Bishop's Theorem and Differentiability of a subspace of $C_b(K)$

Abstract

Let $K$ be a Hausdorff space and $C_b(K)$ be the Banach algebra of all complex bounded continuous functions on $K$. We study the Gâteaux and Fréchet differentiability of subspaces of $C_b(K)$. Using this, we show that the set of all strong peak functions in a nontrivial separating separable subspace $H$ of $C_b(K)$ is a dense $G_{\delta }$ subset of $H$, if $K$ is compact. This gives a generalized Bishop's theorem, which says that the closure of the set of strong peak point for $H$ is the smallest closed norming subset of $H$. The classical Bishop's theorem was proved for a separating subalgebra $H$ and a metrizable compact space $K$. In the case that $X$ is a complex Banach space with the Radon-Nikodým property, we show that the set of all strong peak functions in $A_b(B_X)=\{f\in C_b(B_X):f{|}_{B_X^{\circ }}isholomorphic\}$ is dense. As an application, we show that the smallest closed norming subset of $A_b(B_X)$ is the closure of the set of all strong peak points for $A_b(B_X)$. This implies that the norm of $A_b(B_X)$ is Gâteaux differentiable on a dense subset of $A_b(B_X)$, even though the norm is nowhere Fréchet differentiable when $X$ is nontrivial. We also study the denseness of norm attaining holomorphic functions and polynomials. Finally we investigate the existence of numerical Shilov boundary.

References in corpus (1)

• Boundaries for algebras of holomorphic functions on Banach spaces

Cited by in corpus (1)

• Norm and Numerical Peak Holomorphic Functions on Banach Spaces