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On a class of subalgebras of $C(X)$ 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 $X$ be a Tychonoff space and $A$ a subalgebra of $C(X) $ containing $C^{*}(X) $. Suppose that $C_{K}(X) $ is the set of all functions in $C(X) $ with compact support. Kohls has shown that $C_{K}(X) $ is precisely the intersection of all the free ideals in $C(X) $ or in $C^{*}(X) $. In this paper we have proved the validity of this result for the algebra $A$. Gillman and Jerison have proved that for a realcompact space $X$, $C_{K}(X) $ is the intersection of all the free maximal ideals in $C(X) $. In this paper we have proved that this result does not hold for the algebra $A$, in general. However we have furnished a characterisation of the elements that belong to all the free maximal ideals in $A$. The paper terminates by showing that for any realcompact space $X$, there exists in some sense a minimal algebra $A_m$ for which $X$ becomes $A_m$-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.


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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