Theorem 41.3.17 (Choi).label Let $n \in \natp$, $B$ be a $C^{*}$-algebra, and $\phi: M_{n}(\complex) \to B$ be a linear map. For each $1 \le i, j \le n$, let $E_{ij}$ be the standard matrix units for $M_{n}(\complex)$, then $C_{\phi} = (\phi(E_{ij})) \in M_{n}(B)$ is the Chi matrix of $\phi$. The mapping $\phi \mapsto C_{\phi}$ is an isomorphism from $L(M_{n}(\complex); B)$ to $M_{n}(B)$, and the following are equivalent:
- (1)
$\phi$ is completely positive.
- (2)
$\phi$ is $n$-positive.
- (3)
$C_{\phi}$ is positive in $M_{n}(B)$.
Proof. (2) $\Rightarrow$ (3): The matrix $E = (E_{ij})$ is positive in $M_{n}(\complex)$.
(3) $\Rightarrow$ (1):
$\phi, \psi \in S(G)$, $K \subset G$ be compact, and $V \in \cn_{G}(1_{G})$ be relatively compact, then
\begin{align*}\end{align*}
$\square$
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