Noncommutative weighted individual ergodic theorems with continuous time

TL;DR

This paper proves that ergodic flows in noncommutative symmetric spaces, generated by continuous semigroups of positive operators and modulated by almost periodic functions, converge almost uniformly, extending ergodic theorems to a noncommutative continuous-time setting.

Contribution

It establishes noncommutative weighted ergodic theorems with continuous time, involving semigroups of positive operators and Besicovitch almost periodic functions, which is a novel extension.

Findings

Almost uniform convergence of ergodic flows in noncommutative symmetric spaces.

Extension of local ergodic theorems to noncommutative continuous-time frameworks.

Abstract

We show that ergodic flows in noncommutative fully symmetric spaces (associated with a semifinite von Neumann algebra) generated by continuous semigroups of positive Dunford-Schwartz operators and modulated by bounded Besicovitch almost periodic functions converge almost uniformly. The corresponding local ergodic theorem is also discussed.