On the best constants in noncommutative Khintchine-type inequalities

Haagerup, Uffe; Musat, Magdalena

doi:10.1016/j.jfa.2007.05.014

Cited by 24 publications

(20 citation statements)

References 19 publications

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“…Note that these lower bounds are trivial for the sequence (ε j ), e.g. the fact that c 1 ≥ √ 2 is trivial already from the classical case (just consider ε 1 + ε 2 in (5.4)), but then in that case the proof of Theorem 9.1 from [54] only gives us c 1 ≤ √ 3. So the best value of c 1 is still unknown !…”

Section: Best Constants (Non-commutative Case)mentioning

confidence: 99%

Grothendieck’s Theorem, past and present

Pisier

2012

Bull. Amer. Math. Soc.

213

195

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Abstract. Probably the most famous of Grothendieck's contributions to Banach space theory is the result that he himself described as "the fundamental theorem in the metric theory of tensor products". That is now commonly referred to as "Grothendieck's theorem" ("GT" for short), or sometimes as "Grothendieck's inequality". This had a major impact first in Banach space theory (roughly after 1968), then, later on, in C * -algebra theory (roughly after 1978). More recently, in this millennium, a new version of GT has been successfully developed in the framework of "operator spaces" or non-commutative Banach spaces. In addition, GT independently surfaced in several quite unrelated fields: in connection with Bell's inequality in quantum mechanics, in graph theory where the Grothendieck constant of a graph has been introduced and in computer science where the Grothendieck inequality is invoked to replace certain NP hard problems by others that can be treated by "semidefinite programming" and hence solved in polynomial time. This expository paper (where many proofs are included), presents a review of all these topics, starting from the original GT. We concentrate on the more recent developments and merely outline those of the first Banach space period since detailed accounts of that are already available, for instance the author's 1986 CBMS notes.

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Section: Best Constants (Non-commutative Case)mentioning

confidence: 99%

Grothendieck’s Theorem, past and present

Pisier

2012

Bull. Amer. Math. Soc.

213

195

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“…This paper relates to the general interest of finding the best constants in non-commutative probability inequalities. In particular, major achievements have been made considering the best constants of Burkholder/Gundy inequalities [13], [21], Doob and Stein inequalities [13] and Khintchine inequalities [10], [11].…”

Section: Introductionmentioning

confidence: 99%

The best constants for operator Lipschitz functions on Schatten classes

Caspers

Montgomery‐Smith

Potapov

et al. 2014

Journal of Functional Analysis

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Abstract. Suppose that f is a Lipschitz function on R with f Lip ≤ 1. Let A be a bounded self-adjoint operator on a Hilbert space H. Let p ∈ (1, ∞) and suppose that x ∈ B(H) is an operator such that the commutator [A, x] is contained in the Schatten class S p . It is proved by the last two authors, that then also [f (A), x] ∈ S p and there exists a constant C p independent of x and f such thatThe main result of this paper is to give a sharp estimate for C p in terms of p. Namely, we show that C p ∼ p 2 p−1 . In particular, this gives the best estimates for operator Lipschitz inequalities. We treat this result in a more general setting. This involves commutators of n self-adjoint operators A 1 , . . . , A n , for which we prove the analogous result. The case described here in the abstract follows as a special case.

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“…We refer to [P98] for non commutative vector-valued L p -spaces. The above definition is justified by the non commutative Khintchine inequalities: [HM07]) Let ε k be independent Rademacher random variables, then for 1 ≤ p < ∞,…”

Section: Notations and Preliminariesmentioning

confidence: 99%

Free Hilbert transforms

Mei¹,

Ricard²

2017

Duke Math. J.

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We study analogues of classical Hilbert transforms as fourier multipliers on free groups. We prove their complete boundedness on non commutative L p spaces associated with the free group von Neumann algebras for all 1 < p < ∞. This implies that the decomposition of the free group F∞ into reduced words starting with distinct free generators is completely unconditional in L p . We study the case of Voiculescu's amalgamated free products of von Neumann algebras as well. As by-products, we obtain a positive answer to a compactness-problem posed by Ozawa, a length independent estimate for Junge-Parcet-Xu's free Rosenthal inequality, a Littlewood-Paley-Stein type inequality for geodesic paths of free groups, and a length reduction formula for L p -norms of free group von Neumann algebras.

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On the best constants in noncommutative Khintchine-type inequalities

Cited by 24 publications

References 19 publications

Grothendieck’s Theorem, past and present

Grothendieck’s Theorem, past and present

The best constants for operator Lipschitz functions on Schatten classes

Free Hilbert transforms

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