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