Convergence Factors and Compactness in Weighted Convolution Algebras

We study convergence in weighted convolution algebras L1(ω) on R+, with the weights chosen such that the corresponding weighted space M(ω) of measures is also a Banach algebra and is the dual space of a natural related space of continuous functions. We determine convergence factor ɳ for which weak*-convergence of {λn} to λ in M(ω) implies norm convergence of λn * f to λ * f in L1(ωɳ). We find necessary and sufficent conditions which depend on ω and f and also find necessary and sufficent conditions for ɳ to be a convergence factor for all L1(ω) and all f in L1(ω). We also give some applications to the structure of weighted convolution algebras. As a preliminary result we observe that ɳ is a convergence factor for ω and f if and only if convolution by f is a compact operator from M(ω) (or L1(ω)) to L1(ωɳ).