The Exposed Points of the Set of Invariant Means

Abstract

Let $G$ be a $\sigma$-compact infinite locally compact group, and let $LIM$ be the set of left invariant means on ${L^\infty }(G)$. We prove in this paper that if $G$ is amenable as a discrete group, then $LIM$ has no exposed points. We also give another proof of the Granirer theorem that the set $LIM(X,G)$ of $G$-invariant means on ${L^\infty }(X,\beta ,p)$ has no exposed points, where $G$ is an amenable countable group acting ergodically as measure-preserving transformations on a nonatomic probability space $(X,\beta ,p)$.