A Uniform Algebra of Analytic Functions on a Banach Space

Abstract

Let A(B) be the uniform algebra on the unit ball of a dual Banach space Z = ff* generated by the weak-star continuous linear functionals.We focus on three related problems: (i) to determine when A(B) is a tight uniform algebra; (ii) to describe which functions in H(B) are approximate pointwise on B by bounded nets in A(B) ; and (iii) to describe the weak topology of B regarded as a subset of the dual of A(B).With respect to the second problem, we show that any polynomial in elements of y* can be approximated pointwise on B by functions in A(B) of the same norm.This can be viewed as a generalization of Goldstine's theorem.In connection with the third problem, we introduce a class of Banach spaces, called A-spaces, with the property that if (Xj) is a bounded sequence in ? such that P(x) -> 0 for any mhomogeneous analytic function P on 3?, m > 1, then Xj - 0 in norm.We show for instance that a Banach space has the Schur property if and only if it is a A-space with the Dunford-Pettis property.