A look into homomorphisms between uniform algebras over a Hilbert space

We study the vector-valued spectrum Mu,∞(Bl2 , Bl2 ) which is the set of nonzero 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 into these sets.