On a geometric property of the set of invariant means on a group

Abstract

If G is a discrete group and x ∈ G x \in G then x ∼ x^\sim denotes the homeomorphism of β G \beta G onto β G \beta G induced by left multiplication by x. A subset K of β G \beta G is said to be invariant if it is closed, nonempty and x ∼ ∅ K ⊂ K x^\sim \emptyset K \subset K for each x ∈ G x \in G . Let M L ( G ) ML(G) denote the set of left invariant means on G. (They can be considered as measures on β G \beta G .) Let G be a countably infinite amenable group and let K be an invariant subset of β G \beta G . Then the nonempty w ∗ {w^ \ast } -compact convex set M ( G , K ) = { ϕ ∈ M L ( G ) : suppt ϕ ⊂ K } M(G,K) = \{ \phi \in ML(G):{\text {suppt}}\phi \subset K\} has no exposed points (with respect to w ∗ {w^ \ast } -topology). Therefore, it is infinite dimensional.