Individual ergodic theorems in noncommutative symmetric spaces

TL;DR

This paper extends the convergence of ergodic averages to a broad class of noncommutative symmetric spaces, demonstrating bilaterally almost uniform convergence under specific conditions.

Contribution

It generalizes known convergence results from noncommutative $L^p$ and Orlicz spaces to all noncommutative symmetric spaces with order continuous norm.

Findings

Ergodic averages converge bilaterally almost uniformly in these spaces.

Convergence holds when the non-increasing rearrangement tends to zero.

Results unify convergence behavior across various noncommutative symmetric spaces.

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 bilaterally 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 $x$. In particular, these averages converge bilaterally almost uniformly in all noncommutative symmetric spaces with order continuous norm.