Proposition 41.3.7.label Let $A, B$ be unital $C^{*}$-algebras, $S \subset A$ be an operator system, $\phi: S \to B$ be a unital $2$-positive map, then $\norm{\phi}_{L(S; B)}\le 1$.
Proof, [Proposition 3.2, Po02]. Let $x \in S$ with $\norm{x}_{A} \le 1$, then
\[\phi_{2}\begin{bmatrix}1&x \\ x^{*}&1\end{bmatrix} = \begin{bmatrix}1&\phi(x) \\ \phi(x)^{*}&1\end{bmatrix} \ge 0\]
By Lemma 41.3.6, $\norm{\phi(x)}_{B} \le 1$.$\square$
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