Directed Abelian Groups, Countably Continuous Rings, and Rickart C* -Algebras

Abstract

We establish the following result: Any partially ordered abelian group with an order unit that is monotone σ-complete and satisfies the Riesz decomposition property can be order-embedded in the group of continuous functions on a compact Hausdorff space, equipped with the pointwise ordering. As corollaries, we deduce that if R is either an ℵ0-continuous ring or a finite Rickart C*-algebra, then the intersection of its maximal two-sided ideals is zero. Some of the results of this paper have been announced in [16].