Weak$^{*}$ properties of weighted convolution algebras

Abstract

Suppose that L 1 (w) is a weighted convolution algebra on R + = [0, ∞) with the weight w(t) normalized so that the corresponding space M(w) of measures is the dual space of the space C 0 (1/w) of continuous functions. Suppose that Φ: L 1 (w) → L 1 (w') is a continuous nonzero homomorphism, where L 1 (w') is also a convolution algebra. If L 1 (w)* f is norm dense in L 1 (w), we show that L 1 (w')*Φ(f) is (relatively) weak* dense in L 1 (w'), and we identify the norm closure of L 1 (w') * Φ(f) with the convergence set for a particular semigroup. When Φ is weak* continuous it is enough for L 1 (w) * f to be weak* dense in L 1 (w). We also give sufficient conditions and characterizations of weak* continuity of Φ. In addition, we show that, for all nonzero f in L 1 (w), the sequence f n /∥f∥ converges weak* to 0. When w is regulated, f n+1 /∥f n ∥ converges to 0 in norm.