# Boundaries for Spaces of Holomorphic Functions on <em>C</em>(<em>K</em>)

### 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 <em>C</em>(<em>K</em>). Publ. Res. Inst. Math. Sci. 42 (2006), no. 1, pp. 27–44

DOI 10.2977/PRIMS/1166642057