On the sup-norm closure of the \(L^ \infty\)-representation algebra \(\mathcal{R}\mathcal(S)\) of a foundation semigroup \(\mathcal S\) (Q1914087)

The paper aims to establish a result previously known for discrete semigroups and locally compact groups. We assume that \(S\) is a commutative locally compact semigroup which is separative and foundation, so that \(S\) has plenty of continuous semicharacters and \(M(S)\) contains a subalgebra \(M_a (S)\) playing a role analogous to the group algebra of a locally compact group. It is shown that a bounded continuous function is in the uniform closure of the \(L^\infty\)-representation algebra if and only if \(\int fd \lambda_n \to 0\) whenever \((\lambda_n)\) is a sequence in the unit ball of \(M_a(S)\) whose Gelfand transforms tend to 0 pointwise on \(\widehat S\).