This is an announcement for the paper "The little Grothendieck theorem and Khintchine inequalities for symmetric spaces of measurable operators" by Fran\c coise Lust-Piquard and Quanhua Xu.
Abstract: 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.
Archive classification: Functional Analysis; Operator Algebras
Mathematics Subject Classification: Primary 46L52; Secondary 46L50; 47A63
Remarks: 14 pages. To appear in J. Funct. Anal
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Submitted from: qx@math.univ-fcomte.fr
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