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On uniqueness of invariant means
Authors:M. B. Bekka
Affiliation:Département de Mathématiques, Université de Metz, F--57045 Metz, France
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 $L^{infty } (G/Gamma )$.

Keywords:Invariant means on compact groups   $p$--adic groups   Selberg inequality   lattices in semisimple Lie groups   linear algebraic groups
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