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arXiv · 2110.07196

Boundaries for Gelfand transform images of Banach algebras of holomorphic functions

Yun Sung Choi·Mingu Jung
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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.

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BibTeXRIS

Yun Sung Choi, Mingu Jung. 2021-10-14. Boundaries for Gelfand transform images of Banach algebras of holomorphic functions. https://arxiv.org/abs/2110.07196

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