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

Authors:
Takeshi Miura and Kazuki Niijima

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

MSC (2000):
Primary 46J10

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

Published electronically:
December 30, 2002

MathSciNet review:
1974344

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Abstract: Let be the algebra of all complex-valued continuous functions on a compact Hausdorff space . We say that is algebraically closed if each monic polynomial equation over has a continuous solution. We give a necessary and sufficient condition for to be algebraically closed for a locally connected compact Hausdorff space . In this case, it is proved that is algebraically closed if each element of is the square of another. We also give a characterization of a first-countable compact Hausdorff space such that 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:
https://doi.org/10.1090/S0002-9939-02-06835-1

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