Rademacher averages on noncommutative symmetric spaces

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 $(\epsilon_k)_k$ be a Rademacher sequence, on some probability space $\Omega$. For finite sequences $(x_k)_k of E(M), we consider the Rademacher averages$\sum_k \epsilon_k\otimes x_k$as elements of the noncommutative function space$E(L^\infty(\Omega)\otimes M)$and study estimates for their norms$\Vert \sum_k \epsilon_k \otimes x_k\Vert_E$calculated in that space. We establish general Khintchine type inequalities in this context. Then we show that if E is 2-concave, the latter norm is equivalent to the infimum of$\Vert (\sum y_k^*y_k)^{{1/2}}\Vert + \Vert (\sum z_k z_k^*)^{{1/2}}\Vert$over all$y_k,z_k$in E(M) such that$x_k=y_k+z_k$ for any k. Dual estimates are given when E is 2-convex and has a non trivial upper Boyd index. We also study Rademacher averages for doubly indexed families of E(M).