# Operator system

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Given a [unital](/source/Unital_algebra) [C*-algebra](/source/C*-algebra) A {\displaystyle {\mathcal {A}}} , a [*-closed](/source/C*-algebra) [subspace](/source/Linear_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](/source/Completely_positive_map).

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

## See also

- [Operator space](/source/Operator_space)

## References

1. **[^](#cite_ref-1)** Choi M.D., Effros, E.G. Injectivity and operator spaces. Journal of Functional Analysis 1977

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