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

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

2007-08-30

arxiv.org/abs/0708.4069v1

Mathematics

Functional Analysis

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Bishop's Theorem and Differentiability of a subspace of $C_b(K)$ does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Bishop's Theorem and Differentiability of a subspace of $C_b(K)$, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Bishop's Theorem and Differentiability of a subspace of $C_b(K)$ will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-205542