Fibers and Gleason parts for the maximal ideal space of $\mathcal A_u(B_{\ell_p})$

TL;DR

This paper explores the structure of the maximal ideal space of certain Banach algebras of holomorphic functions on $ ext{ell}_p$ spaces, revealing complex fiber and Gleason part configurations influenced by space geometry.

Contribution

It provides new descriptions of fibers and Gleason parts for the spectrum of these algebras, especially for $p=1$ and $p eq 1$, highlighting geometric influences.

Findings

Fibers over points in $B_{ ext{ell}_p}$ contain large sets with distinct Gleason parts for $p eq 1$.

Fibers over points outside the unit ball in $ ext{ell}_1$ are non-singleton.

Different fibers over points in the sphere $S_{ ext{ell}_1''}$ do not share Gleason parts.

Abstract

In the early nineties, R. M. Aron, B. Cole, T. Gamelin and W.B. Johnson initiated the study of the maximal ideal space (spectrum) of Banach algebras of holomorphic functions defined on the open unit ball of an infinite dimensional complex Banach space. Within this framework, we investigate the fibers and Gleason parts of the spectrum of the algebra of holomorphic and uniformly continuous functions on the unit ball of $\ell_p$ ($1\le p<\infty$). We show that the inherent geometry of these spaces provides a fundamental ingredient for our results. We prove that whenever $p\in\mathbb N$ ($p\ge 2$), the fiber of every $z\in B_{\ell_p}$ contains a set of cardinal $2^{\mathfrak c}$ such that any two elements of this set belong to different Gleason parts. For the case $p=1$, we complete the known description of the fibers, showing that, for each $z\in\overline B_{\ell_1''}\setminus S_{\ell_1}$, the fiber over $z$ is not a singleton. Also, we establish that different fibers over elements in $S_{\ell_1''}$ cannot share Gleason parts.