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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Singly generated homogeneous $ F$-algebras


Author: Ronn Carpenter
Journal: Trans. Amer. Math. Soc. 150 (1970), 457-468
MSC: Primary 46.50
MathSciNet review: 0262829
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Abstract: With each point $ m$ in the spectrum of a singly generated $ F$-algebra we associate an algebra $ {A_m}$ of germs of functions. It is shown that if $ {A_m}$ is isomorphic to the algebra of germs of analytic functions of a single complex variable, then the spectrum of $ A$ contains an analytic disc about $ m$. The algebra $ A$ is called homogeneous if the algebras $ {A_m}$ are all isomorphic. If $ A$ is homogeneous and none of the algebras $ {A_m}$ have zero divisors, we show that $ A$ is the direct sum of its radical and either an algebra of analytic functions or countably many copies of the complex numbers. If $ A$ is a uniform algebra which is homogeneous, then it is shown that $ A$ is either the algebra of analytic functions on an open subset of the complex numbers or the algebra of all continuous functions on its spectrum.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1970-0262829-8
PII: S 0002-9947(1970)0262829-8
Keywords: $ F$-algebra, singly generated, homogeneous, analytic disc, uniform algebra
Article copyright: © Copyright 1970 American Mathematical Society