Noncommutative Maximal Ergodic Theorems

The connection between ergodic theory and the theory of von Neumann algebras goes back to the very beginning of the theory of “rings of operators”. Maximal inequalities in ergodic theory provide an important tool in classical analysis. In this paper we prove the noncommutative analogue of the classical Dunford-Schwartz maximal ergodic theorem, thereby connecting these different aspects of ergodic theory. At the early stage of noncommutative ergodic theory, only mean ergodic theorems have been obtained (cf., e.g., [Ja1, Ja2] for more information). The study of individual ergodic theorems really took off with Lance’s pioneering work [L]. Lance proved that the ergodic averages associated with an automorphism of a σ-finite von Neumann algebra which leaves invariant a normal faithful state converge almost uniformly. Lance’s ergodic theorem was extensively extended and improved by (among others) Conze, Dang-Ngoc [CoN], Kümmerer [Kü] (see [Ja1, Ja2] for more references). On the other hand, Yeadon [Ye] obtained a maximal ergodic theorem in the preduals of semifinite von Neumann algebras. Yeadon’s theorem provides a maximal ergodic inequality which might be understood as a weak type (1, 1) inequality. This inequality is the ergodic analogue of Cuculescu’s [Cu] result obtained previously for noncommutative martingales. We should point out that in