In this paper, we show that the action of a characteristically simple, non-extremely amenable (non-strongly amenable, non-amenable) group on its universal minimal (minimal proximal, minimal strongly proximal) flow is effective. We present necessary and sufficient conditions, for the action of a topological group with trivial center on its universal minimal proximal flow, to be effective. A theorem of Furstenberg about the isomorphism of the universal minimal proximal flows of a discrete group and its subgroups of finite index ([Proximal flows, Springer-Verlag, Berlin-New York, 1976]) is strengthened. Finally, for a pair of groups H < G the same method is applied in order to extend the action of H on its universal minimal proximal flow to an action of its commensurator group CommG(H).
- Effective action
- Strong/extreme amenability
- Universal minimal proximal flow