Homomorphisms between algebras of holomorphic functions on the infinite polydisk

TL;DR

This paper investigates the structure of algebra homomorphisms between spaces of holomorphic functions on the infinite polydisk, revealing a rich fiber structure with multiple disjoint analytic copies and exploring their relation to Gleason parts.

Contribution

It extends previous results to the vector-valued spectrum, showing the existence of multiple disjoint analytic Gleason isometric copies on each fiber, and analyzes the connection between fibers and Gleason parts.

Findings

Existence of 2^c disjoint analytic Gleason isometric copies on each fiber.

Extension of scalar-valued spectrum results to vector-valued spectra.

Analysis of the relationship between fibers and Gleason parts.

Abstract

We study the vector-valued spectrum $\mathcal{M}_\infty(B_{c_0},B_{c_0})$, that is, the set of non null algebra homomorphisms from $\mathcal H^\infty(B_{c_0})$ to $\mathcal H^\infty(B_{c_0})$ which is naturally projected onto the closed unit ball of $\mathcal H^\infty(B_{c_0}, \ell_\infty)$, likewise the scalar-valued spectrum $\mathcal M_\infty(B_{c_0})$ which is projected over $\bar{B}_{\ell_\infty}$. Our itinerary begins in the scalar-valued spectrum $\mathcal{M}_\infty(B_{c_0})$: by expanding a result by Cole, Gamelin and Johnson (1992) we prove that on each fiber there are $2^c$ disjoint analytic Gleason isometric copies of $B_{\ell_\infty}$. For the vector-valued case, building on the previous result we obtain $2^c$ disjoint analytic Gleason isometric copies of $B_{\mathcal{H}^\infty(B_{c_0},\ell_\infty)}$ on each fiber. We also take a look at the relationship between fibers and Gleason parts for both vector-valued spectra $\mathcal{M}_{u,\infty}(B_{c_0},B_{c_0})$ and $\mathcal{M}_\infty(B_{c_0},B_{c_0})$.