Individual ergodic theorems for semifinite von Neumann algebras

TL;DR

This paper extends individual ergodic theorems to a broad class of noncommutative symmetric spaces, demonstrating almost uniform convergence of ergodic averages and establishing multiparameter and Wiener-Wintner type results.

Contribution

It introduces new convergence results for ergodic averages in noncommutative symmetric spaces, generalizing previous known cases and including multiparameter and Wiener-Wintner theorems.

Findings

Almost uniform convergence in noncommutative symmetric spaces.

Extension of ergodic theorems to multiparameter settings.

Almost uniform convergence in noncommutative Wiener-Wintner theorem.

Abstract

It is known that, for a positive Dunford-Schwartz operator in a noncommutative $L^p-$space, $1\leq p<\infty$ or, more generally, in a noncommutative Orlicz space with order continuous norm, the corresponding ergodic averages converge bilaterally almost uniformly. We show that these averages converge almost uniformly in each noncommutative symmetric space $E$ such that $\mu_t(x) \to 0$ as $t \to 0$ for every $x \in E$, where $\mu_t(x)$ is a non-increasing rearrangement of $x$. Noncommutative Dunford-Schwartz-type multiparameter ergodic theorems are studied. A wide range of noncommutative symmetric spaces for which Dunford-Schwartz-type individual ergodic theorems hold is outlined. Also, almost uniform convergence in noncommutative Wiener-Wintner theorem is proved.