Characterizations of Ideals in Intermediate C-Rings A(X) via the A-Compactifications of X - Joshua Sack; Saleem Watson (International Journal of Mathematics and Mathematical Sciences)
Let X be a completely regular topological space. An intermediate ring is a ring A(X) of continuous functions satisfying C*(X)⊆A(X)⊆C(X). In Redlin and Watson (1987) and in Panman et al. (2012), correspondences 𝒵A and ℨA are defined between ideals in A(X) and z-filters on X, and it is shown that these extend the well-known correspondences studied separately for C∗(X) and C(X), respectively, to any intermediate ring. Moreover, the inverse map 𝒵A← sets up a one-one correspondence between the maximal ideals of A(X) and the z-ultrafilters on X. In this paper, we define a function 𝔎A that, in the case that A(X) is a C-ring, describes ℨA in terms of extensions of functions to realcompactifications of X. For such rings, we show that ℨA← maps z-filters to ideals. We also give a characterization of the maximal ideals in A(X) that generalize the Gelfand-Kolmogorov theorem from C(X) to A(X).