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 be the uniform algebra on the unit ball of a dual Banach space generated by the weak-star continuous linear functionals. We focus on three related problems: (i) to determine when is a tight uniform algebra; (ii) to describe which functions in are approximable pointwise on by bounded nets in ; and (iii) to describe the weak topology of regarded as a subset of the dual of . With respect to the second problem, we show that any polynomial in elements of can be approximated pointwise on by functions in 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 -spaces, with the property that if is a bounded sequence in such that for any -homogeneous analytic function on , then in norm. We show for instance that a Banach space has the Schur property if and only if it is a -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