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Article Dans Une Revue Journal of Functional Analysis Année : 2007

The little Grothendieck theorem and Khintchine inequalities for symmetric spaces of measurable operators

Résumé

We prove the little Grothendieck theorem for any 2-convex noncommutative symmetric space. Let $\M$ be a von Neumann algebra equipped with a normal faithful semifinite trace $\t$, and let $E$ be an r.i. space on $(0, \8)$. Let $E(\M)$ be the associated symmetric space of measurable operators. Then to any bounded linear map $T$ from $E(\M)$ into a Hilbert space $\mathcal H$ corresponds a positive norm one functional $f\in E_{(2)}(\M)^*$ such that $$\forall x\in E(\M)\quad \|T(x)\|^2\le K^2 \|T\|^2 f(x^*x+xx^*),$$ where $E_{(2)}$ denotes the 2-concavification of $E$ and $K$ is a universal constant. As a consequence we obtain the noncommutative Khintchine inequalities for $E(\M)$ when $E$ is either 2-concave or 2-convex and $q$-concave for some $q<\8$. We apply these results to the study of Schur multipliers from a 2-convex unitary ideal into a 2-concave one.

Dates et versions

hal-00477042 , version 1 (27-04-2010)

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Françoise Lust-Piquard, Quanhua Xu. The little Grothendieck theorem and Khintchine inequalities for symmetric spaces of measurable operators. Journal of Functional Analysis, 2007, 244, pp.488-503. ⟨hal-00477042⟩
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