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Transactions of the American Mathematical Society

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



Algebras of analytic germs

Author: William R. Zame
Journal: Trans. Amer. Math. Soc. 174 (1972), 275-288
MSC: Primary 32E25
MathSciNet review: 0313545
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Abstract: Let S be a Stein-Riemann domain with global local coordinates $ {\sigma _1}, \cdots ,{\sigma _n}$. Let X be a compact subset of S. Denote by $ \mathcal{O}(X)$ the algebra of germs on X of functions analytic near X. A subalgebra of $ \mathcal{O}(X)$ containing the germs of $ {\sigma _1}, \cdots ,{\sigma _n}$ and the constants is stable if it is closed under differen tiation with respect to the coordinates $ {\sigma _1}, \cdots ,{\sigma _n}$. In this paper the relationship of a stable algebra to its spectrum is investigated. In general, there is no natural imbedding of the spectrum into a Stein manifold. We give necessary and sufficient conditions that such an imbedding exists, and show that a stable algebra whose spectrum admits such an imbedding has a simple description. More generally, we show that a stable algebra is determined by its spectrum. This leads to certain approximation theorems.

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Keywords: Riemann domains, germs of analytic functions, differentiably stable algebras, spectrum, approximation of analytic functions
Article copyright: © Copyright 1972 American Mathematical Society