The weighted compactification of a locally compact group and topological centres

Abstract

Let G be a locally compact group. It is well-known that G^LUC, the spectrum of the algebra of left uniformly continuous functions on G, the so-called LUC-compactification of G, is a semigroup with product restricted from the Arens product on LUC(G)^*. Now consider the algebra of weighted left uniformly continuous functions on G, LUC(G,w^-1). The spectrum G^LUC_w is a compactification of G homeomorphic to G^LUC, but is not a semigroup unless the weight is a homomorphism (in which case G^LUC_w = \G^LUC). We study the algebraic and topological properties of G^LUC_w and the semigroup it generates in [0,1]G^LUC_w, including characterizing when it is dense, and use the results to attempt to extend some topological centre and determination results for G^LUC of Budak, Isik, and Pym to G^LUC_w and present some partial results. We also partially characterize the isometric isomorphisms of Beurling (weighted group) algebras. Finally, we show that the topological centre of the Fourier algebra of the Fell group is strongly Arens irregular.