On ergodic theorems for free group actions on noncommutative spaces

Abstract

Abstract. We extend in a noncommutative setting the individual ergodic theorem of Nevo and Stein concerning measure preserving actions of free groups and averages on spheres s 2n of even radius. Here we study state preserving actions of free groups on a von Neumann algebra A and the behaviour of (s 2n (x)) for x in noncommutative spaces L p (A). For the Cesàro means 1 n n−1 k=0 s k and p = +∞, this problem was solved by Walker. Our approach is based on ideas of Bufetov. We prove a noncommutative version of Rota "Alternierende Verfahren" theorem. To this end, we introduce specific dilations of the powers of some noncommutative Markov operators.

A Markov dilation for self-adjoint Schur multipliers

“…In our concrete setting, one can avoid this abstract construction (and free products). We follow the notation of [1] and classical notation for infinite tensor products (we drop the completion symbol). Let M = B(…”

“…Therefore Claire Anantharaman-Delaroche introduces the notion of factorizable maps in [1], which we describe precisely now. We will use classical notation for von Neumann algebras as in [13], [1].…”

Maximal ergodic theorems for some group actions

“…Specializing all these to free group actions, we deduce the noncommutative analogues of Nevo-Stein's results. These noncommutative results for a free group were proved independently and almost at the same time by Anantharaman [1]. Her arguments follow the approach set up by Bufetov and Grigorchuk mentioned previously, so they are completely different from ours.…”

“…Applying the results about the convergences as above to free group actions in Example 2.5, we can get the noncommutative Nevo-Stein ergodic theorem, which was proved by Anantharaman-Delaroche [1] independently at the same time. While we use some theories in analysis (for example, spectral theory) through the Nevo-Stein method, her work is based on the method of Bufetov [2], which uses Markov chains and some knowledge in probability.…”

“…Let x ∈ L 0 (N , τ ) and x = u|x| be the polar decomposition of x, where |x| = (x * x) 1 2 denotes the absolute value of x. Then x ∈ L p (M) if and only if x ∈ M and |x| p ∈ L 1 (M).…”