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On uniqueness of invariant means

Author: M. B. Bekka
Journal: Proc. Amer. Math. Soc. 126 (1998), 507-514
MSC (1991): Primary 43A07, 22E40
MathSciNet review: 1415573
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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 $L^{\infty } (G/\Gamma )$.

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Additional Information

M. B. Bekka
Affiliation: Département de Mathématiques, Université de Metz, F–57045 Metz, France

Keywords: Invariant means on compact groups, $p$--adic groups, Selberg inequality, lattices in semisimple Lie groups, linear algebraic groups
Received by editor(s): May 6, 1996
Received by editor(s) in revised form: August 12, 1996
Dedicated: Dedicated to Professor Eberhard Kaniuth on the occasion of his 60th birthday
Communicated by: Roe Goodman
Article copyright: © Copyright 1998 American Mathematical Society

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