A look into homomorphisms between uniform algebras over a Hilbert space

TL;DR

This paper investigates the structure of homomorphisms between uniform algebras over a Hilbert space, extending classical concepts to vector-valued spectra and analyzing the relationship between fibers and cluster sets.

Contribution

It introduces vector-valued cluster sets for the first time and explores their connection with fibers in the spectrum of uniform algebras over Hilbert spaces.

Findings

Established the existence of analytic balls within vector-valued cluster sets.

Extended classical cluster set theory to vector-valued spectra.

Analyzed the fibering structure of the vector-valued spectrum.

Abstract

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.