Noncommutative maximal ergodic theorems

This paper is devoted to the study of various maximal ergodic theorems in noncommutative -spaces. In particular, we prove the noncommutative analogue of the classical Dunford-Schwartz maximal ergodic inequality for positive contractions on and the analogue of Stein's maximal inequality for symmetric positive contractions. We also obtain the corresponding individual ergodic theorems. We apply these results to a family of natural examples which frequently appear in von Neumann algebra theory and in quantum probability.