# Boundaries for Spaces of Holomorphic Functions on $C(K)$

### María D. Acosta

Universidad de Granada, Spain

## Abstract

We consider the Banach space $A_{u}(X)$ 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 $B$ of the unit ball of $X$ is a boundary for $A_{u}(X)$ if for every $F∈A_{u}(X)$, the norm of $F$ is given by $‖F‖=sup_{x∈B}∣F(x)∣$. We prove that for every compact $K$, the subset of extreme points in the unit ball of $C(K)$ is a boundary for $A_{u}(C(K))$. If the covering dimension of $K$ is at most one, then every norm attaining function in $A_{u}(C(K))$ must attain its norm at an extreme point of the unit ball of $C(K)$. We also show that for any inﬁnite $K$, there is no Shilov boundary for $A_{u}(C(K))$, that is, there is no minimal closed boundary, a result known before for $K$ scattered.

## Cite this article

María D. Acosta, Boundaries for Spaces of Holomorphic Functions on $C(K)$. Publ. Res. Inst. Math. Sci. 42 (2006), no. 1, pp. 27–44

DOI 10.2977/PRIMS/1166642057