Almost uniform convergence in noncommutative Dunford-Schwartz ergodic theorem

TL;DR

This paper proves that ergodic Cesàro averages generated by positive Dunford-Schwartz operators in noncommutative L^p spaces converge almost uniformly, extending Yeadon's 1977 results from p=1 to all 1≤p<∞.

Contribution

It provides an affirmative solution to the longstanding open problem of almost uniform convergence for these averages in noncommutative L^p spaces.

Findings

Established almost uniform convergence for 1<p<∞

Extended Yeadon's bilaterally almost uniform convergence result from p=1

Solved a problem posed since 1977 in noncommutative ergodic theory

Abstract

This article gives an affirmative solution to the problem whether the ergodic Ces\'aro averages generated by a positive Dunford-Schwartz operator in a noncommutative space $L^p(\mathcal M,\tau)$, $1\leq p<\infty$, converge almost uniformly (in Egorov's sense). This problem goes back to the original paper of Yeadon, published in 1977, where bilaterally almost uniform convergence of these averages was established for $p=1$.