Bishop’s theorem and differentiability of a subspace of C b (K)

Abstract

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 δ 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 inX 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 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. IntroductionLet K be a Hausdorff topological space. A function algebra A on K will be understood to be a closed subalgebra of C b (K) which is the Banach algebra of all bounded complex-valued continuous functions on K. The norm f of a bounded continuous function f on K is defined to be sup x∈K |f (x)|. A function algebra A is called separating if for two distinct points s, t in K, there is f ∈ A such that f (s) = f (t).In this paper, a subspace means a closed linear subspace. For each t ∈ K, let δ t be an evaluation functional onis called separating if for distinct points t, s in K we have αδ t = βδ s for any complex numbers α, β of modulus 1 as a linear functional on A. This definition of a separating subspace is a natural extension of the definition of a separating function algebra. In fact, given a separating function algebra A 2000 Mathematics Subject Classification. 46B04, 46G20, 46G25, 46B22.

Strong peak points and strongly norm attaining points with applications to denseness and polynomial numerical indices

“…Concerning the converse, it is shown in [6] that if the set of all strong peak functions is dense in A, then the set of all strong peak points is a norming subset of A. This completes the proof.…”

“…Following the definition of Globevnik in [11], the smallest closed norming subset of A is called the Shilov boundary for A and it is shown in [6] that if the set of strong peak functions is dense in A then the Shilov boundary of A exists and it is the closure of ρA. The variation method used in [7] gives the partial converse of the above mentioned result.…”

“…It is shown in [6] that if f ∈ A is a strong peak function and f = 1, then f is a smooth point of B A . Then Theorem 2.1 completes the proof.…”

“…It is shown in [6] that if X is locally uniformly convex, then the set of norm attaining elements is…”

Denseness of Norm-Attaining Mappings on Banach Spaces

“…It is also worth remarking that it is shown in [9] that ρA(B X ) is a norming subset for A(B X ) if X has the Radon-Nikodým property. Further, very recently, it has been shown in [18] that the set of all strong peak functions is dense in A(B X : Y ) if ρA(B X ) is a norming subset for A(B X ).…”