Boundaries for Gelfand transform images of Banach algebras of holomorphic functions

TL;DR

This paper characterizes the Shilov boundary of the Gelfand transform images of certain Banach algebras of bounded holomorphic functions on the unit ball of complex Banach spaces, with implications for the Corona theorem.

Contribution

It provides an explicit description of the Shilov boundary for classical Banach spaces and specific Banach algebras of holomorphic functions, advancing understanding of their spectral properties.

Findings

Explicit description of the Shilov boundary for classical Banach spaces

Application insights to the Corona theorem

Analysis of Gelfand transform images of specific Banach algebras

Abstract

Let $\mathcal{A}$ be a Banach algebra of bounded holomorphic functions on the open unit ball $B_X$ of a complex Banach space $X$. Considering the Gelfand transform image $\widehat{\mathcal{A}}$ of the Banach algebra $\mathcal{A}$, which is a uniform algebra on the spectrum of $\mathcal{A}$, we obtain an explicit description of the Shilov boundary for $\widehat{\mathcal{A}}$ for classical Banach spaces $X$ in the case where $\mathcal{A}$ is a certain Banach algebra, for instance, $\mathcal{A}_\infty (B_X)$, $\mathcal{A}_u (B_X)$ or $\mathcal{A}_{wu} (B_X)$. Some possible application of our result to the famous Corona theorem is also briefly discussed.