Strong Arens irregularity of Beurling algebras with a locally convex topology

Abstract.

Let G be a locally compact group with a weight function ω. Recently, we have shown that the Banach space L ^∞₀ (G,1/ω) can be identified with the strong dual of L¹(G, ω)equipped with some locally convex topologies τ. Here we use this duality to introduce an Arens multiplication on (L¹(G, ω), τ)**, and prove that the topological center of (L¹(G, ω), τ)** is (L¹(G, ω); this enables us to conclude that (L¹(G, ω), τ) is Arens regular if and only if G is discrete. We also give a characterization for Arens regularity of L ^∞₀ (G, 1/ω)¹.