Characterization of amenable semigroups with a unique invariant mean
An amenable semigroup S has the property that, for any a,\(b\in S\), a\(S\cap bS\) and S\(a\cap Sb\) are non-void, and one can define a congruence \(a\sim b\) if and only if \(as=bs\) and \(ta=tb\) for some s,\(t\in S\). The author's main theorem here states that, for amenable S, the following are equivalent: (1) S has a unique left invariant mean. (2) S contains a finite two-sided ideal which is also a group. (3) S has a unique right invariant mean. (4) S/\(\sim\) is a finite group and ker \(\sim\) has a zero element.
scientific article; zbMATH DE number 3865926