On uniqueness of invariant means

Abstract

The following results on uniqueness of invariant means are shown: (i) Let G \mathbb {G} be a connected almost simple algebraic group defined over Q \mathbb {Q} . Assume that G ( R ) \mathbb {G}(\mathbb {R}) , the group of the real points in G \mathbb {G} , is not compact. Let p p be a prime, and let G ( Z p ) \mathbb {G}({\mathbb {Z}}_{p}) be the compact p p -adic Lie group of the Z p {\mathbb {Z}}_{p} –points in G \mathbb {G} . Then the normalized Haar measure on G ( Z p ) \mathbb {G}({\mathbb {Z}}_{p}) is the unique invariant mean on L ∞ ( G ( Z p ) ) L^{\infty }(\mathbb {G}({\mathbb {Z}}_{p})) . (ii) Let G G be a semisimple Lie group with finite centre and without compact factors, and let Γ \Gamma be a lattice in G G . Then integration against the G G –invariant probability measure on the homogeneous space G / Γ G/\Gamma is the unique Γ \Gamma –invariant mean on L ∞ ( G / Γ ) L^{\infty } (G/\Gamma ) .