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Let X X be a completely regular space, and let A (X) A (X) be a subalgebra of C (X) C (X) containing C ∗ (X) C^ * (X). We study the maximal ideals in A (X) A (X) by associating a filter Z (f) Z (f) to each f ∈ A (X) f A (X). This association extends to a one-to-one correspondence between M (A) M (A) (the set of maximal ideals of A (X) A (X) ) and β X 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 υ X X and β X X.
Redlin et al. (Thu,) studied this question.