Arch. Math. 86 (2006) 437–448 0003–889X/06/050437–12 DOI 10.1007/s00013-005-1496-6 © Birkh¨ auser Verlag, Basel, 2006 Archiv der Mathematik Strong Arens irregularity of Beurling algebras with a locally convex topology By S. Maghsoudi, R. Nasr-Isfahani and A. Rejali 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 L 1 (G, ω) equipped with some locally convex topologies τ . Here we use this duality to introduce an Arens multiplication on (L 1 (G, ω), τ ) ∗∗ , and prove that the topological center of (L 1 (G, ω), τ ) ∗∗ is L 1 (G, ω); this enables us to conclude that (L 1 (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 . 1. Introduction. Throughout this paper, G denotes a locally compact group with a fixed left Haar measure λ. We also assume that ω be a weight function on G, that is a continuous function ω : G → [1, ∞) with ω(xy)  ω(x) ω(y) (x,y ∈ G). The Beurling algebra L 1 (G, ω) is defined to be the space of all measurable functions ϕ such that ωϕ ∈ L 1 (G), the group algebra of G as defined in [12]. Then L 1 (G, ω) with the convolution product ∗ and the norm . 1,ω defined by ϕ 1,ω =ωϕ 1 (ϕ ∈ L 1 (G, ω)) is a Banach algebra. We follow Dales and Lau [4] in our definitions and notations for Beurling algebras. Also, let L ∞ (G, 1/ω) denote the space of all measurable functions f with f/ω ∈ L ∞ (G), the Lebesgue space as defined in [12]. Then L ∞ (G, 1/ω) with the product . ω defined by f. ω g = fg/ω (f, g ∈ L ∞ (G, 1/ω)), the norm . ∞,ω defined by f  ∞,ω =f/ω ∞ (f ∈ L ∞ (G, 1/ω)), Mathematics Subject Classification (2000): Primary 43A20, 46H05; Secondary 43A10, 43A15, 46A03.