Noncommutative maximal ergodic theorems for positive contractions

LetMbe a von Neumann algebra equipped with a normal semifinite faithful trace τ. Let T be a positive linear contraction on M such that τ ◦ T τ and such that the numerical range of T as an operator on L2(M) is contained in a Stoltz region with vertex 1. We show that Junge and Xu’s noncommutative Stein maximal ergodic inequality holds for the powers of T on Lp(M), 1 < p ∞. We apply this result to obtain the noncommutative analogue of a recent result of Cohen concerning the iterates of the product of a finite number of conditional expectations. © 2008 Elsevier Inc. All rights reserved