On a characterization of the maximal ideal spaces of algebraically closed commutative 𝐶*-algebras

Miura, Takeshi; Niijima, Kazuki

doi:10.1090/S0002-9939-02-06835-1

On a characterization of the maximal ideal spaces of algebraically closed commutative $C^{\ast }$-algebras
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by Takeshi Miura and Kazuki Niijima

Proc. Amer. Math. Soc. 131 (2003), 2869-2876

DOI: https://doi.org/10.1090/S0002-9939-02-06835-1

Published electronically: December 30, 2002

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Abstract:

Let $C(X)$ be the algebra of all complex-valued continuous functions on a compact Hausdorff space $X$. We say that $C(X)$ is algebraically closed if each monic polynomial equation over $C(X)$ has a continuous solution. We give a necessary and sufficient condition for $C(X)$ to be algebraically closed for a locally connected compact Hausdorff space $X$. In this case, it is proved that $C(X)$ is algebraically closed if each element of $C(X)$ is the square of another. We also give a characterization of a first-countable compact Hausdorff space $X$ such that $C(X)$ is algebraically closed.

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