Standard homomorphisms and regulated weights on weighted convolution algebras

Abstract

In this paper we introduce a class of homomorphisms between weighted convolution algebras which we call standard homomorphisms and derive various equivalents of standardness. We also introduce the convergence ideal of a homomorphism. We find various descriptions of the convergence ideal together with its relation to standardness. We show that every continuous homomorphism from a weighted convolution algebra into another weighted convolution algebra, with a regulated weight, is standard, and when the algebras have both regulated weights the extension of a homomorphism to the weighted measure algebras satisfies additional continuity properties.