Individual ergodic theorems for semifinite von Neumann algebras

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 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.

arXiv commentarXiv admin note: text overlap with arXiv:1604.00851 The last section of the three last versions contains a mistake. Additionally, we are planning to submit a version with a better presentation of the study