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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Maximal ideals in subalgebras of $ C(X)$


Authors: Lothar Redlin and Saleem Watson
Journal: Proc. Amer. Math. Soc. 100 (1987), 763-766
MSC: Primary 54C40; Secondary 46E25, 46J20
DOI: https://doi.org/10.1090/S0002-9939-1987-0894451-2
MathSciNet review: 894451
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Abstract: Let $ X$ be a completely regular space, and let $ A(X)$ be a subalgebra of $ C(X)$ containing $ {C^ * }(X)$. We study the maximal ideals in $ A(X)$ by associating a filter $ Z(f)$ to each $ f \in A(X)$. This association extends to a one-to-one correspondence between $ \mathcal{M}(A)$ (the set of maximal ideals of $ A(X)$) and $ \beta X$. We use the filters $ Z(f)$ to characterize the maximal ideals and to describe the intersection of the free maximal ideals in $ A(X)$. Finally, we outline some of the applications of our results to compactifications between $ \upsilon X$ and $ \beta X$.


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DOI: https://doi.org/10.1090/S0002-9939-1987-0894451-2
Keywords: Algebras of continuous functions, maximal ideal, compactifications
Article copyright: © Copyright 1987 American Mathematical Society