EQUIVALENT WEIGHTS AND STANDARD HOMOMORPHISMS FOR CONVOLUTION ALGEBRAS ON R+

We take a second look at two basic topics in the study of weighted convolution algebras L-1(omega) on R+. An early result showed that one could replace the weight w with a very well-behaved weight without changing the space L-1(omega) as long as L-1(omega) was an algebra. We prove the analogous result for measure algebras when M(omega) is an algebra. This allows us to preserve not only the norm topology but also the relative weak* topology on L-1(omega). 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.