Γ-amenability of locally compact groups

Let G be a locally compact group and let Γ be a closed subgroup of G × G. Pier introduced the notion of Γ-amenability which gives a new classification of groups. This concept generalizes the concept of amenability and inner amenability for locally compact groups. In this paper, among other things, we extend some standard results for amenable groups to Γ-amenable groups and give various characterizations for Γ-amenable groups. A sequence of characterizations of Γ-amenable groups is given here by developing the well-known Følner’s conditions for amenable locally compact groups. Several characterizations of inner amenability are also given.