Operator system

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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.[1]


  1. ^ Choi M.D., Effros, E.G. Injectivity and operator spaces. Journal of Functional Analysis 1977