Boundaries for Spaces of Holomorphic Functions on $\mathcal{C}(K)$

Abstract

We consider the Banach space ${\mathcal{A}}_u(X)$ of holomorphic functions on the open unit ball of a (complex) Banach space $X$ which are uniformly continuous on the closed unit ball, endowed with the supremum norm. A subset $\mathcal{B}$ of the unit ball of $X$ is a boundary for ${\mathcal{A}}_u(X)$ if for every $F\in {\mathcal{A}}_u(X)$, the norm of $F$ is given by $\Vert F\Vert ={\sup }_{x\in \mathcal{B}}|F(x)|$. We prove that for every compact $K$, the subset of extreme points in the unit ball of $\mathcal{C}(K)$ is a boundary for ${\mathcal{A}}_u(\mathcal{C}(K))$. If the covering dimension of $K$ is at most one, then every norm attaining function in ${\mathcal{A}}_u(\mathcal{C}(K))$ must attain its norm at an extreme point of the unit ball of $\mathcal{C}(K)$. We also show that for any infinite $K$, there is no Shilov boundary for ${\mathcal{A}}_u(\mathcal{C}(K))$, that is, there is no minimal closed boundary, a result known before for $K$ scattered.