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39
Noncommutative Burkholder/Rosenthal inequalities
 Ann. Probab
, 2000
"... Abstract. We show norm estimates for the sum of independent random variables in noncommutative Lp spaces for 1 < p < ∞ following previous work by the authors. These estimates generalize Rosenthal’s inequalities in the commutative case. Among other applications, we derive a formula for pnorm of the ..."
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Cited by 46 (25 self)
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Abstract. We show norm estimates for the sum of independent random variables in noncommutative Lp spaces for 1 < p < ∞ following previous work by the authors. These estimates generalize Rosenthal’s inequalities in the commutative case. Among other applications, we derive a formula for pnorm of the eigenvalues for matrices with independent entries, and characterize those symmetric subspaces and unitary ideal spaces which can be realized as subspaces of noncommutative Lp for 2 < p < ∞. 0. Introduction and Notation Martingale inequalities have a long tradition in probability. The applications of the work of Burkholder and his collaborators [B73,?, BDG72, B71a, B71b, BGS71, BG70, B66] ranges from classical harmonic analysis to stochastical differential equations and the geometry of Banach spaces. When proving the estimates for the ‘little square function ’ Burkholder
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.
Noncommutative L p modules
 J. Operator Theory
"... Abstract. We construct classes of von Neumann algebra modules by considering “column sums ” of noncommutative L p spaces. Our abstract characterization is based on an L p/2valued inner product, thereby generalizing Hilbert C*modules and representations on Hilbert space. While the (single) represen ..."
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Cited by 7 (2 self)
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Abstract. We construct classes of von Neumann algebra modules by considering “column sums ” of noncommutative L p spaces. Our abstract characterization is based on an L p/2valued inner product, thereby generalizing Hilbert C*modules and representations on Hilbert space. While the (single) representation theory is similar to the L 2 case, the concept of L p bimodule (p ̸ = 2) turns out to be nearly trivial. Noncommutative L p spaces, by now, are standard objects in the theory of operator algebras. Starting with a von Neumann algebra M, there are a variety of equivalent methods for producing the (quasi)Banach space L p (M). If M is L ∞ (X, µ), the result is (isometric to) L p (X, µ), so this can rightfully be thought