Abstract
Let X X be a completely regular space, and let A ( X ) A(X) be a subalgebra of C ( X ) C(X) containing C â ( X ) {C^ * }(X) . We study the maximal ideals in A ( X ) A(X) by associating a filter Z ( f ) Z(f) to each f â A ( X ) f \in A(X) . This association extends to a one-to-one correspondence between M ( A ) \mathcal {M}(A) (the set of maximal ideals of A ( X ) A(X) ) and β X \beta X . We use the filters Z ( f ) Z(f) to characterize the maximal ideals and to describe the intersection of the free maximal ideals in A ( X ) A(X) . Finally, we outline some of the applications of our results to compactifications between Ď X \upsilon X and β X \beta X .
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