On noncommutative ergodic theorems for semigroup and free group actions

TL;DR

This paper develops noncommutative ergodic theorems for actions of semigroups and free groups on von Neumann algebras, including maximal inequalities and pointwise convergence results in the noncommutative setting.

Contribution

It introduces noncommutative maximal ergodic inequalities and pointwise ergodic theorems for semigroup and free group actions on von Neumann algebras, extending classical results.

Findings

Established noncommutative maximal ergodic inequalities.

Proved noncommutative pointwise ergodic theorems for finite von Neumann algebras.

Extended ergodic theory to actions of alZ, alR, and free groups.

Abstract

In this article, we consider actions of \mathcal{Z}_+^d, \mathcal{R}_+^d and finitely generated free groups on a von Neumann algebras $M$ and prove a version of maximal ergodic inequality. Additionally, we establish non-commutative analogues of pointwise ergodic theorems for associated actions in the predual when M is finite.