Bishop’s theorem and differentiability of a subspace

Abstract. Let K be a Hausdorff space and Cb(K) be the Banach algebra of all complex bounded continuous functions on K. We study the Gâteaux and Fréchet differentiability of subspaces of Cb(K). The interesting theorems are the following: Theorem 0.1. Let A be a separating subspace of C(K) on a compact Hausdorff space K and f be a nonzero element of A. Then the norm of A is Gâteaux differentiable at f if and only if f is a peak function. With the aid of Mazur’s theorem, we get the following another version of Bishop’s theorem: Theorem 0.2. Let A be a nontrivial separating separable subspace of C(K) on a compact Hausdorff space K. Then the set of all peak functions is a dense Gδ-subset of A. In particular, the set of all peak points for A is a norming subset for A and its closure is the Shilov boundary for A. Let BX and SX be the closed unit ball and unit sphere of a complex Banach space X, respectively. We apply the results on the differentiability of subspaces of Cb(K) to the following two Banach algebras on BX, Ab(BX) = {f ∈ Cb(BX) : f is analytic on the interior of BX} Au(BX) = {f ∈ Ab(BX) : f is uniformly continuous on BX}, and A(BX) will represent either Au(BX) or Ab(BX). For a Banach space X with the Radon-Nikodým property we get Bishop’s theorem for the Banach algebra A(BX): Theorem 0.3. Suppose that a complex Banach space X has the Radon-Nikodým property. Then the set of all strong peak functions in A(BX) is dense in A(BX). In particular, (i) the set of all strong peak points for A(BX) is a norming subset for A(BX) and its closure is the smallest closed norming subset for A(BX), (ii) the set of all smooth points of BA(BX) is dense in SA(BX). Theorem 0.4. The norm of A(BX) is nowhere Fréchet differentiable, if X is a nontrivial complex Banach space