On a class of subalgebras of and the intersection of their free maximal ideals

Authors:
S. K. Acharyya, K. C. Chattopadhyay and D. P. Ghosh

Journal:
Proc. Amer. Math. Soc. **125** (1997), 611-615

MSC (1991):
Primary 54C40; Secondary 46E25

DOI:
https://doi.org/10.1090/S0002-9939-97-03871-9

MathSciNet review:
1396969

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Abstract: Let be a Tychonoff space and a subalgebra of containing . Suppose that is the set of all functions in with compact support. Kohls has shown that is precisely the intersection of all the free ideals in or in . In this paper we have proved the validity of this result for the algebra . Gillman and Jerison have proved that for a realcompact space , is the intersection of all the free maximal ideals in . In this paper we have proved that this result does not hold for the algebra , in general. However we have furnished a characterisation of the elements that belong to all the free maximal ideals in . The paper terminates by showing that for any realcompact space , there exists in some sense a minimal algebra for which becomes -compact. This answers a question raised by Redlin and Watson in 1987. But it is still unsettled whether such a minimal algebra exists with respect to set inclusion.

**1.**C. E. Aull,*Rings of Continuous Functions*, Dekker, New York, 1985. MR**86g:54001****2.**L. Gillman and M. Jerison,*Rings of Continuous Functions*, Springer-Verlag, New York, 1976. MR**53:11352****3.**C. W. Kohls,*Ideals in Rings of Continuous Functions*, Fund. Math., 45(1957), 28-50. MR**21:1517****4.**D. Plank,*On a class of subalgebras of with application to*, Fund. Math., 64(1969), 41-54. MR**39:6266****5.**L. Redlin and S. Watson,*Maximal ideals in subalgebras of*, Proc. Amer. Math. Soc., 100(1987), 763-766. MR**88f:54031****6.**R. C. Walker,*The Stone-\v{C}ech compactification*, Springer- Verlag, Berlin, Heidelberg, New York, 1974. MR**52:1595**

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Additional Information

**S. K. Acharyya**

Affiliation:
Department of Pure Mathematics, University of Calcutta, 35, Ballygunge Circular Road, Calcutta 700019, India

**K. C. Chattopadhyay**

Affiliation:
Department of Mathematics, University of Burdwan, Burdwan 713104, India

DOI:
https://doi.org/10.1090/S0002-9939-97-03871-9

Keywords:
Algebra of continuous functions,
maximal ideal,
compactification,
realcompactification

Received by editor(s):
February 11, 1994

Received by editor(s) in revised form:
January 30, 1995

Communicated by:
Franklin D. Tall

Article copyright:
© Copyright 1997
American Mathematical Society