Interpolating sequences and the Shilov boundary of H∞( Δ)

Abstract

Let H ∞( Δ) denote the Banach algebra of bounded analytic functions on the open unit disc, let M denote its maximal ideal space, and let ∂ denote its Shilov boundary. D. J. Newman has shown that a homomorphism ϑ in M will be in ∂ if and only if ϑ is unimodular on all Blaschke products. We answer a question of K. Hoffman by showing that ϑ will be in ∂ if and only if ϑ is unimodular on every Blaschke product whose zero set is an interpolating sequence. Our method is based on a construction due to L. Carleson, originally developed for the proof of the Corona theorem.