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40
On the structure of subspaces of noncommutative Lpspaces
 C. R. Acad. Sci. Paris
"... Abstract: We study some structural aspects of the subspaces of the noncommutative (Haagerup) Lpspaces associated with a general (non necessarily semifinite) von Neumann algebra a. If a subspace X of Lp(a) contains uniformly the spaces ℓn p, n ≥ 1, it contains an almost isometric, almost 1complem ..."
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Cited by 33 (3 self)
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Abstract: We study some structural aspects of the subspaces of the noncommutative (Haagerup) Lpspaces associated with a general (non necessarily semifinite) von Neumann algebra a. If a subspace X of Lp(a) contains uniformly the spaces ℓn p, n ≥ 1, it contains an almost isometric, almost 1complemented copy of ℓp. If X contains uniformly the finite dimensional Schatten classes Sn p, it contains their ℓpdirect sum too. We obtain a version of the classical KadecPe̷lczyński dichotomy theorem for Lpspaces, p ≥ 2. We also give operator space versions of these results. The proofs are based on previous structural results on the ultrapowers of Lp(a), together with a careful analysis of the elements of an ultrapower Lp(a) U which are disjoint from the subspace Lp(a). These techniques permit to recover a recent result of N. Randrianantoanina concerning a Subsequence Splitting Lemma for the general noncommutative Lp spaces. Various notions of pequiintegrability are studied (one of which is equivalent to Randrianantoanina’s one) and some results obtained by Haagerup, Rosenthal and Sukochev for Lpspaces based on finite von Neumann algebras concerning subspaces of Lp(a) containing ℓp are extended to the general case.
Gundy’s decomposition for noncommutative martingales and applications
 Proc. London Math. Soc
"... Abstract. We provide an analogue of Gundy’s decomposition for L1bounded noncommutative martingales. An important difference from the classical case is that for any L1bounded noncommutative martingale, the decomposition consists of four martingales. This is strongly related with the row/column na ..."
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Cited by 10 (7 self)
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Abstract. We provide an analogue of Gundy’s decomposition for L1bounded noncommutative martingales. An important difference from the classical case is that for any L1bounded noncommutative martingale, the decomposition consists of four martingales. This is strongly related with the row/column nature of noncommutative Hardy spaces of martingales. As applications, we obtain simpler proofs of the weak type (1, 1) boundedness for noncommutative martingale transforms and the noncommutative analogue of Burkholder’s weak type inequality for square functions. A sequence (xn)n≥1 in a normed space X is called 2colacunary if there exists a bounded linear map from the closed linear span of (xn)n≥1 to l2 taking each xn to the nth vector basis of l2. We prove (using our decomposition) that any relatively weakly compact martingale difference sequence in L1(M, τ) whose sequence of norms is bounded away from zero is 2colacunary, generalizing a result of Aldous and Fremlin to noncommutative L1spaces.
ROSENTHAL’S THEOREM FOR SUBSPACES OF NONCOMMUTATIVE Lp
, 2006
"... Abstract. We show that a reflexive subspace of the predual of a von Neumann algebra embeds into a noncommutative Lp space for some p> 1. This is a noncommutative version of Rosenthal’s result for commutative Lp spaces. Similarly for 1 ≤ q < 2, an infinite dimensional subspace X of a noncommutative L ..."
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Cited by 10 (5 self)
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Abstract. We show that a reflexive subspace of the predual of a von Neumann algebra embeds into a noncommutative Lp space for some p> 1. This is a noncommutative version of Rosenthal’s result for commutative Lp spaces. Similarly for 1 ≤ q < 2, an infinite dimensional subspace X of a noncommutative Lq space either contains ℓq or embeds in Lp for some q < p < 2. The novelty in the noncommutative setting is a double sided change of density.
On ergodic theorems for free group actions on noncommutative spaces
, 2005
"... Abstract. We extend in a noncommutative setting the individual ergodic theorem of Nevo and Stein concerning measure preserving actions of free groups and averages on spheres s2n of even radius. Here we study state preserving actions of free groups on a von Neumann algebra A and the behaviour of (s2n ..."
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Cited by 10 (0 self)
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Abstract. We extend in a noncommutative setting the individual ergodic theorem of Nevo and Stein concerning measure preserving actions of free groups and averages on spheres s2n of even radius. Here we study state preserving actions of free groups on a von Neumann algebra A and the behaviour of (s2n(x)) for x in noncommutative spaces Lp (A). For ∑ n−1 k=0 sk and p = +∞, this problem was solved by the Cesàro means 1 n Walker. Our approach is based on ideas of Bufetov. We prove a noncommutative version of Rota “Alternierende Verfahren ” theorem. To this end, we introduce specific dilations of the powers of some noncommutative Markov operators. 1.
THE NORM OF SUMS OF INDEPENDENT NONCOMMUTATIVE RANDOM VARIABLES IN Lp(ℓ1)
, 2004
"... Abstract. We investigate the norm of sums of independent vectorvalued random variables in noncommutative Lp spaces. This allows us to obtain a uniform family of complete embeddings of the Schatten class S n q in Sp(ℓm q) with optimal order m ∼ n 2. Using these embeddings we show the surprising fact ..."
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Cited by 8 (6 self)
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Abstract. We investigate the norm of sums of independent vectorvalued random variables in noncommutative Lp spaces. This allows us to obtain a uniform family of complete embeddings of the Schatten class S n q in Sp(ℓm q) with optimal order m ∼ n 2. Using these embeddings we show the surprising fact that the sharp type (cotype) index in the sense of operator spaces for Lp[0,1] is min(p, p ′ ) (max(p, p ′)). Similar techniques are used to show that the operator space notions of Bconvexity and Kconvexity are equivalent.
Rosenthal type inequalities for free chaos
, 2005
"... Let A denote the reduced amalgamated free product of a family ..."
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Cited by 8 (5 self)
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Let A denote the reduced amalgamated free product of a family
Operator Valued Hardy Spaces
, 2003
"... We give a systematic study on the Hardy spaces of functions with values in the noncommutative L pspaces associated with a semifinite von Neumann algebra M. This is motivated by the works on matrix valued Harmonic Analysis (operator weighted norm inequalities, operator Hilbert transform), and on th ..."
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Cited by 8 (4 self)
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We give a systematic study on the Hardy spaces of functions with values in the noncommutative L pspaces associated with a semifinite von Neumann algebra M. This is motivated by the works on matrix valued Harmonic Analysis (operator weighted norm inequalities, operator Hilbert transform), and on the other hand, by the recent development on the noncommutative martingale inequalities. Our noncommutative Hardy spaces are defined by the noncommutative Lusin integral function. The main results of this paper include: (i) The analogue in our setting of the classical Fefferman duality theorem between H 1 and BMO. (ii) The atomic decomposition of our noncommutative H 1. (iii) The equivalence between the norms of the noncommutative Hardy spaces and of the noncommutative L pspaces (1 < p < ∞). (iv) The noncommutative HardyLittlewood maximal inequality. (v) A description of BMO as an intersection of two dyadic BMO. (vi) The interpolation results on these Hardy spaces. Plan:
H∞ FUNCTIONAL CALCULUS AND SQUARE FUNCTIONS ON Noncommutative L^Pspaces
, 2006
"... In this work we investigate semigroups of operators acting on noncommutative L pspaces. We introduce noncommutative square functions and their connection to sectoriality, variants of Rademacher sectoriality, and H ∞ functional calculus. We discuss several examples of noncommutative diffusion semig ..."
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Cited by 8 (3 self)
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In this work we investigate semigroups of operators acting on noncommutative L pspaces. We introduce noncommutative square functions and their connection to sectoriality, variants of Rademacher sectoriality, and H ∞ functional calculus. We discuss several examples of noncommutative diffusion semigroups. This includes Schur multipliers, qOrnsteinUhlenbeck semigroups, and the noncommutative Poisson semigroup on free groups.
A reduction method for noncommutative Lpspaces and applications
, 806
"... We consider the reduction of problems on general noncommutative Lpspaces to the corresponding ones on those associated with finite von Neumann algebras. The main tool is a unpublished result of the first named author which approximates any noncommutative Lpspace by tracial ones. We show that under ..."
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Cited by 7 (2 self)
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We consider the reduction of problems on general noncommutative Lpspaces to the corresponding ones on those associated with finite von Neumann algebras. The main tool is a unpublished result of the first named author which approximates any noncommutative Lpspace by tracial ones. We show that under some natural conditions a map between two von Neumann algebras extends to their crossed products by a locally compact abelian group or to their noncommutative Lpspaces. We present applications of these results to the theory of noncommutative martingale inequalities by reducing most recent general noncommutative martingale/ergodic inequalities to those in the tracial case. 0
On the best constants in some noncommutative martingale inequalities
 Bull. London Math. Soc
"... Abstract. We determine the optimal orders for the best constants in the noncommutative BurkholderGundy, Doob and Stein inequalities obtained recently in the noncommutative martingale theory. AMS Classification: 46L53, 46L51 Key words: Noncommutative martingale, inequality, optimal order, triangu ..."
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Cited by 7 (1 self)
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Abstract. We determine the optimal orders for the best constants in the noncommutative BurkholderGundy, Doob and Stein inequalities obtained recently in the noncommutative martingale theory. AMS Classification: 46L53, 46L51 Key words: Noncommutative martingale, inequality, optimal order, triangular projection † Marius Junge is partially supported by the NSF 1