This is an announcement for the paper "Bishop's theorem and differentiability of a subspace of $C_b(K)$" by Yun Sung Choi, Han Ju Lee, and Hyun Gwi Song.

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 $A_b(B_X)={ f\in C_b(B_X) : f|_{B_X^\circ} \mbox{ is holomorphic}}$ 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.

Archive classification: math.FA

Mathematics Subject Classification: 46B04; 46G20; 46G25; 46B22

The source file(s), bishop-070130.tex: 87264 bytes, is(are) stored in gzipped form as 0708.4069.gz with size 25kb. The corresponding postcript file has gzipped size 157kb.

Submitted from: hahnju@postech.ac.kr

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