Results 1  10
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28
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
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.
Ultraproducts in Analysis
 IN ANALYSIS AND LOGIC, VOLUME 262 OF LONDON MATHEMATICAL SOCIETY LECTURE NOTES
, 2002
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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
On the structure of isometries between noncommutative L p spaces, preprint
"... Abstract. We prove some structure results for isometries between noncommutative L p spaces associated to von Neumann algebras. We find that an isometry T: L p (M1) → L p (M2) (1 ≤ p < ∞, p ̸ = 2) can be canonically expressed in a certain simple form whenever M1 has variants of Watanabe’s extension ..."
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Cited by 4 (1 self)
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Abstract. We prove some structure results for isometries between noncommutative L p spaces associated to von Neumann algebras. We find that an isometry T: L p (M1) → L p (M2) (1 ≤ p < ∞, p ̸ = 2) can be canonically expressed in a certain simple form whenever M1 has variants of Watanabe’s extension property [W2]. Conversely, this form always defines an isometry provided that M1 is “approximately semifinite ” (defined below). Although neither of these properties is fully understood, we show that they are enjoyed by all semifinite algebras and hyperfinite algebras (with no summand of type I2), plus others. Thus the classification is stronger than Yeadon’s theorem [Y1] for semifinite algebras (and its recent improvement in [JRS]), and the proof uses independent techniques. Related to this, we examine the modular theory for positive projections from a von Neumann algebra onto a Jordan image of another von Neumann algebra, and use such projections to construct new L p isometries by interpolation. Some complementary results and questions are also presented. 1.
On the structure of noncommutative white noises
, 2004
"... Abstract. We consider the concepts of continuous Bernoulli systems and noncommutative white noises. We address the question of isomorphism of continuous Bernoulli systems and show that for large classes of quantum Lévy processes one can make quite precise statements about the time behaviour of thei ..."
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Cited by 2 (1 self)
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Abstract. We consider the concepts of continuous Bernoulli systems and noncommutative white noises. We address the question of isomorphism of continuous Bernoulli systems and show that for large classes of quantum Lévy processes one can make quite precise statements about the time behaviour of their moments. 1.
IMPROVED BOUNDS IN THE METRIC COTYPE INEQUALITY FOR BANACH SPACES
"... Abstract. It is shown that if (X, ‖·‖X) is a Banach space with Rademacher cotype q then 1 1+ for every integer n there exists an even integer m � n q such that for every f: Zn m → X we have n∑ ∥∥∥f ( Ex x + m ..."
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Cited by 2 (2 self)
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Abstract. It is shown that if (X, ‖·‖X) is a Banach space with Rademacher cotype q then 1 1+ for every integer n there exists an even integer m � n q such that for every f: Zn m → X we have n∑ ∥∥∥f ( Ex x + m
NONCOMMUTATIVE L p STRUCTURE ENCODES EXACTLY JORDAN STRUCTURE
, 2004
"... Abstract. We prove that for all 1 ≤ p ≤ ∞, p ̸ = 2, the L p spaces associated to two von Neumann algebras M, N are isometrically isomorphic if and only if M and N are Jordan *isomorphic. This follows from a noncommutative L p BanachStone theorem: a specific decomposition for surjective isometries ..."
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Cited by 1 (0 self)
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Abstract. We prove that for all 1 ≤ p ≤ ∞, p ̸ = 2, the L p spaces associated to two von Neumann algebras M, N are isometrically isomorphic if and only if M and N are Jordan *isomorphic. This follows from a noncommutative L p BanachStone theorem: a specific decomposition for surjective isometries of noncommutative L p spaces. 1.