On individual ergodic theorems for semifinite von Neumann algebras

摘要

It is known that, for a positive Dunford-Schwartz operator in a noncommutative L-p-space, 1 <= p < infinity, 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 blaterally almost uniformly in each noncommutative symmetric space E such that mu(t)(x) -> 0 as t -> infinity for every x is an element of E, where mu(t)(x) is the 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. (C) 2020 Elsevier Inc. All rights reserved.