Dimension-free maximal inequalities for noncommutative spherical means over cyclic groups

TL;DR

This paper proves dimension-free $L_p$ bounds for noncommutative spherical means over cyclic groups, extending spectral techniques and applying results to automorphism actions on von Neumann algebras.

Contribution

It introduces a noncommutative spectral technique to establish dimension-free maximal inequalities for spherical means over cyclic groups.

Findings

Dimension-free $L_p$ estimates for noncommutative spherical means.

Extension of spectral techniques to noncommutative settings.

Application to automorphism actions on von Neumann algebras.

Abstract

In this paper, we establish dimension-free $L_p$-estimates for operator-valued maximal spherical means over cyclic groups $\Z_{m+1}^d$ for all $p>1$ and $m\geq1$. The key ingredient is a noncommutative extension of the spectral technique developed by Nevo and Stein. As an application, we obtain a noncommutative spherical maximal inequality for automorphism actions on von Neumann algebras, along with several concrete examples.