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A uniform algebra of analytic functions on a Banach space


Authors: T. K. Carne, B. Cole and T. W. Gamelin
Journal: Trans. Amer. Math. Soc. 314 (1989), 639-659
MSC: Primary 46J15; Secondary 46B20, 46G20
DOI: https://doi.org/10.1090/S0002-9947-1989-0986022-0
MathSciNet review: 986022
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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.


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DOI: https://doi.org/10.1090/S0002-9947-1989-0986022-0
Article copyright: © Copyright 1989 American Mathematical Society

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