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
DOI:
https://doi.org/10.1090/S0002-9939-1970-0264404-3
MathSciNet review:
0264404
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Abstract | References | Similar Articles | Additional Information
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 $f|{x_1} \in A|{x_1}$ and $f|{x_2} \in A|{x_2}$, 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 ($(I)$).
- T. W. Gamelin, Embedding Riemann surfaces in maximal ideal spaces, J. Functional Analysis 2 (1968), 123–146. MR 0223894, DOI https://doi.org/10.1016/0022-1236%2868%2990014-1
- Kenneth Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1962. MR 0133008
- Walter Rudin, The closed ideals in an algebra of analytic functions, Canadian J. Math. 9 (1957), 426–434. MR 89254, DOI https://doi.org/10.4153/CJM-1957-050-0
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Keywords:
Function algebra,
maximal ideal space,
peak point
Article copyright:
© Copyright 1970
American Mathematical Society