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

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.