The norm of sums of independent noncommutative random variables in Lp(ℓ1)

Marius Junge; Javier Parcet

doi:10.1016/j.jfa.2004.09.002

The norm of sums of independent noncommutative random variables in L_p(ℓ₁)

Research output: Contribution to journal › Article › peer-review

Abstract

We investigate the norm of sums of independent vector-valued random variables in noncommutative L_p spaces. This allows us to obtain a uniform family of complete embeddings of the Schatten class S_qⁿ in S_p (ℓ_q ^m) with optimal order m ∼ n². Using these embeddings we show the surprising fact that the sharp type (cotype) index in the sense of operator spaces for L_p [0, 1] is min (p, p′) (max(p, p′)). Similar techniques are used to show that the operator space notions of B-convexity and K-convexity are equivalent.

Original language	English (US)
Pages (from-to)	366-406
Number of pages	41
Journal	Journal of Functional Analysis
Volume	221
Issue number	2
DOIs	https://doi.org/10.1016/j.jfa.2004.09.002
State	Published - Apr 15 2005
Externally published	Yes

Keywords

K-convexity
Noncommutative random variable
Operator space
Type and cotype

ASJC Scopus subject areas

Analysis

Online availability

10.1016/j.jfa.2004.09.002

Library availability

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Cite this

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title = "The norm of sums of independent noncommutative random variables in Lp(ℓ1)",

abstract = "We investigate the norm of sums of independent vector-valued random variables in noncommutative Lp spaces. This allows us to obtain a uniform family of complete embeddings of the Schatten class Sqn in Sp (ℓq m) with optimal order m ∼ n2. 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 B-convexity and K-convexity are equivalent.",

keywords = "K-convexity, Noncommutative random variable, Operator space, Type and cotype",

author = "Marius Junge and Javier Parcet",

note = "Funding Information: ∗ Corresponding author. Fax: +34 91 497 48 89. E-mail addresses: [email protected] (M. Junge), [email protected] (J. Parcet). 1Partially supported by the NSF DMS-0301116. 2Partially supported by the Project BFM 2001/0189, Spain.",

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AU - Junge, Marius

AU - Parcet, Javier

N1 - Funding Information: ∗ Corresponding author. Fax: +34 91 497 48 89. E-mail addresses: [email protected] (M. Junge), [email protected] (J. Parcet). 1Partially supported by the NSF DMS-0301116. 2Partially supported by the Project BFM 2001/0189, Spain.

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N2 - We investigate the norm of sums of independent vector-valued random variables in noncommutative Lp spaces. This allows us to obtain a uniform family of complete embeddings of the Schatten class Sqn in Sp (ℓq m) with optimal order m ∼ n2. 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 B-convexity and K-convexity are equivalent.

AB - We investigate the norm of sums of independent vector-valued random variables in noncommutative Lp spaces. This allows us to obtain a uniform family of complete embeddings of the Schatten class Sqn in Sp (ℓq m) with optimal order m ∼ n2. 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 B-convexity and K-convexity are equivalent.

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