## Tokyo Journal of Mathematics

### On a Characterization of Compact Hausdorff Space $X$ for Which Certain Algebraic Equations Are Solvable in $C(X)$

#### 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.

#### Article information

Source
Tokyo J. Math., Volume 30, Number 2 (2007), 403-416.

Dates
First available in Project Euclid: 4 February 2008

https://projecteuclid.org/euclid.tjm/1202136685

Digital Object Identifier
doi:10.3836/tjm/1202136685

Mathematical Reviews number (MathSciNet)
MR2376518

Zentralblatt MATH identifier
1179.46044

#### Citation

HONMA, Dai; MIURA, Takeshi. On a Characterization of Compact Hausdorff Space $X$ for Which Certain Algebraic Equations Are Solvable in $C(X)$. Tokyo J. Math. 30 (2007), no. 2, 403--416. doi:10.3836/tjm/1202136685. https://projecteuclid.org/euclid.tjm/1202136685

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