zbMATH — the first resource for mathematics

Compressions on partially ordered abelian groups. (English) Zbl 1063.47003
A retraction on a partially ordered abelian additive group \(G\) with order unit \(u\) is (as defined in the paper under review) an idempotent order-preserving group endomorphism \(J:G\to G\) such that \(J(u)\leq u\) and \(J(e)=e\) whenever \(0\leq e\leq J(u)\). As a generalization of unital \(C^*\)-algebras, the author introduces effect-ordered rings as rings \(A\) with unit \(1\) such that \(A\) is (under addition) a partially ordered abelian group with positive cone \(A^+\ni 1\), its (additive) subgroup \(G(A):=A^+-A^+\) is a unital group with unit \(1\) and positive cone \(G(A)=A^+\) and, for any \(a,b\in A^+\), \(ab\) (if \(ab=ba\)), \(aba\), \((a-b)^2\in A^+\), and \(ab=ba=0\) if \(aba=0\). \(A\) is archimedean if \(G(A)\) is archimedean. The main result of the paper shows that if \(A\) is an archimedean effect-ordered ring, then any retraction \(J\) on \(G(A)\) with focus \(p=J(1)\) is a compression and has the form \(g\mapsto pgp\).

47A20 Dilations, extensions, compressions of linear operators
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
06F25 Ordered rings, algebras, modules
Full Text: DOI