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Local invertibility in subrings of C*(X)

Published online by Cambridge University Press:  17 April 2009

H. Linda Byun ,
Lothar Redlin  and
Saleem Watson
H. Linda Byun
Affiliation:
Department of Mathematics, California State University, Long Beach, CA 90840, United States of America
Lothar Redlin
Affiliation:
Department of Mathematics, The Pennsylvania State University, Abington, PA 19001, United States of America
Saleem Watson
Affiliation:
Department of Mathematics, California State University, Long Beach, CA 90840, United States of America
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Abstract

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It is known that the maximal ideals in the rings C(X) and C*(X) of continuous and bounded continuous functions on X, respectively, are in one-to-one correspondence with βX. We have shown previously that the same is true for any ring A(X) between C(X) and C*(X). Here we consider the problem for rings A(X) contained in C*(X) which are complete rings of functions (that is, they contain the constants, separate points and closed sets, and are uniformly closed). For every noninvertible f ∈ A(X), we define a z–filter ZA(f) on X which, in a sense, provides a measure of where f is ‘locally invertible’. We show that the map ZA generates a correspondence between ideals of A(X) and z–filters on X. Using this correspondence, we construct a unique compactification of X for every complete ring of functions. Each such compactification is explicitly identified as a quotient of βX. In fact, every compactification of X arises from some complete ring of functions A(X) via this construction. We also describe the intersections of the free ideals and of the free maximal ideals in complete rings of functions.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 46 , Issue 3 , December 1992 , pp. 449 - 458
Copyright
Copyright © Australian Mathematical Society 1992

References

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