2-Co-lacunary sequences in noncommutative symmetric Banach spaces
aa r X i v : . [ m a t h . OA ] S e p FEDOR SUKOCHEV AND DEJIAN ZHOU ∗ Abstract.
We characterize noncommutative symmetric Banach spaces forwhich every bounded sequence admits either a convergent subsequence, or a2-co-lacunary subsequence. This extends the classical characterization, due toR¨abiger. Introduction
Let ( X, k · k X ) be a Banach space. A sequence ( x n ) n ≥ in X is said to be 2-co-lacunary if there is a constant C > λ n ) n ≥ ofcomplex numbers,(1.1) (cid:16) X n ≥ | λ n | (cid:17) / ≤ C (cid:13)(cid:13)(cid:13) X n ≥ λ n x n (cid:13)(cid:13)(cid:13) X . The following remarkable result about 2-co-lacunary subsequences was proved byAldous and Fremlin [2, Theorem 6]. Let (Ω , F , P ) be a probability space. If ( x n ) n ≥ is a bounded sequence in L (Ω), then either ( x n ) n ≥ admits a convergent subse-quence or a 2-co-lacunary subsequence. R¨abiger [27] showed that for any Banachlattice E , the following properties are equivalent.(1) Every bounded sequence in E has a subsequence which is either convergentin norm, or is 2-co-lacunary.(2) Every semi-normalized disjoint sequence in E has a subsequence which is2-co-lacunary.A sequence ( x n ) n ≥ in E is said to be semi-normalized if inf n ≥ k x n k E > n ≥ k x n k E < ∞ .Here we are interested in the extension of such results to noncommutative sym-metric spaces, such as those studied in [21]. We now state our first main resultbelow. Theorem 1.1.
Suppose that E is an order continuous symmetric Banach func-tion space such that E ⊂ ( L + L )(0 , ∞ ) . Let M be a hyperfinite and semifinitevon Neumann algebra acting on a separable Hilbert space H , equipped with a faith-ful normal semifinite trace τ , and let E ( M ) be the symmetric space associated to ( M , τ ) , and E .Then the following assertions are equivalent: Mathematics Subject Classification.
Primary 46L52; Secondary 46L53,46E30.
Key words and phrases. ∗ corresponding author. (i) Every bounded sequence in E ( M ) admits either a convergent subsequence ora -co-lacunary subsequence. (ii) Every sequence of pairwise orthogonal elements in E ( M ) either converges tozero or contains a -co-lacunary subsequence. A sequence of operators ( x n ) n ≥ ⊂ E ( M ) is said to be pairwise orthogonal if, forall m = n, we have x n x m = x ∗ n x m = 0 . In other words, our result extends R¨abiger’scharacterization to symmetric spaces associated with semifinite and hyperfinite vonNeumann algebras. Moreover, if E ( M ) = L ( M ) , then Theorem 1.1 goes back to[25, Corollary 3.7]. The proof of Theorem 1.1, given in Section 3, heavily dependson the recent result [20, Lemma 38].The next theorem is our second main result. See Definition 2.2 for the notion of E -equi-integrability. Theorem 1.2.
Let E be an order continuous symmetric Banach function spacesuch that E ⊂ ( L + L )(0 , ∞ ) . Suppose that M is a semifinite von Neumannalgebra, and ( x n ) n ≥ is a martingale difference sequence in E ( M ) such that (i) A = inf {k x n k E ( M ) : n ≥ } > ; (ii) ( x n ) n ≥ is E -equi-integrable in E ( M ) .Then the martingale difference sequence ( x n ) n ≥ is -co-lacunary in E ( M ) . It was proved by Akemann [1, Theorem II.2] (see also [36, Page 149, Theorem5.4]) that for K ⊂ L ( M ), K is L -equi-integrable if and only if K is relativelyweakly compact. Then, if E ( M ) = L ( M ) in the above theorem, Theorem 1.2 isjust [25, Theorem 3.6].In [25], to prove Theorem 1.1 for E = L (see Corollary 3.7 there), Parcet andRandrianantoanina firstly proved [25, Theorem 3.6]. At the end of [25], a simpleproof of [25, Theorem 3.6] due to Pisier, was given for finite von Neumann algebras.Pisier’s argument mainly depends on the fact that the spaces L p ( M ), 0 < p < ,
1) inequality for martingaletransformations, established in [28]. As mentioned by the authors of [25, p. 251], itwas unknown that if one could extend Pisier’s argument to semifinite von Neumannalgebra M . Our method of proof for Theorem 1.1 is completely distinct from thatof [25, Theorem 3.6], and may be seen as a development of Pisier’s ideas.Preliminaries are given in Section 2. Sections 3 and 4 are devoted to the proofsof Theorems 1.1 and 1.2, respectively. At the end of Section 3, we give someconcluding remarks and an example demonstrating that Theorem 1.1 holds truefor a concrete Orlicz function space.Throughout this paper, we write A . B if there is a constant c > A ≤ cB. We write A ≈ B if both A . B and B . A hold, possibly with differentconstants. 2. Preliminaries
Noncommutative symmetric spaces.
Throughout the paper, M is a semifi-nite von Neumann algebra equipped with a distinguished faithful normal semifinitetrace τ . Assume that M is acting on a Hilbert space H . A closed densely definedoperator x on H is said to be affiliated with M if x commutes with the commutant M ′ of M . If a is self-adjoint and if a = R R sde as is its spectral decomposition, thenfor any Borel subset B ⊆ R , we denote by χ B ( a ) the corresponding spectral projec-tion R R χ B ( s ) de as . An operator x affiliated with M is called τ -measurable if there ONCOMMUTATIVE 2-CO-LACUNARY SEQUENCES 3 exists s > τ ( χ ( s, ∞ ) ( | x | )) < ∞ (see e.g. [16, Definition 1.2]). Denote by S ( M ) the topological ∗ -algebra of all τ -measurable operators.For x ∈ S ( M ), the generalized singular-valued function µ ( x ) is defined by µ t ( x ) = inf { s > τ (cid:0) χ ( s, ∞ ) ( | x | ) (cid:1) ≤ t } , t > . The function t µ t ( x ) is decreasing and right-continuous. For more detailed studyof the generalized singular-value function, see for example [16]. If M = L ∞ (0 , ∞ ) isthe abelian von Neumann algebra, then, for a measurable function f , the function µ ( f ) is just the decreasing rearrangement of f (see [4, Page 39]).Let L (0 , ∞ ) denote the set of all equivalence classes of Lebesgue measurablefunctions on (0 , ∞ ). A Banach (or quasi-Banach) function space ( E, k · k E ) on theinterval (0 , ∞ ) is called symmetric if, for every g ∈ E and for every measurablefunction f ∈ L (0 , ∞ ) with µ ( f ) ≤ µ ( g ), we have f ∈ E and k f k E ≤ k g k E . Wesay E (0 , ∞ ) is order continuous if k x β k ↓ ≤ x β ↓ ⊂ E (0 , ∞ ). E isorder continuous if and only if it is separable [7].Following [21], for a given symmetric Banach (or quasi-Banach) function space( E, k·k E ) , we define the corresponding noncommutative space on ( M , τ ) by setting E ( M ) := { x ∈ S ( M ) : µ ( x ) ∈ E } . The associated quasi-norm is(2.1) k x k E ( M ) = k µ ( x ) k E . It is shown in [21] that if E (0 , ∞ ) is a symmetric Banach space, then E ( M ) isBanach. This result is extended to quasi-Banach spaces in [34]. That is it isestablished in [34] that if E (0 , ∞ ) is a symmetric quasi-Banach space, then E ( M )is quasi-Banach.A useful quasi-Banach space is the weak space L , ∞ (0 , ∞ ) defined by L , ∞ (0 , ∞ ) = { f ∈ L (0 , ∞ ) : k f k L , ∞ < ∞} , where k f k L , ∞ = sup t> tµ t ( f ). Then L , ∞ ( M ) can be defined according to (2.1).As usual, for 0 < p, q ≤ ∞ , ( L p + L q )(0 , ∞ ) is the sum of the quasi-Banachspaces L p (0 , ∞ ) and L q (0 , ∞ ). Here, the quasi-norm is given by the formula k f k L p + L q = inf {k g k p + k g k q : f = g + h } . The space ( L p ∩ L q )(0 , ∞ ) is the intersection of the quasi-Banach spaces L p (0 , ∞ )and L q (0 , ∞ ). Here, the quasi-norm is given by the formula k f k L p ∩ L q = max {k f k p , k f k q } . According to [18, Theorem 4.1], for 0 < p < q < ∞ , we have(2.2) k f k L p + L q ≈ (cid:16) Z µ t ( f ) p dt (cid:17) /p + (cid:16) Z ∞ µ t ( f ) q dt (cid:17) /q . The spaces ( L p + L q )( M ) and ( L p ∩ L q )( M ) are defined by (2.1). For the casewhere E (0 , ∞ ) is a symmetric Banach function space, the inclusions(2.3) ( L ∩ L ∞ )( M ) ⊂ E ( M ) ⊂ ( L + L ∞ )( M )hold with the continuous embeddings. F. SUKOCHEV AND D. ZHOU
Let x, y ∈ ( L + L ∞ )( M ). The operator y is said to be submajorized by x ,denoted by y ≺≺ x , if for all t ≥ Z t µ s ( y ) ds ≤ Z t µ s ( x ) ds. This definition is taken from [24, Definition 3.3.1], and we also refer the reader to[24, Chapter 3.3] for more information. We say that E ( M ) is fully symmetric if y ∈ E ( M ) and µ ( x ) ≺≺ µ ( y ) imply x ∈ E ( M ) and k x k E ( M ) ≤ k y k E ( M ) . We say that E ( M ) has the Fatou property if ( x n ) n ≥ ⊂ E ( M ) and x ∈ S ( M ) such that x n → x for the measure topology, then x ∈ E ( M ) and k x k E ( M ) ≤ lim inf n →∞ k x n k E ( M ) . Remark 2.1. If E (0 , ∞ ) is an order continuous symmetric Banach function space,then E (0 , ∞ ) is fully symmetric (see e.g. [10] , [13, Page 112] , or [22] ). Hence, E ( M ) is fully symmetric ( [14, Page 245] ).If E (0 , ∞ ) has the Fatou property, then E ( M ) also has the Fatou property (see [14, Theorem 54(iii)] ). We need the following definition introduced by Randrianantoanina [29]. We alsopoint out that related notions had been considered earlier in [7], and were studiedextensively in [15], [30], [31] and [35].
Definition 2.2 ([29, Definition 2.5] and [12, Definition 3.3]) . Let E (0 , ∞ ) be asymmetric quasi-Banach function space and K be a bounded subset of E ( M ) . Wewill say that K is E -equi-integrable if for every decreasing sequence ( e n ) n ≥ ofprojections with e n ↓ , lim n →∞ sup x ∈ K k e n xe n k E ( M ) = 0 . The following lemma is taken from [15] (see also [12] and [29]).
Lemma 2.3 ([15, Theorem 3.4]) . Let E (0 , ∞ ) be an order continuous symmetricBanach function space. If ( x n ) n ≥ ⊂ E ( M ) is bounded and E -equi-integrable, then k x n k E ( M ) → if and only if x n → in measure topology. Hyperfinite von Neumann algebras.
A von Neumann algebra is calledhyperfinite if coincides with the weak closure of a increasing net of finite dimensionalsubalgebras (see e.g. [8] or [32, Page 49]).Consider M a hyperfinite and infinite von Neumann algebra acting on a separableHilbert space H . Denote by R the hyperfinite II factor (see for example [8]). Then M is trace preserving ∗ -isomorphic to a von Neumann subalgebra of R ¯ ⊗ B ( H ).Indeed, we have that (see e.g. [36, Theorem V.1.19]) M = M I ⊕ M II ⊕ M II ∞ . By applying [8, Proposition 6.5], we have that (see also [17, Page 59]) every finitehyperfinite von Neumann algebra N is ( ∗ -isomorphic to) a countable direct sumof von Neumann algebras of the form A ¯ ⊗B , where A is abelian and B is either B ( ℓ n ) for some n < ∞ or R . Note that A can be realized as a subalgebra of L ∞ (0 , ∞ ) = L ∞ (0 ,
1) ¯ ⊗ ℓ ∞ . Hence, A can be realised as a subalgebra in R ¯ ⊗ B ( H ) . Obviously, B is a subalgebra in B ( H ) . Thus, A ¯ ⊗B can be realised as a subalgebrain R ¯ ⊗ B ( H ) . Then N is a subalgebra of R ¯ ⊗ B ( H ) ¯ ⊗ ℓ ∞ , which is trace preserving ∗ -isomorphic to a subalgebra of R ¯ ⊗ B ( H ) ¯ ⊗ B ( H ) = R ¯ ⊗ B ( H ). ONCOMMUTATIVE 2-CO-LACUNARY SEQUENCES 5
According to [8] (see also [17, Page 60]), M II ∞ is N ¯ ⊗ B ( H ) where N is a finitehyperfinite von Neumann algebra. Hence, M II ∞ is trace preserving ∗ -isomorphicto a von Neumann subalgebra of R ¯ ⊗ B ( H ) ¯ ⊗ B ( H ) = R ¯ ⊗ B ( H ).If M I is infinite, then M I is ∗ -isomorphic to a countable direct sum of von Neu-mann algebras A ¯ ⊗B , where A is abelian and B is B ( ℓ n ) for some n < ∞ or B ( H )([36, Theorem V.1.27]), and consequently, M I is trace preserving ∗ -isomorphic toa von Neumann subalgebra of R ¯ ⊗ ℓ ∞ ¯ ⊗ B ( H ), which is a subalgebra of R ¯ ⊗ B ( H ).2.3. Noncommutative martingale differences.
In this subsection, we reviewthe basics of noncommutative martingales. Let ( M n ) n ≥ be an increasing sequenceof von Neumann subalgebras of M such that the union of the M n ’s is weaklydense in M . Assume that for every n ≥
1, there exists a normal τ -invariantconditional expectation from M onto M n . In fact, for the case where M is finitethen such conditional expectations always exist (see [32, Lemma 3.6.2] or [36]). Ifthe restriction of τ on M n remains semifinite, then such conditional expectationsexist ([36, Page 332]). Since E n is τ -invariant, it extends to a contractive projectionfrom L p ( M , τ ) onto L p ( M n , τ n ) for all 1 ≤ p ≤ ∞ , where τ n denotes the restrictionof τ on M n . Definition 2.4.
A sequence x = ( x k ) k ≥ in ( L + L ∞ )( M ) is said to be a sequenceof martingale differences if x k ∈ M k for each k ≥ and E k − ( x k ) = 0 for every k ≥ . The following important result is a corollary of [28, Theorem 3.1].
Lemma 2.5.
Let ( x k ) k ≥ ⊂ L ( M ) be a sequence of martingale differences. Then (cid:13)(cid:13)(cid:13) X k ≥ r k ⊗ x k (cid:13)(cid:13)(cid:13) L , ∞ ( L ∞ (0 ,
1) ¯ ⊗M ) ≤ c abs sup n (cid:13)(cid:13)(cid:13) n X k =1 x k (cid:13)(cid:13)(cid:13) , where ( r k ) k ≥ is the sequence of Rademacher functions (see, for example, [23] ) and L ∞ (0 ,
1) ¯ ⊗M is the tensor product von Neumann algebra (see for example [36, Page183] ).Proof. Since ( x k ) k ≥ is a martingale difference sequence in L ( M ), so is ( χ (0 , ⊗ x k ) k ≥ in L ( L ∞ (0 , ⊗ M ) with respect to the filtration ( L ∞ (0 , ⊗ M k ) k ≥ .Then, it follows from [28, Theorem 3.1] that (cid:13)(cid:13)(cid:13) X k ≥ r k ⊗ x k (cid:13)(cid:13)(cid:13) L , ∞ ( L ∞ (0 ,
1) ¯ ⊗M ) = (cid:13)(cid:13)(cid:13) X k ≥ ( r k ⊗ χ (0 , ⊗ x k ) (cid:13)(cid:13)(cid:13) L , ∞ ( L ∞ (0 ,
1) ¯ ⊗M ) ≤ c abs sup n (cid:13)(cid:13)(cid:13) n X k =1 χ (0 , ⊗ x k (cid:13)(cid:13)(cid:13) L ( L ∞ (0 ,
1) ¯ ⊗M ) = c abs sup n (cid:13)(cid:13)(cid:13) n X k =1 x k (cid:13)(cid:13)(cid:13) . (cid:3) Proof of Theorem 1.1
In this section, we prove Theorem 1.1. We will need the following perturbationlemma from [2, Lemma 2].
F. SUKOCHEV AND D. ZHOU
Lemma 3.1.
Let X be a normed space, and let ( x n ) n ≥ be a bounded sequence in X . Then (i) if ( x n ) n ≥ is -co-lacunary and x ∈ X , then there exists m ∈ N such that ( x n − x ) n ≥ m is -co-lacunary; (ii) if ( x n ) n ≥ is -co-lacunary and ( y n ) n ≥ ⊂ X with P n ≥ k x n − y n k X beingconvergent, then there exists m ∈ N such that ( y n ) n ≥ m is -co-lacunary. The following assertion can be found in [25].
Theorem 3.2 ([25, Corollary 3.7]) . Let M be hyperfinite and semifinite. Everybounded sequence ( x n ) n ≥ in L ( M ) admits either a convergent subsequence or a -co-lacunary subsequence. Corollary 3.3.
Let H be a separable Hilbert space and R be the hyperfinite II factor. Every bounded sequence ( x n ) n ≥ in ( L + L )( R ¯ ⊗ B ( H )) admits either aconvergent subsequence or a -co-lacunary subsequence.Proof. It was proved in [20, Lemma 38] that there exists an isomorphic embedding T : ( L + L )( R ¯ ⊗ B ( H )) → L ( R ) . Obviously, the sequence ( x n ) n ≥ (in ( L + L )( M )) is 2-co-lacunary (respectively, convergent) if and only if the sequence( T x n ) n ≥ (in L ( R )) is 2-co-lacunary (respectively, convergent). Now the assertionfollows from Theorem 3.2. (cid:3) Corollary 3.4.
Let M be a hyperfinite and semifinite von Neumann algebra actingon a separable Hilbert space H . Every bounded sequence ( x n ) n ≥ in ( L + L )( M ) admits either a convergent subsequence or a -co-lacunary subsequence.Proof. If M is finite, then the corollary is just Theorem 3.2. We now considerthe case when M is infinite. Note that M is trace preserving ∗ -isomorphic to asubalgebra of R ¯ ⊗ B ( H ) (see Subsection 2.2). Then ( L + L )( M ) is isomorphic toa subspace of ( L + L )( R ¯ ⊗ B ( H )), and the assertion of the corollary follows fromCorollary 3.3. (cid:3) We need the following result from [6].
Lemma 3.5 ([6, Theorem 2.5]) . Let ( M , τ ) be semifinite and E (0 , ∞ ) be an ordercontinuous symmetric Banach function space. If ( x n ) n ≥ ⊂ E ( M ) is a sequenceof elements which converges to zero in the measure topology, then there exists asubsequence ( x n k ) k ≥ and two sequences of mutually orthogonal projections ( p k ) k ≥ and ( q k ) k ≥ in M such that k x n k − p k x n k q k k E ( M ) → . Lemma 3.6.
Suppose that E satisfies the Assumption (ii) in Theorem 1.1. Everybounded sequence ( x n ) n ≥ ⊂ E ( M ) which converges to zero in the measure topologyadmits either a convergent subsequence or a -co-lacunary subsequence.Proof. If the sequence ( x n ) n ≥ converges to 0 in E ( M ) , then the assertion follows.Otherwise, passing to a subsequence if needed, we may assume that the sequence( x n ) n ≥ is semi-normalised.By Lemma 3.5, there are a subsequence ( x n k ) k ≥ and sequences ( p k ) k ≥ and( q k ) k ≥ of mutually orthogonal projections in M such that(3.1) k x n k − p k x n k q k k E ( M ) → . ONCOMMUTATIVE 2-CO-LACUNARY SEQUENCES 7
Denote for brevity u k = p k x n k q k . According to (3.1), passing to a further subse-quence if it is necessary, we may assume X l ≥ k x n k − u k k E ( M ) < ∞ . Since the sequence ( x n ) n ≥ is semi-normalised, ( u k ) k ≥ is also semi-normalized.So is the sequence ( | u k | ) k ≥ . The latter sequence consists of pairwise orthogonalelements. By Assumption (ii) of Theorem 1.1, there exists a 2-co-lacunary subse-quence ( | u k l | ) l ≥ . Observe that for any ( λ l ) l ≥ ⊂ C , | X l λ l u k l | = X l ,l λ l λ l q k l x ∗ n kl p k l p k l x n kl q k l = X l | λ l | q k l x ∗ n kl p k l x n kl q k l = X l | λ l | | u k l | = | X l λ l | u k l || . Since ( | u k l | ) l ≥ is 2-co-lacunary, it follows that (cid:16) X l ≥ | λ l | (cid:17) / ≤ c (cid:13)(cid:13)(cid:13) X l λ l | u k l | (cid:13)(cid:13)(cid:13) E ( M ) = c (cid:13)(cid:13)(cid:13) X l λ l u k l (cid:13)(cid:13)(cid:13) E ( M ) . This means the sequence ( u k l ) l ≥ is 2-co-lacunary.By Lemma 3.1(ii), the subsequence ( x n kl ) l ≥ is 2-co-lacunary. Thus ( x n ) n ≥ contains a 2-co-lacunary subsequence. (cid:3) Lemma 3.7.
Let E (0 , ∞ ) be an order continuous symmetric Banach functionspace. If every sequence of pairwise orthogonal elements in E (0 , ∞ ) admits ei-ther a convergent subsequence or a -co-lacunary subsequence, then E (0 , ∞ ) hasthe Fatou property.Proof. By the assumption, no sequence of disjointly supported elements in E spans c . By [15, Theorem 6.5(v) ⇒ (i)], E has the Fatou property. (cid:3) We are now ready to prove our main result Theorem 1.1.Recall that E is an order continuous symmetric Banach function space such that E ⊂ ( L + L )(0 , ∞ ) . Assume that M is semifinite and hyperfinite. Proof of Theorem 1.1.
The implication (i) ⇒ (ii) is easy. Indeed, let ( x n ) n ≥ in E ( M ) be a bounded sequences, and both right and left disjointly supported. If( x n ) n ≥ is convergent in E ( M ) , then it converges to zero. However, ( x n ) n ≥ issemi-normalized; hence, it does not have a subsequence which converges to 0 . Thus,(ii) follows from (i).We now concentrate on the implication (ii) ⇒ (i) . It suffices to consider a se-quence ( x n ) n ≥ in E ( M ) which is semi-normalised. Case A : Suppose that the sequence ( x n ) n ≥ converges to zero in ( L + L )( M ) . In this case, the sequence ( x n ) n ≥ converges to zero in measure, and hence theapplication of Lemma 3.6 yields the assertion of Theorem 1.1. Case B : Suppose that the sequence ( x n ) n ≥ does not converge to zero in ( L + L )( M ) . Choose δ > x n k such thatlim k →∞ k x n k k L + L = δ. F. SUKOCHEV AND D. ZHOU
By Corollary 3.4, the sequence ( x n k ) k ≥ either contains a subsequence ( x n kl ) whichis either 2-co-lacunary in ( L + L )( M ) or converges in ( L + L )( M ) . If ( x n kl ) is 2-co-lacunary in ( L + L )( M ) , then for any complex sequence ( λ l ) l , (cid:13)(cid:13)(cid:13) X l ≥ λ l x n kl (cid:13)(cid:13)(cid:13) E ≥ (cid:13)(cid:13)(cid:13) X l ≥ λ l x n kl (cid:13)(cid:13)(cid:13) L + L & (cid:16) X l ≥ | λ l | (cid:17) / . Hence, the sequence ( x n kl ) is 2-co-lacunary in E ( M ) . If ( x n kl ) is convergent in ( L + L )( M ) , then we denote the limit by x. ByAssumption (ii) of Theorem 1.1 and Lemma 3.7, E has the Fatou property and,hence, so does E ( M ) (see Remark 2.1). By the Fatou property, x ∈ E ( M ) . Set y l = x n kl − x ∈ E ( M ) . Note that ( y l ) l ≥ converges to 0 in measure. If y l → E ( M ) , then x n kl → x in E ( M ) . If ( y l ) l ≥ does not converge to 0 in E ( M ) , then, by Lemma 3.6, we constructa 2-co-lacunary subsequence of the sequence ( y l ) l ≥ . By Lemma 3.1(i), there is a2-co-lacunary subsequence of the sequence ( x n kl ) l ≥ . This completes the proof inCase B. (cid:3)
At the end of this section, we present some concluding remarks for Theorem 1.1.First of all, we demonstrate that the assumption E ⊂ ( L + L )(0 , ∞ ) is necessaryfor the validity of Theorem 1.1.Let E (0 , ∞ ) be a symmetric Banach function space. If every bounded sequence( f n ) n ≥ ⊂ E (0 , ∞ ) admits either a convergent subsequence or a 2-co-lacunary sub-sequence, then E ⊂ ( L + L )(0 , ∞ ). Indeed, take f ∈ E (0 , ∞ ). We have k f k E = k µ ( f ) k E ≥ (cid:13)(cid:13)(cid:13) X k ≥ µ k +1 ( f ) χ [ k,k +1) (cid:13)(cid:13)(cid:13) E . Since the sequence ( χ [ k,k +1) ) k ≥ is pairwise disjoint and semi-normalized, it doesnot have a convergent subsequence. Hence, by the assumption, it is 2-co-lacunary.Then we have k f k E & (cid:16) X k ≥ µ k +1 ( f ) (cid:17) / ≥ (cid:16) X k ≥ Z k +2 k +1 µ s ( f ) ds (cid:17) / = (cid:16) Z ∞ µ s ( f ) ds (cid:17) / Observe that k µ ( f ) χ (0 , k ≤ k µ ( f ) χ (0 , k E ≤ k µ ( f ) k E = k f k E . Hence, by (2.2),we have k f k L + L ≈ Z µ s ( f ) χ (0 , ds + (cid:16) Z ∞ µ s ( f ) ds (cid:17) / . k f k E , which means E ⊂ ( L + L )(0 , ∞ ).We give an example to show that there exists an order continuous symmetricfunction space E ⊂ ( L + L )(0 , ∞ ) cannot contain a 2-co-lacunary subsequence.Let us assume that E = M ψ is a “separable part” of Marcinkiewicz space M ψ ,where ψ is continuous concave function on [0 , ∞ ) such that ψ (0) = 0 (see e.g. [22]and [3]) such that M ψ ⊂ ( L + L )(0 , ∞ ). By [3, Proposition 2.1], we know thatany normalized disjointly supported sequence in E for which every member is ascalar multiple of a characteristic function of a measurable subset of (0 ,
1) containsa subsequence equivalent a standard vector basis in c . Hence, such a subsequencecannot contain a 2-co-lacunary subsequence. ONCOMMUTATIVE 2-CO-LACUNARY SEQUENCES 9
Recall that a Banach lattice X is said to satisfy a lower p -estimate (1 < p < ∞ )(see, e.g., [23, Definition 1.f.4]) if there is a constant M such that for every choiceof pairwise disjoint elements ( x i ) ni =1 ⊂ X , we have (cid:16) n X i =1 k x i k pX (cid:17) /p ≤ M (cid:13)(cid:13)(cid:13) n X i =1 x i (cid:13)(cid:13)(cid:13) X . Now consider the von Neumann algebra B ( H ), where H is a separable Hilbertspace, and the standard trace tr. Our symmetric spaces E ( B ( H ) , tr) associatedto the algebra become ideals of B ( H ). For every symmetric sequence space E satisfying lower 2-estimate, the ideal C E = E ( B ( H )) is contained between theSchatten classes C and C . See, for example, [24, Part II] for extensive discussionof the correspondence between sequence spaces and ideals of B ( H ). We note that by[33, Proposition 2.3], we have that for a symmetric sequence space E , any pairwisedisjoint sequence in C E is isometrically isomorphic to disjoint basic sequence in E .Theorem 1.1 may now be rephrased in terms of ideals. Corollary 3.8.
Let symmetric sequence space E be separable. If E satisfies lower -estimate, then every bounded sequence ( x n ) n ≥ in the ideal C E admits either aconvergent subsequence, or a -co-lacunary subsequence. Proof of Theorem 1.2
In this section, we prove Theorem 1.2, which builds upon Pisier’s argument in[25, Page 251].Let 0 < q < ∞ . A quasi-Banach space ( X, k · k X ) is q -concave (see e.g. [23,Definition 1.d.3]) if for every finite sequence ( x k ) nk =1 ⊂ X there exists a constant K q such that (cid:16) n X i =1 k x i k qX (cid:17) /q ≤ K q (cid:13)(cid:13)(cid:13)(cid:16) n X i =1 | x i | q (cid:17) /q (cid:13)(cid:13)(cid:13) X . We need to show that for each 0 < p <
1, ( L p + L )( M ) is 2-concave. It wasshown in [13, Theorem 3.8] that if a symmetric Banach space E (0 , ∞ ) is q -concave,then E ( M ) is q -concave. However, we may not apply [13, Theorem 3.8] here, since( L p + L )(0 , ∞ ) is quasi-Banach. Lemma 4.1.
Suppose that < p ≤ and M is a semifinite von Neumann algebra.Then ( L p + L )( M ) is -concave, i.e., that is there exists a constant c p such thatfor all n ≥ and ( x k ) nk =1 ⊂ ( L p + L )( M ) , we have (cid:16) n X k =1 k x k k L p + L )( M ) (cid:17) / ≤ c p (cid:13)(cid:13)(cid:13)(cid:16) n X k =1 | x k | (cid:17) / (cid:13)(cid:13)(cid:13) ( L p + L )( M ) . Proof.
Using the fact that L w ( M ) is cotype 2 for any 0 < w ≤ Remark 3.3]), we have (cid:16) n X k =1 k x k k w (cid:17) / ≤ c w Z (cid:13)(cid:13)(cid:13) n X k =1 r k ( t ) x k (cid:13)(cid:13)(cid:13) w dt ≤ c w (cid:16) Z (cid:13)(cid:13)(cid:13) n X k =1 r k ( t ) x k (cid:13)(cid:13)(cid:13) ww dt (cid:17) /w ≤ c w (cid:13)(cid:13)(cid:13)(cid:16) n X k =1 | x k | (cid:17) / (cid:13)(cid:13)(cid:13) w , (4.1)where { r k } k ≥ is the sequence of Rademacher functions.On the other hand, it is well known that for any symmetric quasi-Banachspace E , the map J : E ( M , ℓ c ) → E ( M ¯ ⊗ B ( ℓ )) defined by setting J (( a k ) k ≥ ) = P k ≥ a k ⊗ e k, is an isometry.It now follows from (4.1) that for every 0 < w ≤
2, the map Θ : L w ( M ¯ ⊗ B ( ℓ )) → ℓ ( L w ( M )) defined by Θ(( a i,j ) i,j ≥ ) = ( a i, ) i ≥ is bounded. In particular, we have that Θ is simultaneously bounded as maps Θ : L p ( M ¯ ⊗ B ( ℓ )) → ℓ (( L p + L )( M )) and Θ : L ( M ¯ ⊗ B ( ℓ )) → ℓ (( L p + L )( M )).This implies that Θ : ( L p + L )( M ¯ ⊗ B ( ℓ )) → ℓ (( L p + L )( M ))is bounded. Composing this with the map J , we concludeΘ ◦ J : ( L p + L )( M ¯ ⊗ B ( ℓ )) → ℓ (( L p + L )( M ))is bounded. This actually means ( L p + L )( M ) is 2-concave. (cid:3) We shall apply the noncommutative Khintchine inequality established in [5].
Lemma 4.2.
Suppose that < p ≤ and M is a semifinite von Neumann algebra.For every finite sequence ( x k ) ⊂ ( L p + L )( M ) , we have (cid:13)(cid:13)(cid:13) X k ≥ r k ⊗ x k (cid:13)(cid:13)(cid:13) ( L p + L )( L ∞ (0 ,
1) ¯ ⊗M ) ≥ c p (cid:16) X k ≥ k x k k L p + L )( M ) (cid:17) , where { r k } k ≥ is the sequence of Rademacher functions.Proof. According to [5, Remark 3.3], the noncommutative Khintchine inequalityholds in the space L p + L . That is (cid:13)(cid:13)(cid:13) X k ≥ r k ⊗ x k (cid:13)(cid:13)(cid:13) ( L p + L )( L ∞ (0 ,
1) ¯ ⊗M ) ≈ inf x k = y k + z k n(cid:13)(cid:13)(cid:13)(cid:16) X k ≥ | y k | (cid:17) / (cid:13)(cid:13)(cid:13) ( L p + L )( M ) + (cid:13)(cid:13)(cid:13)(cid:16) X k ≥ | z ∗ k | (cid:17) / (cid:13)(cid:13)(cid:13) ( L p + L )( M ) o . By Lemma 4.1, we have that (cid:16) X k ≥ k y k k L p + L )( M ) (cid:17) / ≤ c p (cid:13)(cid:13)(cid:13)(cid:16) X k ≥ | y k | (cid:17) / (cid:13)(cid:13)(cid:13) ( L p + L )( M ) and (cid:16) X k ≥ k z k k L p + L )( M ) (cid:17) / ≤ c p (cid:13)(cid:13)(cid:13)(cid:16) X k ≥ | z ∗ k | (cid:17) / (cid:13)(cid:13)(cid:13) ( L p + L )( M ) . ONCOMMUTATIVE 2-CO-LACUNARY SEQUENCES 11
Thus (cid:13)(cid:13)(cid:13) X k ≥ r k ⊗ x k (cid:13)(cid:13)(cid:13) ( L p + L )( L ∞ (0 ,
1) ¯ ⊗M ) & inf x k = y k + z k n(cid:16) X k ≥ k y k k L p + L )( M ) (cid:17) / + (cid:16) X k ≥ k z k k L p + L )( M ) (cid:17) / o . By the quasi-triangle inequality, we have that (cid:16) X k ≥ k y k k L p + L )( M ) (cid:17) / + (cid:16) X k ≥ k z k k L p + L )( M ) (cid:17) / & (cid:16) X k ≥ k x k k L p + L )( M ) (cid:17) / . The assertion follows by combining the last two inequalities. (cid:3)
Lemma 4.3.
Let M be a semifinite von Neumann algebra. Assume that ( M n ) n ≥ is an increasing sequence of von Neumann subalgebras of M such that the union ofthe M n ’s is w ∗ -dense in M and ( E n ) n ≥ are the corresponding τ -invariant condi-tional expectations. Let x ∈ ( L + L )( M ) and x k = E k ( x ) − E k − ( x ) , k ≥ . Then,there is a positive constant c p depends on p such that (cid:16) X k ≥ k x k k L p + L )( M ) (cid:17) ≤ c p k x k ( L + L )( M ) , < p < . Proof.
Since x ∈ ( L + L )( M ), there exist y ∈ L ( M ) and z ∈ L ( M ) such that x = y + z , k y k ≤ k x k ( L + L )( M ) and k z k ≤ k x k ( L + L )( M ) . For every k ≥
1, set y k = E k ( y ) − E k − ( y ) and z k = E k ( z ) − E k − ( z ) . Now, x k = y k + z k . Since L p + L is a quasi-space for 0 < p <
1, there is a constant c p > k x k k L p + L ≤ c p ( k y k k L p + L + k z k k L p + L ) ≤ c p ( k y k k L p + L + k z k k ) . Thus c − p (cid:16) X k ≥ k x k k L p + L (cid:17) ≤ (cid:16) X k ≥ k y k k L p + L (cid:17) + (cid:16) X k ≥ k z k k (cid:17) . Observing that 0 < p <
1, we infer that L , ∞ (0 , ∞ ) ⊂ ( L p + L )(0 , ∞ ). Then, byLemma 4.2 and Lemma 2.5, we have (cid:16) X k ≥ k y k k L p + L )( M ) (cid:17) ≤ c p (cid:13)(cid:13)(cid:13) X k ≥ r k ⊗ y k (cid:13)(cid:13)(cid:13) ( L p + L )( L ∞ (0 ,
1) ¯ ⊗M ) ≤ c p c ′ p (cid:13)(cid:13)(cid:13) X k ≥ r k ⊗ y k (cid:13)(cid:13)(cid:13) L , ∞ ( L ∞ (0 ,
1) ¯ ⊗M ) ≤ c p c ′ p k y k . Combining the above estimates, we obtain c − p (cid:16) X k ≥ k x k k L p + L )( M ) (cid:17) ≤ c p c ′ p k y k + k z k ≤ c p c ′ p ) k x k ( L + L )( M ) . The proof is complete. (cid:3)
Now we are in a position to prove Theorem 1.2.
Proof of Theorem 1.2.
Fix 0 < p <
1. Set δ = inf {k x n k ( L p + L )( M ) : n ≥ } . We claim that δ >
0. Assume that lim inf n →∞ k x n k ( L p + L )( M ) = 0. Then thereexists a subsequence ( x n k ) k ≥ which converges to zero in ( L p + L )( M ), and con-sequently, ( x n k ) k ≥ converges to zero in measure (see e.g. [19, Lemma 2.4] or[11, Lemma 4.4]). By the second assumption of the theorem, the martingaledifference sequence ( x n ) n ≥ is E -equi-integrable in E ( M ). Then it follows fromLemma 2.3 that lim k →∞ k x n k k E ( M ) = 0, which contradicts our initial assumption.Hence δ >
0. Choose a square-summable sequence ( λ n ) n ≥ of scalars such that P k ≥ λ n x n ∈ E ( M ). Then we have δ (cid:16) X n ≥ | λ n | (cid:17) / ≤ (cid:16) X n ≥ k λ n x n k L p + L )( M ) (cid:17) / . By Lemma 4.3, we have (cid:16) X n ≥ | λ n | (cid:17) / . (cid:13)(cid:13)(cid:13) X n ≥ λ n x n (cid:13)(cid:13)(cid:13) ( L + L )( M ) ≤ (cid:13)(cid:13)(cid:13) X n ≥ λ n x n (cid:13)(cid:13)(cid:13) E ( M ) . This means the martingale difference sequence ( x n ) n ≥ is 2-co-lacunary in E ( M )and the proof is complete. (cid:3) Acknowledgement.
Most of the work was completed when the second authorwas visiting UNSW. He would like to express his gratitude to the School of Mathe-matics and Statistics of UNSW for its warm hospitality. The authors would like tothank Dmitriy Zanin for useful communication. The authors would like to thankThomas Scheckter for his careful reading and helpful comments that significantlyimprove the presentation of the paper. The authors would also like to thank theanonymous reviewer who suggested the current proofs of Lemmas 2.5 and 4.1 whichare substantially shorter than their initial versions.
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Fedor Sukochev, School of Mathematics and Statistics, University of New SouthWales, Kensington 2052, Australia
E-mail address : [email protected] Dejian Zhou, School of Mathematics and Statistics, Central South University, Chang-sha 410083, China
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