Compactness and weak-star continuity of derivations on weighted convolution algebras

Abstract

Let ω be a continuous weight on R+ and let L1(ω) be the corresponding convolution algebra. By results of Grønbæk and Bade & Dales the continuous derivations from L1(ω) to its dual space L∞(1/ω) are exactly the maps of the form (Dφf)(t)=∫0∞f(s)st+sφ(t+s)ds(t∈R+ and f∈L1(ω)) for some φ∈L∞(1/ω). Also, every Dφ has a unique extension to a continuous derivation D¯φ:M(ω)→L∞(1/ω) from the corresponding measure algebra. We show that a certain condition on φ implies that D¯φ is weak-star continuous. The condition holds for instance if φ∈L0∞(1/ω). We also provide examples of functions φ for which D¯φ is not weak-star continuous. Similarly, we show that Dφ and D¯φ are compact under certain conditions on φ. For instance this holds if φ∈C0(1/ω) with φ(0)=0. Finally, we give various examples of functions φ for which Dφ and D¯φ are not compact.