Uniqueness of the invariant mean on Abelian topological semigroups

Abstract

|lv|| = 1 = v(e) where e is the function which is identically one. An element v in C(S)* is called left-invariant if l*v = v for every a in S and is called right-invariant if *V= v for every a C S. We say that v is invariant if l *v = v= r *v for every a in S. S is called amenable if there exists an invariant mean. It is known that an Abelian topological semigroup is amenable. In my earlier paper [21 1 proved that a discrete Abelian semigroup has a unique invariant mean if and only if it has a finite ideal. It is quite reasonable to conjecture that in general an Abelian topological semigroup has a unique invariant mean if and only if the semigroup has a compact ideal. In this paper we prove the conjecture in certain special situations.