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Maximal ideals in subalgebras of C(X). (English) Zbl 0622.54011
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 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.

##### MSC:
 54C40 Algebraic properties of function spaces in general topology 46E25 Rings and algebras of continuous, differentiable or analytic functions 54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
maximal ideals
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