On the second conjugate of the weighted semigroup algebra for a locally compact semigroup

Abstract

Let S be a topological semigroup, ω be a weight function on S, and M _a (S, ω) be the weighted semigroup algebra of S. Let (S; M _a (S, ω)) be the C *-algebra of all M _a (S, ω)-measurable functions g on S such that g/ω vanishes at infinity. Here, we consider (S; M _a (S, ω))* as a closed subalgebra of M _a (S, ω)** equipped with the first Arens product, and study the relation between these two algebras. In particular, we show that (S; M _a (S, ω))* is a closed ideal of M _a (S, ω)** only when S is compact.