Rademacher averages on noncommutative symmetric spaces

TL;DR

This paper develops Khintchine type inequalities for Rademacher averages in noncommutative symmetric spaces, providing norm equivalences and dual estimates for these averages in the context of von Neumann algebras.

Contribution

It introduces new inequalities and norm estimates for Rademacher averages in noncommutative symmetric function spaces, extending classical results to a noncommutative setting.

Findings

Established general Khintchine type inequalities in noncommutative symmetric spaces.

Proved norm equivalences for Rademacher averages when E is 2-concave.

Provided dual estimates for E spaces with 2-convexity and non-trivial Boyd index.

Abstract

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).