Boundaries for Gelfand transform images of Banach algebras of holomorphic functions

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 $\hat{\mathcal{A}}$ 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 $\hat{\mathcal{A}}$ 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.

18 pages