A look into homomorphisms between uniform algebras over a Hilbert space

We study the vector-valued spectrum \(\mathcal{M}_{u,\infty}(B_{\ell_2},B_{\ell_2})\) which is the set of nonzero algebra homomorphisms from \(\mathcal{A}_u(B_{\ell_2})\) (the algebra of uniformly continuous holomorphic functions on \(B_{\ell_2}\)) to \(\mathcal {H}^\infty(B_{\ell_2})\) (the algebra of bounded holomorphic functions on \(B_{\ell_2}\)). This set is naturally projected onto the closed unit ball of \(\mathcal {H}^\infty(B_{\ell_2}, \ell_2)\) 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.