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December 2007 On a Characterization of Compact Hausdorff Space $X$ for Which Certain Algebraic Equations Are Solvable in $C(X)$
Dai HONMA, Takeshi MIURA
Tokyo J. Math. 30(2): 403-416 (December 2007). DOI: 10.3836/tjm/1202136685

Abstract

Let $X$ be a compact Hausdorff space and $C(X)$ the Banach algebra of all complex-valued continuous functions on $X$. We consider the following property of $C(X)$: for each $f \in C(X)$ there exist a $g \in C(X)$ and positive integers $p$ and $q$ such that $p$ does not divide $q$ and $f^{q} = g^{p}$. When $X$ is locally connected, we give a necessary and sufficient condition for $C(X)$ to have this property. We also give a characterization of a first-countable compact Hausdorff space $X$ for which $C(X)$ has the property above. As a corollary, we prove that if $X$ is locally connected, or first-countable, then $C(X)$ has the property above if and only if $C(X)$ is algebraically closed.

Citation

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Dai HONMA. Takeshi MIURA. "On a Characterization of Compact Hausdorff Space $X$ for Which Certain Algebraic Equations Are Solvable in $C(X)$." Tokyo J. Math. 30 (2) 403 - 416, December 2007. https://doi.org/10.3836/tjm/1202136685

Information

Published: December 2007
First available in Project Euclid: 4 February 2008

zbMATH: 1179.46044
MathSciNet: MR2376518
Digital Object Identifier: 10.3836/tjm/1202136685

Subjects:
Primary: 46J10

Rights: Copyright © 2007 Publication Committee for the Tokyo Journal of Mathematics

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Vol.30 • No. 2 • December 2007
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