Noncommutative maximal ergodic inequalities for amenable groups

TL;DR

This paper establishes noncommutative maximal ergodic inequalities and a pointwise ergodic theorem for amenable group actions, introducing new techniques like a well-behaved filtration and square function bounds.

Contribution

It introduces novel noncommutative maximal inequalities and ergodic theorems for amenable groups, utilizing quasi-tilings and non-doubling Calderón-Zygmund decompositions.

Findings

Proved a pointwise ergodic theorem for amenable groups in noncommutative spaces.

Established maximal inequalities and variational ergodic inequalities.

Developed new technical tools like a well-behaved filtration and square function estimates.

Abstract

We prove a pointwise ergodic theorem and a maximal inequality for actions of amenable groups on noncommutative measure spaces. To do so, we establish a square function estimate quantifying the difference between ergodic averages and some conditional expectations. Our main technical results are the construction of a well-behaved filtration, based on the quasi-tilings of Ornstein and Weiss, and the square function bound, which we derive from non-doubling noncommutative Calder\'on-Zygmund decomposition. For actions on usual measure spaces, we obtain new variational ergodic inequalities and jump estimates.