A look into homomorphisms between uniform algebras over a Hilbert space
Volume 265 / 2022
Abstract
We study the vector-valued spectrum $\mathcal {M}_{u,\infty }(B_{\ell _2},B_{\ell _2})$, which is the set of non-zero 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 in those sets.
