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Transactions of the American Mathematical Society

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



Generators for $ A(\Omega )$

Authors: N. Sibony and J. Wermer
Journal: Trans. Amer. Math. Soc. 194 (1974), 103-114
MSC: Primary 32E25; Secondary 46J15
MathSciNet review: 0419838
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Abstract: We consider a bounded domain $ \Omega $ in $ {{\mathbf{C}}^n}$ and the Banach algebra $ A(\Omega )$ of all continuous functions on $ \bar \Omega $ which are analytic in $ \Omega $. Fix $ {f_1}, \ldots ,{f_k}$ in $ A(\Omega )$. We say they are a set of generators if $ A(\Omega )$ is the smallest closed subalgebra containing the $ {f_i}$. We restrict attention to the case when $ \Omega $ is strictly pseudoconvex and smoothly bounded and the $ {f_i}$ are smooth on $ \bar \Omega $. In this case, Theorem 1 below gives conditions assuring that a given set $ {f_i}$ is a set of generators.

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Keywords: Domain in $ {{\mathbf{C}}^n}$, generators, Banach algebra, polynomial convexity
Article copyright: © Copyright 1974 American Mathematical Society

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