Operator system

This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these messages) This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Operator system" – news · newspapers · books · scholar · JSTOR (September 2014) (Learn how and when to remove this message) This article possibly contains original research. Please improve it by verifying the claims made and adding inline citations. Statements consisting only of original research should be removed. (May 2024) (Learn how and when to remove this message) (Learn how and when to remove this message)

Given a unital C*-algebra ${\displaystyle {\mathcal {A}}}$, a *-closed subspace S containing 1 is called an operator system. One can associate to each subspace ${\displaystyle {\mathcal {M}}\subseteq {\mathcal {A}}}$ of a unital C*-algebra an operator system via ${\displaystyle S:={\mathcal {M}}+{\mathcal {M}}^{*}+\mathbb {C} 1}$.

The appropriate morphisms between operator systems are completely positive maps.

By a theorem of Choi and Effros, operator systems can be characterized as *-vector spaces equipped with an Archimedean matrix order.^[1]

• Operator space