Directed Abelian Groups, Countably Continuous Rings, and Rickart C * ‐Algebras | Zendy

Handelman David | Zendy; Higgs Denis | Zendy; Lawrence John | Zendy

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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].

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