Proposition 35.1.4.label Let $n \in \natp$. For each $y \in M_{n}(\complex)$, let
\[\phi_{y}: M_{n}(\complex) \to \complex \quad x \mapsto \dpn{x, y}{F}= \text{tr}(y^{*}x)\]
then the following are equivalent:
- (1)
$\phi_{y} \in S(M_{n}(\complex))$.
- (2)
$y \ge 0$ and $\text{tr}(y) = 1$.
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