Almost Uniform Convergence in Noncommutative Dunford-Schwartz Ergodic Theorem for $p>1$

TL;DR

This paper proves that ergodic Cesàro averages generated by positive Dunford-Schwartz operators in noncommutative L^p spaces for p>1 converge almost uniformly, extending previous results from the case p=1 to p>1.

Contribution

It establishes almost uniform convergence of ergodic averages in noncommutative L^p spaces for p>1, generalizing earlier results for p=1.

Findings

Almost uniform convergence of ergodic averages for p>1

Extension of Yeadon's results from p=1 to p>1

Advancement in noncommutative ergodic theory

Abstract

We prove that the ergodic Ces\' aro averages generated by a positive Dunford-Schwartz operator in a noncommutative space $L^p(\mathcal M,\tau)$, $1<p<\infty$, converge almost uniformly (in Egorov's sense). This problem goes back to the original paper of Yeadon \cite{ye}, where bilaterally almost uniform convergence of these averages was established for $p=1$.