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 $\mathcal {Z} = {\mathcal {Y}^\ast }$ 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 ${H^\infty }(B)$ are approximable 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 ${\mathcal {Y}^{\ast \ast }}$ 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 $\Lambda$-spaces, with the property that if $\{ {x_j}\}$ is a bounded sequence in $\mathcal {X}$ such that $P({x_j}) \to 0$ for any $m$-homogeneous analytic function $P$ on $\mathcal {X}, m \geq 1$, then ${x_j} \to 0$ in norm. We show for instance that a Banach space has the Schur property if and only if it is a $\Lambda$-space with the Dunford-Pettis property.