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]