New insights into Gleason parts for an algebra of holomorphic functions

TL;DR

This paper explores the structure of Gleason parts within the spectrum of an algebra of holomorphic functions on the unit ball of l_p, revealing a vast number of Gleason parts in fibers and advancing understanding of boundary behaviors.

Contribution

It provides new results on the abundance and properties of Gleason parts in fibers of the spectrum for l_p, extending previous research and addressing open questions.

Findings

Fibers over points in l_p contain 2^{\u210b} Gleason parts.

Existence of many strong boundary points within certain fibers.

Results hold for both p > 1 and p = 1 cases, with more complex proofs for p=1.

Abstract

We study the structure of the spectrum of the algebra of uniformly continuous holomorphic functions on the unit ball of $\ell_p$. Our main focus is the relationship between \emph{Gleason parts} and \emph{fibers}. For every $z \in B_{\ell_p}$ with $1 < p < \infty$, we prove that the fiber over $z$ contains $2^{\mathfrak{c}}$ distinct Gleason parts. We also investigate some of the properties of these Gleason parts and show the existence of many strong boundary points in certain fibers. We then examine the case $p = 1$, where similar results on the abundance of Gleason parts within the fibers hold, although the arguments required are more involved. Our results extend and complete earlier work on the subject, providing answers to previously posed questions.