Equivalent weights and standard homomorphisms for convolution algebras on ℝ⁺

Abstract

We take a second look at two basic topics in the study of weighted convolution algebras L¹(ω) on ℝ⁺. An early result showed that one could replace the weight $\omega$ with a very well-behaved weight without changing the space L¹(ω) as long as L¹(ω) was an algebraffi We prove the analogous result for measure algebras when M(ω) is an algebraffi This allows us to preserve not only the norm topology but also the relative weak* topology on L¹(ω). A homomorphism between weighted convolution algebras is said to be standard if it preserves generators of dense principal ideals. The original proofs of standardness and its variants are all based on finding the generator of a particular strongly continuous convolution semigroup. In this paper we give much simpler direct proofs of these results. We also improve the statement and proof of the theorem, giving useful properties equivalent to the standardness of a homomorphism.