Noncommutative Ergodic Theorems for Connected Amenable Groups

TL;DR

This paper establishes noncommutative ergodic theorems for connected amenable groups by transferring the problem to $R^d$ actions and proving maximal inequalities in specific noncommutative function spaces.

Contribution

It extends ergodic theorems to noncommutative settings for connected amenable groups using structure theorems and multi-parameter convergence analysis.

Findings

Maximal ergodic inequalities on $L_1(\\mathcal{M})$ and $L_1\log^{2(d-1)}L(\\mathcal{M})$

Ergodic theorems for group actions along specific sequences

Transfer of results from discrete to continuous group actions

Abstract

This paper is devoted to the study of noncommutative ergodic theorems for connected amenable locally compact groups. For a dynamical system $(\mathcal{M},\tau,G,\sigma)$, where $(\mathcal{M},\tau)$ is a von Neumann algebra with a normal faithful finite trace and $(G,\sigma)$ is a connected amenable locally compact group with a well defined representation on $\mathcal{M}$, we try to find the largest noncommutative function spaces constructed from $\mathcal{M}$ on which the individual ergodic theorems hold. By using the Emerson-Greenleaf's structure theorem, we transfer the key question to proving the ergodic theorems for $\mathbb{R}^d$ group actions. Splitting the $\mathbb{R}^d$ actions problem in two cases according to different multi-parameter convergence types---cube convergence and unrestricted convergence, we can give maximal ergodic inequalities on $L_1(\mathcal{M})$ and on noncommutative Orlicz space $L_1\log^{2(d-1)}L(\mathcal{M})$, each of which is deduced from the result already known in discrete case. Finally we give the individual ergodic theorems for $G$ acting on $L_1(\mathcal{M})$ and on $L_1\log^{2(d-1)}L(\mathcal{M})$, where the ergodic averages are taken along certain sequences of measurable subsets of $G$.