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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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A uniform algebra of analytic functions on a Banach space
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by T. K. Carne, B. Cole and T. W. Gamelin PDF
Trans. Amer. Math. Soc. 314 (1989), 639-659 Request permission


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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Additional Information
  • © Copyright 1989 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 314 (1989), 639-659
  • MSC: Primary 46J15; Secondary 46B20, 46G20
  • DOI:
  • MathSciNet review: 986022