Multipliers on L(S), L(S)**, and LUC(S)* for a locally compact topological semigroup

Abstract

We study compact and weakly compact multipliers on L(S), L(S)**, and LUC(S)*, where the latter is the dual of LUC(S). We show that a left cancellative semigroup S is left amenable if and only if there is a nonzero compact (or weakly compact) multiplier on L(S)**. We also prove that S is left amenable if and only if there is a nonzero compact (or weakly compact) multiplier on LUC(S)*.