Multipliers on \(L(S)\), \(L(S)^{**}\), and \(\mathrm{LUC}(S)^*\) for a locally compact topological semigroup (Q1607859)

Let \(S\) denote a locally compact, Hausdorff topological semigroup and \(M(S)\) the space of complex Borel measures on \(S\). \(L(S)\) denotes the set of all measures \(\mu\) in \(M(S)\) such that convolution on both the left and the right of \(|\mu |\) by the point mass \(\delta_x\) is continuous with respect to \(x\in S\). In the case that \(S\) is a topological group \(L(S)\) is just \(L^1(S)\). \(\mathrm{LUC}(S)\) denotes the set of bounded continuous functions \(f\) on \(S\) such that left translation of \(f,l_xf\), by \(x\) is norm continuous with respect to \(x\). The author shows that for left cancellative \(S\), \(S\) is amenable if and only if there is a nonzero compact (or weakly compact) multiplier on \(L(S)^{**}\). Also it is shown that \(S\) is left amenable if and only if there is a non-zero compact (or weakly compact) multiplier on \(\mathrm{LUC}(S)^*\).