= ∞Φ(µ (x|))d|t. R0 t t | | t (cid:3) | | 0 t R Next result follows immediately from Proposition 2.1. INDIVIDUAL ERGODIC THEOREMS IN NONCOMMUTATIVE ORLICZ SPACES 5 Corollary 2.1. LΦ = x L0 : µ (x) LΦ(0, ) and x = µ (x) for all t Φ t Φ { ∈ ∈ ∞ } k k k k x LΦ. ∈ If (LΦ(0, ), ) is the Orlicz function space on (0, ) for an Orlicz function Φ ∞ k·k ∞ Φ, then, by [8, Ch.2, Proposition 2.1.12], it is a rearrangement invariant function space. Since (LΦ(0, ), ) has the Fatou property [8, Ch.2, Theorem 2.1.11], Φ ∞ k·k Corollary 2.1, [6, Theorem 4.1], and [4, Theorem 3.4] yield the following. Corollary 2.2. (LΦ, ) is a fully symmetric space with the Fatou property and Φ k·k an exact interpolation space for the Banach couple (L1, ). M We will also need the following property of the Luxemburg norm. Proposition 2.2. If x L and x 1, then τ(Φ(x) x . Φ Φ Φ ∈ k k ≤ | | ≤k k Proof. By[8,Ch.2,Proposition2.1.10], ∞Φ(f )dt f forf LΦ(0, )with f Φ 1. Thus the result follows fromRP0ropsi|tio|n 2.≤1 akndkΦCorolla∈ry 2.1. ∞ (cid:3) k k ≤ Definition2.2. AnOrliczfunctionΦissaidtosatisfy∆ condition(δ condition) 2 2 − − if there exist k>0 and u 0 such that 0 ≥ Φ(2u) kΦ(u) u u (respectively, Φ(2u) kΦ(u) u (0,u ]). 0 0 ≤ ∀ ≥ ≤ ∀ ∈ If anOrliczfunction Φsatisfies ∆ conditionandδ conditionsimultaneously, 2 2 − − we will say that Φ satisfies (δ ,∆ ) condition. In this case Φ(2u) cΦ(u) for all 2 2 − ≤ u 0 and some c>0. Clearly, every space Lp, 1 p< , is the Orlicz space for th≥e function Φ(u)= up, u 0, which satisfies (δ ,≤∆ ) c∞ondition. p ≥ 2 2 − Remark 2.3. (i) If an Orlicz function Φ satisfies ∆ condition, then the Boyd 2 − index qLΦ(0,∞) <∞, that is, it is non-trivial (see [14, II, Ch.2, Proposition 2.b.5]). (ii) The function Φ (u) =u lnα(e+u), α 0, is an Orlicz function that satisfies α ≥ (δ2,∆2)−conditionforwhichtheBoydindex pLΦ(0,∞) istrivial,thatis,pLΦ(0,∞) = 1 [21, 5]. § ABanachspace(E, ) L0 issaidtohaveorder continuousnormif x E α E k·k ⊂ k k ↓ 0 for every net x E with x 0. α α { }⊂ ↓ Proposition 2.3. Let an Orlicz function Φ satisfy (δ ,∆ ) condition. Then 2 2 − (i) The fully symmetric space (LΦ, ) has order continuous norm. Φ k·k (ii) The linear subspace L1 is dense in (LΦ, ). Φ ∩M k·k Proof. (i) As shown in [8, Ch.2, 2.1], the fully symmetric space (LΦ(0, ), ) Φ § ∞ k·k hasordercontinuousnorm. Therefore,by[5,Proposition3.6],thenoncommutative fully symmetric space (LΦ, ) also has order continuous norm. Φ k·k (ii) Let x LΦ, n = 1,2,..., and e the spectral projection corresponding to ∈ + n the interval (n−1,n). It is clear that xe and e⊥ 0. Also, by (i) and [7, { n} ⊂ M n ↓ Theorem 3.1], we have x xe = xe⊥ 0 as n . k − nkΦ k nkΦ → →∞ Now, since τ( x > ǫ ) < for all ε > 0 (see proof of Proposition 2.1), it follows { } ∞ that xe L1. n { }⊂ Since, foranarbitraryx LΦ,wehavex=x x +i(x x ),wherex LΦ, i=1,...,4, the assertion fol∈lows. 1− 2 3− 4 i ∈ +(cid:3) 6 VLADIMIRCHILIN, SEMYONLITVINOV 3. Main Results Let be a semifinite von Neumann algebra with a faithful normal semifinite M trace τ, L0 = L0( ,τ) the -algebra of τ-measurable operators affiliated with M ∗ , Lp = Lp( ,τ), 1 p , the noncommutative Lp space associated with M M ≤ ≤ ∞ − ( ,τ). M Definition 3.1. Let (X, ) be a normed space, and let Y X be suchthat the k·k ⊂ neutralelementofX isanaccumulationpointofY. AfamilyofmapsA :X L0, α → α I, is called uniformly equicontinuous in measure (u.e.m) (bilaterally uniformly ∈ equicontinuous in measure (b.u.e.m)) at zero on Y if for every ǫ > 0 and δ > 0 there is γ > 0 such that, given x Y with x < γ, there exists e ( ) such ∈ k k ∈ P M that τ(e⊥) ǫ and sup A (x)e δ (respectively, sup eA (x)e δ). α ∞ α ∞ ≤ k k ≤ k k ≤ α∈I α∈I Remark 3.1. As explained in [15, Introduction], in the commutative case, the notion of uniform equicontinuity in measure at zero of a family An n∈N coincides { } withthecontinuityinmeasureatzeroofthemaximaloperatorassociatedwiththis family. Definition 3.2. A sequence x L0 is said to converge to x L0 almost n { } ⊂ ∈ uniformly(a.u.) (bilaterallyalmostuniformly(b.a.u.)) ifforeveryǫ>0thereexists such a projection e ( ) that τ(e⊥) ǫ and (x x )e 0 (respectively, n ∞ ∈ P M ≤ k − k → e(x x )e 0). n ∞ k − k → A proof of the following fact can be found in [15, Theorem 2.1]. Proposition 3.1. Let (X, ) be a Banach space, A : X L0 a sequence of n k·k → additive maps. If the family A is u.e.m. (b.u.e.m.) at zero on X, then the set n { } x X : A (x) converges a.u. (respectively, b.a.u.) n { ∈ { } } is closed in X. Definition 3.3. A linear map T :L1+L∞ L1+L∞ such that → T(x) x x and T(x) x x L1. ∞ ∞ 1 1 k k ≤k k ∀ ∈M k k ≤k k ∀ ∈ is called a Dunford-Schwartz operator. If T is a Dunford-Schwartz operator (positive Dunford-Schwartz operator), we will write T DS (respectively, T DS+). If T DS, consider its ergodic ∈ ∈ ∈ averages n−1 1 (1) A (x)=A (T,x)= Tk(x), x L1+L∞. n n n ∈ kX=0 Here is a noncommutative maximal ergodic inequality due to Yeadon [23] (for the assumption T DS+, see a clarification given in [3, Proposition 1.1, Remark ∈ 1.2]): Theorem 3.1. Let T DS+ and A : L1 L1, n = 1,2,... be given by (1). n ∈ → Then for every x L1 and ν >0 there exists a projection e ( ) such that ∈ + ∈P M x τ(e⊥) k k1 and sup eA (x)e ν. n ∞ ≤ ν k k ≤ n INDIVIDUAL ERGODIC THEOREMS IN NONCOMMUTATIVE ORLICZ SPACES 7 Now, let Φ be an Orlicz function, LΦ = LΦ( ,τ) the corresponding noncom- M mutative Orlicz space, the Luxemburg norm in L . Φ Φ k·k As LΦ isanexactinterpolationspaceforthe Banachcouple(L1, )(seeCorol- M lary 2.2), (2) T(LΦ)⊂LΦ and kTkLΦ→LΦ ≤1, hold for any T DS, and we have the following. ∈ Proposition 3.2. If T DS+, then the family A :LΦ LΦ, n=1,2,..., given n ∈ → by (1) is b.u.e.m. at zero on (LΦ, ). Φ k·k Proof. It is easy to verify (see [15, Lemma 4.1]) that it is sufficient to show that A is b.u.e.m. at zero on (LΦ, ). { n} + k·kΦ Fix ǫ>0,δ >0. By Lemma 2.1, there exists t>0 such that δ t Φ(λ) λ as soon as λ . · ≥ ≥ 2 Let ν >0 and 0<γ 1 be such that ν δ and γ ǫ. Takex LΦ with≤x γ,andletx≤=2t∞λdeν ≤beitsspectraldecomposition. ∈ + k kΦ ≤ 0 λ Then we can write R δ/2 ∞ ∞ x= λde + λde x +t Φ(λ)de x +t Φ(x), λ λ δ λ δ Z Z ≤ ·Z ≤ · 0 δ/2 δ/2 δ/2 ∞ where x= λde and Φ(x)= Φ(λ)de . 0 λ 0 λ As x R δ and T DS+, wRe have k δk∞ ≤ 2 ∈ δ sup A (x ) . n δ ∞ k k ≤ 2 n Besides, by Proposition 2.2, x 1 implies that Φ(x) x γ. Since Φ 1 M k k ≤ k k ≤ k k ≤ Φ(x) L1, in view of Theorem 3.1, one can find a projection e ( ) such that ∈ + ∈P M Φ(x) γ δ τ(e⊥) k k1 ǫ and sup eA (Φ(x))e ν . n ∞ ≤ ν ≤ ν ≤ k k ≤ ≤ 2t n Consequently, δ δ sup eA (x)e sup eA (x )e +t sup eA (Φ(x))e +t =δ, n ∞ n δ ∞ n ∞ k k ≤ k k · k k ≤ 2 · 2t n n n and the proof is complete. (cid:3) Here is an individual ergodic theorem for noncommutative Orlicz spaces: Theorem3.2. AssumethatanOrliczfunctionΦsatisfy(δ ,∆ ) condition. Then, 2 2 − given T DS+ and x LΦ, the averages (1) converge b.a.u. to some xˆ LΦ. ∈ ∈ ∈ Proof. Since, by Proposition 2.3, the set L1 L2 is dense in LΦ and the ∩M ⊂ averages (1) converge a.u., hence b.a.u., for every x L2 (see, for example, [15, ∈ Theorem 4.1]), it follows from Propositions 3.2 and 3.1 that for any x LΦ the ∈ averages(1) converge b.a.u. to some xˆ L0. ∈ It is clear that a b.a.u. convergent sequence in L0 converges in measure, hence A (x) xˆ, x LΦ, in measure. Since, by Corollary 2.2, LΦ has the Fatou n → ∈ property, its unit ball is closed in the measure topology [6, Theorem 4.1], and (2), hence supkAn(x)kLΦ→LΦ ≤kxkΦ, implies that xˆ∈LΦ. (cid:3) n 8 VLADIMIRCHILIN, SEMYONLITVINOV Remark3.2. Inwasshownin[3,Theorem5.2]thatifE(0, )isafullysymmetric ∞ function space with Fatou property and non-trivial Boyd indices and T DS+, ∈ thenforanyx E( ,τ)theaveragesA (x)convergeb.a.u. tosomex E( ,τ). n ∈ M ∈ M According to Remark 2.3 (ii), there exists an Orlicz function Φ that satisfies (δ2,∆2)−condition for which the Boyd index pLΦ(0,∞) is trivial. Thbus, Theorem 3.2 does not follow from Theorem [3, Theorem 5.2]. Nowwe shallturntoa classofOrliczspacesforwhichthe averages(1)converge a.u. The following fundamental result is crucial. Theorem 3.3 (Kadison’s inequality [11]). If S : is a positive linear M → M operator such that S(1) 1, then S(x)2 S(x2) for every x∗ =x . ≤ ≤ ∈M Definition 3.4. We call a convex function Φ on [0, ) 2 convex if the function ∞ − Φ(u)=Φ(√u) is also convex. e For example, Φ(u) = up, u 0, is 2 convex that satisfies (δ ,∆ ) condition p ≥ − 2 2 − whenever p 2. ≥ It is clear that if Φ is a 2 convex Orlicz function, then Φ is also an Orlicz − function, and it is easy to verify the following. e Proposition 3.3. If Φ be a 2−convex Orlicz function, then x2 ∈L+Φe and kx2kΦe = x 2 for every x LΦ. k kΦ ∈ + Proposition 3.4. Let Φ be a 2 convex Orlicz function. Then the family A n − { } given by (1) is u.e.m. at zero on (LΦ, ). M k·k Proof. As it was noticed earlier,it is sufficient to show that A is u.e.m. at zero n { } on (LΦ, ). + k·kΦ e Fix ǫ > 0, δ > 0. By Proposition 3.2, {An} is be.u.e.m. at zero on (LΦ,k·kΦe). Therefore there exists γ >0 such that, given y LΦ with y e <γ, ∈ k kΦ sup eA (y)e δ2 for some e ( ) with τ(e⊥) ǫ. n ∞ k k ≤ ∈P M ≤ n Now, let x LΦ be such that x < γ1/2. Then, due to Proposition 3.3, e ∈ + k kΦ x2 ∈ LΦ and kx2kΦe = kxk2Φ ≤ γ, implying that there is a projection e ∈ P(M) such that sup eA (x2)e δ2 and τ(e⊥) ǫ. n ∞ n k k ≤ ≤ Then, by Kadison’s inequality, 2 sup A (x)e =sup A (x)e 2 =sup eA (x)2e (cid:20) k n k∞(cid:21) k n k∞ k n k∞ ≤ n n n sup eA (x2)e δ2, n ∞ ≤ k k ≤ n which completes the proof. (cid:3) Now, as in Theorem 3.2, we obtain the following. Theorem3.4. IfanOrliczfunctionΦsatisfies(δ ,∆ ) conditionandis2 convex, 2 2 − − then, given T DS+ and x LΦ, the averages (1) converge a.u. to some xˆ LΦ. ∈ ∈ ∈ INDIVIDUAL ERGODIC THEOREMS IN NONCOMMUTATIVE ORLICZ SPACES 9 References [1] V.Chilin,S.Litvinov,andA.Skalski,Afewremarksinnon-commutativeergodictheory,J. Operator Theory,53(2)(2005), 331-350. [2] V.Chilin,S.Litvinov,Uniformequicontinuityforsequencesofhomomorphismsintothering ofmeasurableoperatorsMethods Funct. Anal. Topology, 12(2006), 124-130. 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