Maximal Inequality Associated to Doubling Condition for State Preserving Actions

TL;DR

This paper establishes maximal inequalities and ergodic theorems for state-preserving actions on von Neumann algebras under doubling condition groups, utilizing Hardy-Littlewood maximal inequality and transference techniques.

Contribution

It introduces new maximal inequality and ergodic theorems for von Neumann algebra actions with groups satisfying the doubling condition, extending classical results to a non-commutative setting.

Findings

Proved maximal inequalities for state-preserving actions.

Established ergodic theorems in the von Neumann algebra context.

Utilized Hardy-Littlewood maximal inequality and transference methods.

Abstract

In this article, we prove maximal inequality and ergodic theorems for state preserving actions on von Neumann algebra by an amenable, locally compact, second countable group equipped with the metric satisfying the doubling condition. The key idea is to use Hardy-Littlewood maximal inequality, a version of the transference principle, and certain norm estimates of differences between ergodic averages and martingales.