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

Published electronically:
December 30, 2002

MathSciNet review:
1974344

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Abstract | References | Similar Articles | Additional Information

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.

**1.**E. M. Čirka,*Approximating continuous functions by holomorphic ones on Jordan arcs in 𝐶ⁿ*, Dokl. Akad. Nauk SSSR**167**(1966), 38–40 (Russian). MR**0201681****2.**R. S. Countryman Jr.,*On the characterization of compact Hausdorff 𝑋 for which 𝐶(𝑋) is algebraically closed*, Pacific J. Math.**20**(1967), 433–448. MR**0208410****3.**Don Deckard and Carl Pearcy,*On matrices over the ring of continuous complex valued functions on a Stonian space*, Proc. Amer. Math. Soc.**14**(1963), 322–328. MR**0147926**, 10.1090/S0002-9939-1963-0147926-1**4.**Don Deckard and Carl Pearcy,*On algebraic closure in function algebras*, Proc. Amer. Math. Soc.**15**(1964), 259–263. MR**0161171**, 10.1090/S0002-9939-1964-0161171-6**5.**Theodore W. Gamelin,*Uniform algebras*, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1969. MR**0410387****6.**Osamu Hatori and Takeshi Miura,*On a characterization of the maximal ideal spaces of commutative 𝐶*-algebras in which every element is the square of another*, Proc. Amer. Math. Soc.**128**(2000), no. 4, 1185–1189. MR**1690991**, 10.1090/S0002-9939-99-05454-4**7.**John G. Hocking and Gail S. Young,*Topology*, 2nd ed., Dover Publications, Inc., New York, 1988. MR**1016814****8.**Takeshi Miura,*On commutative 𝐶*-algebras in which every element is almost the square of another*, Function spaces (Edwardsville, IL, 1998) Contemp. Math., vol. 232, Amer. Math. Soc., Providence, RI, 1999, pp. 239–242. MR**1678337**, 10.1090/conm/232/03401**9.**Kiiti Morita,*Dimension of general topological spaces*, Surveys in general topology, Academic Press, New York-London-Toronto, Ont., 1980, pp. 297–336. MR**564105****10.**Russell C. Walker,*The Stone-Čech compactification*, Springer-Verlag, New York-Berlin, 1974. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 83. MR**0380698**

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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