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

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 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 ⩽ K 2 ‖ T ‖ 2 f ( x ∗ x + x x ∗ ) , 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 < ∞ . We apply these results to the study of Schur multipliers from a 2-convex unitary ideal into a 2-concave one.