Premium
Directed Abelian Groups, Countably Continuous Rings, and Rickart C * ‐Algebras
Author(s) -
Handelman David,
Higgs Denis,
Lawrence John
Publication year - 1980
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-21.2.193
Subject(s) - abelian group , mathematics , hausdorff space , pointwise , monotone polygon , order (exchange) , intersection (aeronautics) , pure mathematics , unit (ring theory) , group (periodic table) , ring (chemistry) , piecewise , discrete mathematics , combinatorics , algebra over a field , mathematical analysis , physics , chemistry , geometry , mathematics education , organic chemistry , finance , engineering , economics , aerospace engineering , quantum mechanics
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 ℵ 0 ‐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].