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

TL;DR

This paper extends Bishop's theorem to subspaces of bounded continuous functions on Hausdorff spaces, analyzing differentiability and peak functions, and exploring the structure of norming sets and boundary properties in complex Banach spaces.

Contribution

It generalizes Bishop's theorem to non-metrizable compact spaces and studies differentiability and peak functions in subspaces of $C_b(K)$ and $A_b(B_X)$, including the Radon-Nikodým property case.

Findings

The set of strong peak functions is dense in certain subspaces.

The smallest closed norming subset is the closure of strong peak points.

The norm is Gteaux differentiable on a dense subset, but nowhere Fre9chet differentiable when $X$ is nontrivial.

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 G\^{a}teaux and Fr\'echet 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-Nikod\'ym property, we show that the set of all strong peak functions in $A_b(B_X)=\{f\in C_b(B_X) : f|_{B_X^\circ} {is holomorphic}\}$ 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 G\^{a}teaux differentiable on a dense subset of $A_b(B_X)$, even though the norm is nowhere Fr\'echet 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.