Gelfand–Kolmogoroff theorem for rings of analytic functions

Abstract

Consider rings of single variable real analytic or complex entire functions, denoted by K〈z〉. We study “discrete z-filters” on K and their connections with the space of maximal ideals of K〈z〉. We characterize the latter as a compact T1 space θK of discrete z-ultrafilters on K. We show that θK is a bijective continuous image of βK∖Q(K), where Q(K) is the set of far points of βK. θK turns out to be the Wallman compactification of the canonically embedded image of K inside θK. Using our characterization of θK, we derive a Gelfand–Kolmogoroff characterization of maximal ideals of K〈z〉 and show that the Krull dimension of K〈z〉 is at least c. We also establish the existence of a chain of prime z-filters on K consisting of at least 2c many elements.