Standard homomorphisms and convergent sequences in weighted convolution algebras
Fereidoun Ghahramani, Sandy Grabiner · Illinois Journal of Mathematics · 1992
In this paper we continue the study, from our joint paper [11] with J.P. McClure, of the relation between homomorphisms, semigroups, types of convergence, and closed ideals in weighted convolution algebras on the half-line R += [0, ).In particular, we are interested in the question of which continuous homomorphisms preserve dense principal ideals, which we showed to be equivalent [13], [14] to the question of which convolution semigroups are strongly continuous.We will call a positive Borel function o(x) on R / a weight if both w and 1/o are bounded on all finite intervals [0, a].The weight w(x) is an algebra weight if, in addition, w(x) is right continuous, o(0)= 1, and o(x + y)< w(x)o(y) for all x and y in R/.For a weight o, we let Ll(og) be the Banach space of those (equivalence classes of) locally integrable functions f on R / for which fo belongs to LI(R+), with the inherited norm Ilfll-Ilfllflf(t)[w(t) dr.The other weighted spaces we consider are defined analo- gously.Thus M(o) is the space of locally finite complex Borel measures on R + for which the norm I1 fR+o(t)dllx I(t) is finite; L(1/o) is the space of f for which f/o is in L(R+), with the inherited norm Ilfll ess sup If(t)l/o(t); and C0(1/o) is the closed subspace of L(1/o) com- prised of continuous functions with lim f(t)/o(t) 0. Occasionally we will consider Ll(o) and M(o) when o(x) is just a bounded non-negative Borel function.When o(x) is an algebra weight, Ll(o) is a Banach algebra under the convolution product f g(x) ff(x t)g(t) dr.Under the analogous con- volution of measures on R/, the space M(o) is also a Banach algebra which, under the usual identification of f with f(t)dt, contains Ll(o) as a closed ideal.Moreover [9, Th. 1.4], [13, Th. 2.2, p. 592], for our algebra weights we