Noncommutative ergodic theorems for action of semisimple Lie groups

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.