On uniqueness of invariant means
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- by M. B. Bekka PDF
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Abstract:
The following results on uniqueness of invariant means are shown: (i) Let $\mathbb {G}$ be a connected almost simple algebraic group defined over $\mathbb {Q}$. Assume that $\mathbb {G}(\mathbb {R})$, the group of the real points in $\mathbb {G}$, is not compact. Let $p$ be a prime, and let $\mathbb {G}({\mathbb {Z}}_{p})$ be the compact $p$-adic Lie group of the ${\mathbb {Z}}_{p}$–points in $\mathbb {G}$. Then the normalized Haar measure on $\mathbb {G}({\mathbb {Z}}_{p})$ is the unique invariant mean on $L^{\infty }(\mathbb {G}({\mathbb {Z}}_{p}))$. (ii) Let $G$ be a semisimple Lie group with finite centre and without compact factors, and let $\Gamma$ be a lattice in $G$. Then integration against the $G$–invariant probability measure on the homogeneous space $G/\Gamma$ is the unique $\Gamma$–invariant mean on $L^{\infty } (G/\Gamma )$.References
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Additional Information
- M. B. Bekka
- Affiliation: Département de Mathématiques, Université de Metz, F–57045 Metz, France
- MR Author ID: 33840
- Email: bekka@poncelet.univ-metz.fr
- Received by editor(s): May 6, 1996
- Received by editor(s) in revised form: August 12, 1996
- Communicated by: Roe Goodman
- © Copyright 1998 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 126 (1998), 507-514
- MSC (1991): Primary 43A07, 22E40
- DOI: https://doi.org/10.1090/S0002-9939-98-04044-1
- MathSciNet review: 1415573
Dedicated: Dedicated to Professor Eberhard Kaniuth on the occasion of his 60th birthday