Noncommutative ergodic theorems for action of semisimple Lie groups

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 $G\mathrm{/}K$, in the setting where $G$ acts by trace-preserving $\ast$-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 Calderón 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 $G\mathrm{/}K$ for certain class of semisimple Lie groups.