On individual ergodic theorems for semifinite von Neumann algebras

It is known that, for a positive Dunford-Schwartz operator in a noncommutative Lp-space, 1≤p<∞, 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 μt(x)→0 as t→∞ for every x∈E, where μt(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.