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) {norm of matrix} T (x) {norm of matrix}² ≤ K² {norm of matrix} T {norm of matrix}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.