On rings of bounded continuous functions with values in a division ring

Abstract

Introduction. Let C*(X, A) denote the ring of bounded continuous functions on a (Hausdorff) topological space2 X with values in a topological division ring A. If, for every maximal (two-sided) ideal M of C*(X, A), we have C*(X, A)/M is isomorphic with A, we say that Stone's theorem holds for C*(X, A). It is well known [9; 6] that Stone's theorem holds for C*(X, A) if A is locally compact and connected, or a finite field. In giving a partial answer to a question of Kaplansky [7], Goldhaber and Wolk showed in [5] that, with restriction on X, and if A is of type V and satisfies the first axiom of countability, then a necessary condition that Stone's theorem holds for C*(X, A) is that A be locally compact. They ask if this condition is also sufficient. Below, we answer this question in the affirmative. In the course of their proof of the necessity of this condition, they construct a class of maximal ideals of C*(X, A). (See ?3). They also ask if every maximal ideal of C*(X, A) is in this class. We show that even if both X and A are the reals, the answer to this question is no.