A look into homomorphisms between uniform algebras over a Hilbert space

Abstract

We study the vector-valued spectrum Mu,∞(Bℓ2,Bℓ2), which is the set of non-zero algebra homomorphisms from Au(Bℓ2) (the algebra of uniformly continuous holomorphic functions on Bℓ2) to H∞(Bℓ2) (the algebra of bounded holomorphic functions on Bℓ2). This set is naturally projected onto the closed unit ball of H∞(Bℓ2,ℓ2) giving rise to an associated fibering. Extending the classical notion of cluster sets introduced by I. J. Schark (1961) to the vector-valued spectrum we define vector-valued cluster sets. The aim of the article is to look at the relationship between fibers and cluster sets obtaining results regarding the existence of analytic balls in those sets.