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

Abstract:

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