Intersections of maximal ideals in algebras between \(C^*(X)\) and \(C(X)\) (Q1807577)

A subalgebra of \(C(X)\) which contains \(C^{*}(X)\) is called an intermediate algebra on \(X\). This paper studies intermediate algebras and a certain type of ideal of these algebras, here called a \(\mathcal B\)-ideal; in \(C^{*}(X)\), these ideals play a role analogous to the \(z\)-ideals of \(C(X)\). Following \textit{L. Redlin} and \textit{S. Watson} [Proc. Am. Math. Soc. 100, 763-766 (1987; Zbl 0622.54011)], for each element \(f\) of an intermediate algebra \(A\) \({\mathcal Z}_A(f)\) is defined to be the family \(\{E: E\) is a zero-set \(:fg|E^{c} =1\) for some \(g\in A\}\) and then if \(I\) is an ideal of \(A\), \({\mathcal Z}_A (I)=\bigcup\{{\mathcal Z}_A(f):f\in I\}\). \(I\) is said to be a \(\mathcal B\)-ideal if whenever \({\mathcal Z}_A(f)\subset{\mathcal Z}_A(I)\), then \(f\in I\). Sections 1 and 2 of the paper are concerned with a number of preliminary technical lemmas and in Section 3 it is shown that the \(\mathcal B\)-ideals of \(A\) are precisely the intersections of maximal ideals. In Section 4, the intersection of the collection of all free maximal ideals of \(A\) is characterized. [A similar theorem was obtained independently by \textit{S. K. Acharyya, K. C. Chattopadhyay} and \textit{D. P. Ghosh} [ibid. 125, No. 2, 611-615 (1997; Zbl 0858.54014)].