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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On a maximal ideal space separated by a peak point

Author: Joseph E. Sommese
Journal: Proc. Amer. Math. Soc. 26 (1970), 471-472
MSC: Primary 46.55
MathSciNet review: 0264404
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Abstract: The purpose of this note is to answer in the negative the following question raised by Gamelin [l]: if $ A$ is a function algebra which has the property that $ X$, the spectrum of $ A$, is expressible as the union of two compact sets $ {X_1}$ and $ {X_2}$ which have as their intersection a peak point $ p$ of $ X$, and if $ f \in (C(X)$ satisfies $ f\vert{x_1} \in A\vert{x_1}$ and $ f\vert{x_2} \in A\vert{x_2}$, then is $ f \in A$? The counterexample is obtained by the use of a construction which is applicable to general function algebras. Let $ A$ be a function algebra and $ I$ a proper closed ideal, denoting by $ A[I]$ the set $ \{ (f,f + s):f \in A,s \in I\} $, it is shown that $ A[I]$ is a function algebra which has as its spectrum two copies of the spectrum of $ A$ identified along hull ($ (I)$).

References [Enhancements On Off] (What's this?)

  • [1] T. W. Gamelin, Embedding Riemann surfaces in maximal ideal spaces, J. Functional Analysis 2 (1968), 123–146. MR 0223894
  • [2] Kenneth Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1962. MR 0133008
  • [3] Walter Rudin, The closed ideals in an algebra of analytic functions, Canad. J. Math. 9 (1957), 426–434. MR 0089254

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Keywords: Function algebra, maximal ideal space, peak point
Article copyright: © Copyright 1970 American Mathematical Society