Noncommutative ergodic theorems for action of semisimple Lie groups

TL;DR

This paper establishes noncommutative ergodic theorems for actions of rank-one semisimple Lie groups, using spectral analysis, maximal inequalities, and convergence techniques in the setting of von Neumann algebras.

Contribution

It introduces new noncommutative maximal inequalities and pointwise ergodic theorems for semisimple Lie group actions, extending classical results to the noncommutative setting.

Findings

Proves noncommutative maximal inequalities for spherical and ball averages.

Establishes pointwise ergodic theorems with bilateral almost uniform convergence.

Extends results to higher-rank groups using Property (T) and spectral gaps.

Abstract

Let $G$ be a connected simple Lie group of real rank one and finite center, and let $K$ be a maximal compact subgroup. We study the families of spherical, ball, and uniform averages $(\sigma_t)_{t>0}$, $(\beta_t)_{t>0}$, and $(\mu_t)_{t>0}$ on $G$ induced by the canonical $G$-invariant metric on $G/K$, in the setting where $G$ acts by trace-preserving $*$-automorphisms on a finite von Neumann algebra $(\mathcal M,\tau)$. For the associated noncommutative $L_p$-spaces $L_p(\mathcal M)$, we consider both local and global noncommutative maximal inequalities for these averages, and corresponding pointwise ergodic theorems in the sense of bilateral almost uniform convergence. Our approach combines a noncommutative Calder\'on transfer principle, spectral analysis for the Gelfand pair $(G,K)$ via Harish--Chandra's spherical functions, fractional integration methods, and Littlewood--Paley $g$-function estimates. This work is complemented by our results for higher-rank semisimple Lie groups, where the presence of Property~(T) and associated spectral gaps serve as the key tools in establishing Wiener-type noncommutative pointwise ergodic theorems and noncommutative $L_p$-maximal inequalities for ball and spherical averages on $G/K$ for certain class of semisimple Lie groups.