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

摘要

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 tau, and let E be an r.i.. space on (0, infinity). 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 R corresponds a positive norm one functional f is an element of E-(2) (M)* such that for all x is an element of E(M) parallel to T(x)parallel to(2) <= K-2 parallel to T parallel to(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 < infinity. We apply these results to the study of Schur multipliers from a 2-convex unitary ideal into a 2-concave one. (c) 2006 Elsevier Inc. All rights reserved.