Ergodic Group Actions with Nonunique Invariant Means

Abstract

Let $M(X,G)$ be the set of $G$-invariant means on ${L^\infty }(X,\mathcal {B},P)$, where $G$ is a countable group acting ergodically as measure preserving transformations on a nonatomic probability space $(X,\mathcal {B},P)$. We show that if there exists $\mu \in M(X,G),\mu \ne P$, then $M(X,G)$ contains an isometric copy of $\beta N\backslash N$, where $\beta N\backslash N$ is considered as a subset of ${({l^\infty })^*}$. This provides an answer to a question raised by J. Rosenblatt in 1981.