Boundaries for spaces of holomorphic functions on C(K)

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 is an element of A(u)(X), the norm of F is given by parallel to F parallel to=sup(x is an element of B) vertical bar F(x)vertical bar. 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 Au(C(K)): that is, there is no minimal closed boundary, a result known before for K scattered.