Theorem 41.3.16 (Stinespring).label Let $X$ be a compact Hausdorff space, $B$ be a $C^{*}$-algebra, $\phi: C(X; \complex) \to B$ be a positive map, then $\phi$ is completely positive.
Proof. Let $P \in C(X; M_{n}(\complex))$ with $P \ge 0$ and $\eps > 0$. By Proposition 6.4.5 and Theorem 5.20.10, there exists a partition of unity $\seqf{\psi_j}\subset C(X; [0, 1])$ and positive matrices $\seqf{p_j}\subset M_{n}(\complex)$ such that
\[\norm{P(x) - \sum_{j = 1}^n p_j\psi_j(x)}_{M_n(\complex)}< \eps\]
for all $x \in X$. For each $1 \le j \le n$, $\phi_{n}(p_{j}\psi_{j}) = p_{j}\phi(\psi_{j}) \ge 0$. As $\eps > 0$ is arbitrary, $\phi_{n}(P) \ge 0$.$\square$
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