Γ-amenability of locally compact groups

Let G be a locally compact group and let Gamma be a closed subgroup of G x G. Pier introduced the notion of Gamma-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 Gamma-amenable groups and give various characterizations for Gamma-amenable groups. A sequence of characterizations of Gamma-amenable groups is given here by developing the well-known Folner's conditions for amenable locally compact groups. Several characterizations of inner amenability are also given.