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), 28692876
MSC (2000):
Primary 46J10
Published electronically:
December 30, 2002
MathSciNet review:
1974344
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Abstract: Let be the algebra of all complexvalued 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 firstcountable compact Hausdorff space such that is algebraically closed.
 1.
E. M. Cirka, Approximation of continuous functions by functions holomorphic on Jordan arcs in , Soviet Math. Doklady 7 (1966), 336338. MR 34:1563
 2.
R. S. Countryman, Jr., On the characterization of compact Hausdorff for which is algebraically closed, Pacific J. Math. 20 (1967), 433448. MR 34:8220
 3.
D. Deckard and C. Pearcy, On matrices over the ring of continuous complexvalued functions on a Stonian space, Proc. Amer. Math. Soc. 14 (1963), 322328. MR 26:5438
 4.
D. Deckard and C. Pearcy, On algebraic closure in function algebras, Proc. Amer. Math. Soc. 15 (1964), 259263. MR 28:4379
 5.
T. W. Gamelin, Uniform algebras, PrenticeHall, N.J. 1969. MR 53:14137
 6.
O. Hatori and T. Miura, On a characterization of the maximal ideal spaces of commutative algebras in which every element is the square of another, Proc. Amer. Math. Soc. 128 (2000), 11851189. MR 2000k:46072a
 7.
J. G. Hocking and G. S. Young, Topology (second edition), Dover Publications, Inc., N. Y. 1988. MR 90h:54001
 8.
T. Miura, On commutative algebras in which every element is almost the square of another, Contemporary Mathematics 232 (1999), 239242. MR 2000k:46072b
 9.
K. Morita, Dimension of general topological spaces, Surveys in general topology (G. M. Reed ed.) Academic Press N. Y. 1980. MR 83b:54049
 10.
Russell Walker, The StoneCech compactification, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 83. SpringerVerlag, 1974. MR 52:1595
 1.
 E. M. Cirka, Approximation of continuous functions by functions holomorphic on Jordan arcs in , Soviet Math. Doklady 7 (1966), 336338. MR 34:1563
 2.
 R. S. Countryman, Jr., On the characterization of compact Hausdorff for which is algebraically closed, Pacific J. Math. 20 (1967), 433448. MR 34:8220
 3.
 D. Deckard and C. Pearcy, On matrices over the ring of continuous complexvalued functions on a Stonian space, Proc. Amer. Math. Soc. 14 (1963), 322328. MR 26:5438
 4.
 D. Deckard and C. Pearcy, On algebraic closure in function algebras, Proc. Amer. Math. Soc. 15 (1964), 259263. MR 28:4379
 5.
 T. W. Gamelin, Uniform algebras, PrenticeHall, N.J. 1969. MR 53:14137
 6.
 O. Hatori and T. Miura, On a characterization of the maximal ideal spaces of commutative algebras in which every element is the square of another, Proc. Amer. Math. Soc. 128 (2000), 11851189. MR 2000k:46072a
 7.
 J. G. Hocking and G. S. Young, Topology (second edition), Dover Publications, Inc., N. Y. 1988. MR 90h:54001
 8.
 T. Miura, On commutative algebras in which every element is almost the square of another, Contemporary Mathematics 232 (1999), 239242. MR 2000k:46072b
 9.
 K. Morita, Dimension of general topological spaces, Surveys in general topology (G. M. Reed ed.) Academic Press N. Y. 1980. MR 83b:54049
 10.
 Russell Walker, The StoneCech compactification, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 83. SpringerVerlag, 1974. MR 52:1595
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Additional Information
Takeshi Miura
Affiliation:
Department of Basic Technology, Applied Mathematics and Physics, Yamagata University, Yonezawa 9928510, Japan
Email:
miura@yz.yamagatau.ac.jp
Kazuki Niijima
Affiliation:
Gumma Prefectural Ôta Technical High School, 380 Motegichou, Ôta 3730809, Japan
DOI:
http://dx.doi.org/10.1090/S0002993902068351
PII:
S 00029939(02)068351
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. TomczakJaegermann
Article copyright:
© Copyright 2002
American Mathematical Society
