Correspondences between ideals and \(z\)-filters for rings of continuous functions between \(C^∗\) and \(C\)

Let \(X\) be a completely regular topological space. Let \(A(X)\) be a ring of continuous functions between \(C^∗(X)\) and \(C(X)\), that is, \(C^∗(X) \subset A(X) \subset C(X)\). In [9], a correspondence \(\mathcal{Z}_A\) between ideals of \(A(X)\) and \(z\)-filters on \(X\) is defined. Here we show that \(\mathcal{Z}_A\) extends the well-known correspondence for \(C^∗(X)\) to all rings \(A(X)\). We define a new correspondence \(\mathcal{Z}_A\) and show that it extends the well-known correspondence for \(C(X)\) to all rings \(A(X)\). We give a formula that relates the two correspondences. We use properties of \(\mathcal{Z}_A\) and \(\mathcal{Z}_A\) to characterize \(C^∗(X)\) and \(C(X)\) among all rings \(A(X)\). We show that \(\mathcal{Z}_A\) defines a one-one correspondence between maximal ideals in \(A(X)\) and the \(z\)-ultrafilters in \(X\).