Singly generated homogeneous -algebras

Author:
Ronn Carpenter

Journal:
Trans. Amer. Math. Soc. **150** (1970), 457-468

MSC:
Primary 46.50

DOI:
https://doi.org/10.1090/S0002-9947-1970-0262829-8

MathSciNet review:
0262829

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Abstract: With each point in the spectrum of a singly generated -algebra we associate an algebra of germs of functions. It is shown that if is isomorphic to the algebra of germs of analytic functions of a single complex variable, then the spectrum of contains an analytic disc about . The algebra is called homogeneous if the algebras are all isomorphic. If is homogeneous and none of the algebras have zero divisors, we show that is the direct sum of its radical and either an algebra of analytic functions or countably many copies of the complex numbers. If is a uniform algebra which is homogeneous, then it is shown that 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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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1970-0262829-8

Keywords:
-algebra,
singly generated,
homogeneous,
analytic disc,
uniform algebra

Article copyright:
© Copyright 1970
American Mathematical Society