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.