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

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 τ, and let E be an r.i. space on (0,∞). 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 H corresponds a positive norm one functional f ∈ E(2)(M)∗ such that ∀x ∈ E(M) T (x) 2 K2 T 2f 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 qconcave for some q <∞. We apply these results to the study of Schur multipliers from a 2-convex unitary ideal into a 2-concave one. © 2006 Elsevier Inc. All rights reserved.