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Let X be a completely regular space, and let A(X) be a subal-gebra of C(X) containing C*{X). We study the maximal ideals in A(X) by associating a filter Z(f) to each / 6 A(X). This association extends to a one-to-one correspondence between M(A) (the set of maximal ideals of A(X)) and ßX. 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 vX and ßX.

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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 / 6 A(X). This association extends to a oneto-one correspondence between M(A) (the set of maximal ideals of A(X)) and ßX. 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 vX and ßX.