THE TOPOLOGICAL CENTER OF WEIGHTED SEMIGROUP ALGEBRAS WITH A STRICT TOPOLOGY

For a family of a locally compact semigroup S with a weight function ω, we have recently introduced and studied some locally convex topologies τ on the weighted semigroup algebra Ma(S, ω) and shown that the strong dual of (Ma(S, ω), τ) can be identified with a Banach space of certain functions on S. In this paper, we shall be concerned with the second dual of (Ma(S, ω), τ); using this duality, we first introduce and study an Arens multiplication on the second dual of (Ma(S, ω), τ). We then investigate the topological center of (Ma(S, ω), τ) for an extensive class of locally compact semigroups S. As a consequence, we conclude some results on Arens regularity and strong Arens irregularity of (Ma(S, ω), τ).