Topological invariant means on locally compact semigroups (Q916952)

The article provides topological analogues of some results concerning discrete semigroups [\textit{T. Mitchell}, Trans. Am. Math. Soc. 119, 244- 261 (1965; Zbl 0146.120); \textit{W. R. Emerson} [ibid. 241, 183-194 (1978; Zbl 0384.43003)]. Let S be a topological semigroup; M(S) denotes the Banach algebra of all bounded regular Borel measures on S and \(M_ 0(S)\) is the subsemigroup of probability measures. If \(F\in M(S)^*\), \(\mu\in M(S)\), let \((\mu \odot F)(\nu)=F(\mu *\nu)\), \(\nu\in M(S)\). The mean M on \(M(S)^*\) is called topologically invariant if \(M(\mu \odot F)=M(F)\) whenever \(F\in M(S)^*\), \(\mu\in M(S)\). Existence of a topologically invariant mean is shown to be equivalent to the existence, for every pair of elements \(\mu_ 1\), \(\mu_ 2\) in M(S), of a mean M on \(M(S)^*\) such that \(M(\mu_ 1\odot F)=M(\mu_ 2\odot F)\) whenever \(F\in M(S)^*\).