On the structures of locally compact groups admitting inner invariant means

Abstract

For locally compact groups G, the author introduced a notion of [IA] groups that is if there exists an inner invariant mean on G. This concept generalizes the concept of amenability for locally compact groups and the concept of inner amenability for discrete groups, hence gives a new classification of locally compact groups. The purpose of this paper is to characterize [IA] groups by generalizing the well-known Folner's conditions. Our main result is the following equivalence: A locally compact group G is [IA] if and only if G admits measurable subsets having finite non-zero Haar measure which are not substantially modified by conjugations of the gorup.