Operator system

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

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.