Intersections of maximal ideals in algebras between C∗(X) and C(X)

Let C(X) be the algebra of all real-valued continuous functions on a completely regular space X, and C∗(X) the subalgebra of bounded functions. There is a known correspondence between a certain class of z-filters on X and proper ideals in C∗(X) that leads to theorems quite analogous to those for C(X). This correspondence has been generalized by Redlin and Watson to any algebra between C∗(X) and C(X). In the process they have singled out a class of ideals that play a similar geometric role to that of z-ideals in the setting of C(X). We show that these ideals are just the intersections of maximal ideals. It is also known that any algebra A between C∗(X) and C(X) is the ring of fractions of C∗(X) with respect to a multiplicatively closed subset. We make use of this representation to characterize the functions that belong to all the free maximal ideals in A. We conclude by applying our characterization to a subalgebra H of C(N) previously studied by Brooks and Plank.