Rademacher averages on noncommutative symmetric spaces
by
Christian Le Merdy
Universite de Franche-Comte, Besancon, France
Coauthors: Fedor Sukochev (Flinders University, Australia)

Let E be a separable (or the dual of a separable) symmetric function space, let M be a semifinite von Neumann algebra and let E(M) be the associated noncommutative function space. Let (e_k)_{k ≥ 1} be a Rademacher sequence, on some probability space W. For finite sequences (x_k)_{k ≥ 1} of E(M), we consider the Rademacher averages ∑_ke_k⊗x_k as elements of the noncommutative function space E(L^∞(W)[`(⊗)] M) and study estimates for their norms ∥∑_k e_k⊗x_k∥_E calculated in that space. Our aim is to establish Khintchine type inequalities in this context. In particular we show that if E is 2-concave, then ∥∑_ke_k⊗x_k∥_E is equivalent to the infimum of ∥(∑y_k^* y_k)^1/2∥+ ∥(∑z_kz_k^*)^1/2∥ over all y_k, z_k in E(M) such that x_k=y_k+z_k for any k ≥ 1. Dual estimates are given when E is 2-convex and has a non trivial upper Boyd index. In this case, ∥∑_k e_k⊗x_k ∥_E is equivalent to ∥(∑x_k^* x_k)^1/2∥+ ∥(∑x_kx_k^*)^1/2∥.

Date received: May 1, 2008