On a theorem of Gelfand and Kolmogoroff concerning maximal ideals in rings of continuous functions

This paper deals with a theorem of Gelfand and Kolmogoroff concerning the ring C= C(X, R) of all continuous real-valued functions on a completely regular topological space X, and the subring C* = C*(X, R) consisting of all bounded functions in C. The theorem in question yields a one-one correspondence between the maximal ideals of C and those of C*; it is stated without proof in [2]. Here we supply a proof (§2), and we apply the theorem to three problems previously considered by Hewitt in [5]. Our first result (§3) consists of two simple constructions of the Q-space vX. The second (§4) exhibits a one-one correspondence between the maximal ideals of C and those of C*, in a manner which may be considered qualitatively different from that expressed by Gelfand and Kolmogoroff. In our final application (§5), we confirm Hewitt's conjecture that every m-closed ideal of C is the intersection of all the maximal ideals that contain it. In this connection, we also examine the corresponding problem for the ring C*; we find that a necessary and sufficient condition for the theorem to hold here is that every function in C be bounded.