• Corpus ID: 208026170

Boundaries for Spaces of Holomorphic Functions on C ( K )

@inproceedings{Maria2006BoundariesFS,
  title={Boundaries for Spaces of Holomorphic Functions on C ( K )},
  author={Di Bari Maria},
  year={2006},
  url={https://api.semanticscholar.org/CorpusID:208026170}
}

• D. Maria
• Published 2006
• Mathematics

We consider the Banach space Au(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 Au(X) if for every F ∈ Au(X), the norm of F is given by ‖F‖ = supx∈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 Au(C(K)). If the covering dimension of K is at most one, then… 

We consider the Banach space Au(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 Au(X) if for every F ∈ Au(X), the norm of F is given by ‖F‖ = supx∈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 Au(C(K)). If the covering dimension of K is at most one, then every norm attaining function in Au(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.