Weak* Properties of Weighted Convolution Algebras

Abstract

Suppose that L1(ω) is a weighted convolution algebra on R+ = [0,∞) with the weight ω(t) normalized so that the corresponding space M(ω) of measures is the dual space of the space C0(1/ω) of continuous functions. Suppose that φ : L1(ω) → L1(ω0 ) is a continuous nonzero homomorphism, where L1(ω0 ) is also a convolution algebra. If L1(ω)∗f is norm dense in L1(ω), we show that L1(ω0 ) ∗ φ(f) is (relatively) weak∗ dense in L1(ω0 ), and we identify the norm closure of L1(ω0 ) ∗ φ(f) with the convergence set for a particular semigroup. When φ is weak∗ continuous it is enough for L1(ω) ∗ f to be weak∗ dense in L1(ω). We also give sufficient conditions and characterizations of weak∗ continuity of φ. In addition, we show that, for all nonzero f in L1(ω), the sequence fn/||fn|| converges weak∗ to 0. When ω is regulated, fn+1/||fn|| converges to 0 in norm.