Noncommutative ergodic averages of balls and spheres over Euclidean spaces

TL;DR

This paper develops a noncommutative analogue of Calderón's transference principle to derive ergodic maximal inequalities and pointwise convergence results for Euclidean spaces, extending classical ergodic theorems to the noncommutative setting.

Contribution

It introduces a noncommutative transference principle and applies it to establish dimension-free ergodic inequalities and individual ergodic theorems in noncommutative analysis.

Findings

Established noncommutative maximal inequalities for Euclidean spaces.

Proved noncommutative Wiener's and Stein-Calderón's ergodic theorems.

Constructed dense subsets for pointwise convergence in noncommutative ergodic theory.

Abstract

In this paper, we establish a noncommutative analogue of Calder\'on's transference principle, which allows us to deduce noncommutative ergodic maximal inequalities from the special case---operator-valued maximal inequalities. As applications, we deduce dimension-free estimates of noncommutative Wiener's maximal ergodic inequality and noncommutative Stein-Calder\'on's maximal ergodic inequality over Euclidean spaces. We also show the corresponding individual ergodic theorems. To show Wiener's pointwise ergodic theorem, we construct a dense subset on which pointwise convergence holds following a somewhat standard way. To show Jones' pointwise ergodic theorem, we use again transference principle together with Littlewood-Paley method, which is different from Jones' original variational method that is still unavailable in the noncommutative setting.