Kadec–Pełczyński decomposition for Haagerup Lp-spaces

Randrianantoanina, Narcisse

doi:10.1017/s0305004101005370

Cited by 16 publications

(20 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As already noticed in the remarks concerning the questions after Corollary 9, Randrianantoanina [26] has proved that the Kadec-Pe lczyński splitting lemma holds for preduals of von Neumann algebras. With this result Komlos' theorem follows almost immediately for von Neumann preduals.…”

Section: ) and Suppose That Zmentioning

confidence: 73%

“…Let (x n ) ⊂ X be bounded. Then by [26] there is a subsequence (x n k ) and there is a decomposition x n k = y k + z k where (z k ) w-converges and such that there is a sequence (ỹ k ) of pairwise orthogonal elements in X with ỹ k − y k → 0.…”

Section: ) and Suppose That Zmentioning

confidence: 99%

“…In a recent preprint Randrianantoanina [26] showed the Kadec-Pe lczyński subsequence decomposition for preduals of von Neumann algebras. Since the latter are known to have the weak Banach-Saks property [2], Komlos' theorem follows almost immediately from Randrianantoanina's result for von Neumann preduals, see Proposition 24 below.…”

Section: §1 Introductionmentioning

confidence: 99%

See 2 more Smart Citations

L-embedded banach spaces and measure topology

Pfitzner

2014

Isr. J. Math.

View full text Add to dashboard Cite

An L-embedded Banach spaace is a Banach space which is complemented in its bidual such that the norm is additive between the two complementary parts. On such spaces we define a topology, called an abstract measure topology, which by known results coincides with the usual measure topology on preduals of finite von Neumann algebras ( From the point of view of Banach space theory, L-embedded Banach spaces provide a natural frame for preduals of von Neumann algebras. So the starting point of this paper is on the one hand the definition of an abstract measure topology, Definition 3, patterned after the just mentionend characterization and on the other hand the easy but important observation, Theorem 4, that every L-embedded space admits such a topology. Although this topology does not come out easily with its properties -at the time of this writing it is not clear whether it is Hausdorff let alone metrizable or whether addition is continuous -it allows to generalize several results on subspaces of L 1 (µ) to subspaces of arbitrary L-embedded spaces. Thus section §4 of the present paper is titled "Section IV.3 of [13] (partly) revisited". For example, Theorem 10 generalizes a theorem of Buhvalov-Lozanovskii which describes the link between L-embeddedness and measure topology for subspaces Y of L 1 (µ), µ finite: Y is L-embedded if and only if its unit ball is closed in measure. (Note in passing that this criterion involves only the space Y itself, not its bidual.) Moreover, as a consequence of this, the closedness in measure of the unit ball of Y is a weak substitute for compactness which could be called "convex sequential compactness", see Corollary 9. We also reprove a result of Godefroy and Li concerning a criterion for L-embedded subspaces which are duals of M-embedded spaces, see Theorem 13. In this vein, that is by substituting arbitrary L-embedded spaces for L 1 (µ), we recover also some results of Godefroy, Kalton, Li [10] in §5. Finally, in §6 it is proved that addition is τ µ -continuous in preduals of von Neumann algebras. §2 Notation, Background: The results are stated for complex scalars. The dual of a Banach space X is denoted by X ′ . B X denotes the unit ball of X. Subspace of a Banach space means norm-closed subspace, bounded always means normbounded. As usual, we consider a Banach space as a subspace of its bidual omitting the canonical embedding. [x n ] denotes the closed linear span of a (finite or infinite) sequence (x n ).Basic properties and definitions which are not explained here can be found in [4] or in [20]-[21] for Banach spaces and in [22,28] for C * -algebras. The standard reference for M-and L-embedded spaces is the monograph [13].

show abstract

Section: ) and Suppose That Zmentioning

confidence: 73%

Section: ) and Suppose That Zmentioning

confidence: 99%

See 1 more Smart Citation

L-embedded banach spaces and measure topology

Pfitzner

2014

Isr. J. Math.

View full text Add to dashboard Cite

show abstract

“…Noncommutative L p spaces have been defined by Dixmier, Kunze and Segal in the semifinite setting (see also Nelson [Nel74]) and by Haagerup [Haa79] in the non-tracial case (see also [Hil81] for Connes' approach). Randrianantoanina [Ran02] has an argument in the semifinite setting which is different from ours and does not provide a good control of the constants. In this paper we use modular theory of operator algebras in conjunction with a noncommutative version of the Peter Jones theorem due to Pisier [Pis92] (related to estimates of Kaftal, Larsen and Weiss [KLW92] for triangular matrices) to solve the problem: Theorem A.…”

Section: Introductionmentioning

confidence: 79%

Rosenthal's theorem for subspaces of noncommutative Lp

Junge

Parcet

2008

Duke Math. J.

View full text Add to dashboard Cite

Abstract. We show that a reflexive subspace of the predual of a von Neumann algebra embeds into a noncommutative L p space for some p > 1. This is a noncommutative version of Rosenthal's result for commutative L p spaces. Similarly for 1 ≤ q < 2, an infinite dimensional subspace X of a noncommutative L q space either contains ℓ q or embeds in L p for some q < p < 2. The novelty in the noncommutative setting is a double sided change of density. IntroductionThe theory of noncommutative L p spaces has a long tradition in Banach space theory and the theory of operator algebras [GK69,Haa79,Hil81,TJ84,Fac87] and provides the background for recent progress in noncommutative analysis and probability [PX97,JLX03,JX03]. In the commutative setting, the work of and Rosenthal [Ros73] on subspaces of L p are corner stones for the understanding of general Banach space properties. In this paper we prove the noncommutative version of Rosenthal's result.Theorem (Rosenthal '73). A reflexive subspace of L 1 embeds into L p for some p > 1.The problem of generalizing Rosenthal theorem to the noncommutative setting is open for at least 20 years. This problem has an interesting history. In his seminal paper [Pis86b] on factorization properties, Pisier described a new approach to some factorization results by Maurey obtained from Nikishin's theorem. In this paper Pisier comes very close to proving the noncommutative version of Rosenthal's result. Indeed, he shows that a reflexive subspace of a von Neumann algebra predual embeds into an interpolation space between an L 1 space and certain (unusual) L 2 space (see below). Since then it has been a mystery how to modify the argument and to obtain a subspace of a noncommutative L p space. Noncommutative L p spaces have been defined by Dixmier, Kunze and Segal in the semifinite setting (see also Nelson [Nel74]) and by Haagerup [Haa79] in the non-tracial case (see also [Hil81] for Connes' approach). Randrianantoanina [Ran02] has an argument in the semifinite setting which is different from ours and does not provide a good control of the constants. In this paper we use modular theory of operator algebras in conjunction with a noncommutative version of the Peter Jones theorem due to Pisier The new interesting point in our proof is the natural change of density argument. We show that there exists a positive density d ∈ L 1 (N) such that tr(d) = 1 and a mapping u : X → L p (N) such thatIn the σ-finite case this completely determines u. For simplicity let us assume that N is finite and d = j d j e j has a countable spectrum. Then the map u is given by the following relationPisier's approach to this result [Pis86b] is used as a starting point in our proof. For subspaces of L q (N) with q > 1 we have a similar result, which extends the most general form of Rosenthal's theorem [Ros73, Theorem 8] to the noncommutative setting.Theorem B. Let N be a von Neumann algebra and fix 1 ≤ q < 2. Given a subspace X of L q (N) not containing ℓ q , there exists a positive density d ∈ L 1 (N) with tr(d) = 1 and a...

show abstract

“…From the definition of the ϕ k 's, the array of operators (f k,j ) ∞,∞ j =1,k=1 satisfies the assumptions of Lemma 3.9. There exists a subsequence of (f n ) ∞ n=1 (which can be assumed to be a subsequence of (f n 2 ) ∞ n=1 and will be denoted again by (f n ) ∞ n=1 ) such that for every increasing sequence (n k ) ∞ k=1 of N, If one considers the measure topology, the subsequence splitting lemma of [19] combined with a non-commutative analogue of the classical Szlenk's theorem on weak convergence in L 1 -spaces by Belanger and Diestel [6] provides the following: Proposition 3.11. Let (M, τ ) be a finite von Neumann algebra and suppose that (f n ) ∞ n=1 is a sequence in L 1 (M, τ ) with sup n f n 1 < ∞ then there exists a subsequence (g n ) ∞…”

Section: Lemma 310 (A) the Seriesmentioning

confidence: 99%

Non-commutative subsequence principles

Randrianantoanina

2003

Mathematische Zeitschrift

Self Cite

View full text Add to dashboard Cite

We prove that if M is a hyperfinite semi-finite von Neumann algebra equipped with a normal semi-finite trace τ and (x n ) ∞ n=1 is a bounded sequence in L 1 (M, τ ) then there exists a subsequence (y n ) ∞ n=1 of (x n ) ∞ n=1 such that for any fur-thus providing a non-commutative analogue of the classical Komlós's subsequence theorem. We also extend a classical result of Menchoff and Marcienkiewicz on convergence almost everywhere of subseries of orthogonal functions in L 2 [0, 1] to orthogonal operators in the non-commutative L 2 -space associated to the type I I 1 hyperfinite factor. (2000): 46L51, 46L52, 40A05, 60F15. Mathematics Subject Classification

show abstract

Kadec–Pełczyński decomposition for Haagerup L^p-spaces

Cited by 16 publications

References 27 publications

L-embedded banach spaces and measure topology

L-embedded banach spaces and measure topology

Rosenthal's theorem for subspaces of noncommutative Lp

Non-commutative subsequence principles

Contact Info

Product

Resources

About