WEAK-STAR PROPERTIES OF HOMOMORPHISMS FROM WEIGHTED CONVOLUTION ALGEBRAS ON THE HALF-LINE

Abstract

AbstractLet L1(ω) be the weighted convolution algebra L1ω(ℝ+) on ℝ+ with weight ω. Grabiner recently proved that, for a nonzero, continuous homomorphism Φ:L1(ω1)→L1(ω2), the unique continuous extension $\widetilde {\Phi }:M(\omega _1)\to M(\omega _2)$ to a homomorphism between the corresponding weighted measure algebras on ℝ+ is also continuous with respect to the weak-star topologies on these algebras. In this paper we investigate whether similar results hold for homomorphisms from L1(ω) into other commutative Banach algebras. In particular, we prove that for the disc algebra $A(\overline {\mathbb D})$ every nonzero homomorphism $\Phi :L^1({\omega })\to A(\overline {\mathbb D})$ extends uniquely to a continuous homomorphism $\widetilde {\Phi }:M(\omega )\to H^{\infty }(\mathbb D)$ which is also continuous with respect to the weak-star topologies. Similarly, for a large class of Beurling algebras A+v on $\overline {\mathbb D}$ (including the algebra of absolutely convergent Taylor series on $\overline {\mathbb D}$) we prove that every nonzero homomorphism Φ:L1(ω)→A+v extends uniquely to a continuous homomorphism $\widetilde {\Phi }:M(\omega )\to A^+_v$ which is also continuous with respect to the weak-star topologies.