# 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 $$g\mapsto pgp$$.

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