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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On a characterization of the maximal ideal spaces of algebraically closed commutative $C^{\ast}$-algebras


Authors: Takeshi Miura and Kazuki Niijima
Journal: Proc. Amer. Math. Soc. 131 (2003), 2869-2876
MSC (2000): Primary 46J10
Published electronically: December 30, 2002
MathSciNet review: 1974344
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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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Additional Information

Takeshi Miura
Affiliation: Department of Basic Technology, Applied Mathematics and Physics, Yamagata University, Yonezawa 992-8510, Japan
Email: miura@yz.yamagata-u.ac.jp

Kazuki Niijima
Affiliation: Gumma Prefectural Ôta Technical High School, 380 Motegi-chou, Ôta 373-0809, Japan

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06835-1
PII: S 0002-9939(02)06835-1
Keywords: Commutative Banach algebras, maximal ideal spaces
Received by editor(s): April 24, 2001
Received by editor(s) in revised form: April 10, 2002
Published electronically: December 30, 2002
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2002 American Mathematical Society