On uniqueness of invariant means

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 $L^{\infty } (G/\Gamma )$.