Non-Commutative Maximal Inequalities for State-Preserving Actions of amenable groups

TL;DR

This paper develops maximal inequalities and ergodic theorems for state-preserving actions of amenable groups on non-commutative L^1 spaces, extending classical ergodic theory into the non-commutative setting.

Contribution

It introduces new maximal inequalities and ergodic theorems for non-commutative spaces under amenable group actions, combining these with Neveu decomposition for stochastic ergodic results.

Findings

Established maximal inequalities for non-commutative L^1 spaces

Derived ergodic theorems for state-preserving actions

Obtained a stochastic ergodic theorem using Neveu decomposition

Abstract

In this article, we establish maximal inequalities and deduce ergodic theorems for state-preserving actions of amenable, locally compact, second-countable groups on tracial non-commutative $L^1$-spaces. As a further consequence, in combination with the Neveu decomposition, we obtain a stochastic ergodic theorem for amenable group actions.