Noncommutative ergodic theorems for action of semisimple Lie groups
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arXiv
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| Format: | Preprint |
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2025
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| author | hong, Guixiang Ray, Samya Kumar |
| author_facet | hong, Guixiang Ray, Samya Kumar |
| contents | 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 $(σ_t)_{t>0}$, $(β_t)_{t>0}$, and $(μ_t)_{t>0}$ 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 $(\mathcal M,τ)$. 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 $G/K$ for certain class of semisimple Lie groups. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_07444 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Noncommutative ergodic theorems for action of semisimple Lie groups hong, Guixiang Ray, Samya Kumar Operator Algebras Functional Analysis 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 $(σ_t)_{t>0}$, $(β_t)_{t>0}$, and $(μ_t)_{t>0}$ 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 $(\mathcal M,τ)$. 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 $G/K$ for certain class of semisimple Lie groups. |
| title | Noncommutative ergodic theorems for action of semisimple Lie groups |
| topic | Operator Algebras Functional Analysis |
| url | https://arxiv.org/abs/2508.07444 |