A characterization of semigroups admitting a unique multiplicative invariant mean (Q1821289)

In a semigroup S the subset T is called left thick if for any finite subset F of S there exists an element s in S such that Fs\(\subset T\). The authors establish the following properties: Let the semigroup S admit a multiplicative left invariant mean on the space of bounded real valued functions defined on S (the semigroup is extremely left amenable); this mean is unique if and only if every nonempty left thick set is a singleton. Any semigroup admits a unique multiplicative left invariant mean if and only if it admits a zero.