Individual ergodic theorems in noncommutative symmetric spaces

Abstract

It is known that, for a positive Dunford-Schwartz operator in a noncommutative $L^p-$space, $1\le p<\infty$ 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.