Injective operator systems

An operator system $R$ is called injectiv if given operator systems $N\subset M$ each completely positive map $\varphi:N\rightarrow R$ has a completely positive extension $\psi:M\rightarrow R$.

If an operator system $R$ is injective then there is a unital complete order isomorphism from $R$ onto a unital $C^*$-algebra. The latter is conditionally complete²³. (cf. [CE77, Theorem 3.1])

Footnotes

... complete²³

An ordered vector space $V$ is conditionally complete if any upward directed subset of $V_\mathrm{sa}$ that is bounded above has a supremum in $V_\mathrm{sa}$.