Noncommutative maximal ergodic theorems for spherical means on the Heisenberg group

TL;DR

This paper establishes noncommutative maximal ergodic theorems for spherical means on the Heisenberg group, extending classical results to noncommutative $L_p$ spaces with optimal scale in the reduced case.

Contribution

It introduces noncommutative maximal ergodic theorems for spherical averages on the Heisenberg group, generalizing classical ergodic theorems to a noncommutative framework.

Findings

Proved maximal ergodic theorems for spherical averages on noncommutative $L_p$ spaces.

Established individual and differential ergodic theorems in the noncommutative setting.

Achieved optimal scale results in the reduced Heisenberg group case.

Abstract

We prove maximal ergodic theorems for spherical averages on the Heisenberg groups acting on $L_p$ spaces over measure spaces not necessarily commutative, that is, on noncommutative $L_p$ spaces. The scale of $p$ is optimal in the reduced Heisenberg group case. We also obtain the corresponding individual ergodic theorems and differential theorems in the noncommutative setting. The results can be regarded as noncommutative analogues of Nevo-Thangavelu's ergodic theorems. The approach of proof involves recent developments in noncommutative $L_p$ spaces and in noncommutative harmonic analysis, in addition to the spectral theory used in the commutative setting.