Theorem 41.9.2.label Let $S$ be an operator system, $\phi \in L(S; M_{n}(\complex))$, and
\[\tilde \phi: M_{n}(S) \to \complex \quad E_{ij}\otimes a \mapsto \phi(a)_{ij}\]
then the following are equivalent:
- (1)
$\phi$ is completely positive.
- (2)
$\phi$ is $n$-positive.
- (3)
$\tilde \phi \in M_{n}(S)^{*}$ is a positive linear functional.
Proof. See Proposition 41.11.6.$\square$
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