Proposition 41.3.8 (Kadison-Schwarz Inequality).label Let $A, B$ be unital $C^{*}$-algebras, $\phi: A \to B$ be a unital $2$-positive map, then for each $x \in A$,
\[\phi(x)^{*}\phi(x) \le \phi(x^{*}x)\]
Proof. Since
\[\begin{bmatrix}1&x \\ x^{*}&x^{*}x\end{bmatrix} = \begin{bmatrix}1 &x \\ 0 &0\end{bmatrix}^{*}\begin{bmatrix}1 &x \\ 0 &0\end{bmatrix} \ge 0\]
and $\phi$ is unital and $2$-positive,
\[\begin{bmatrix}1&\phi(x) \\ \phi(x)^{*}&\phi(x^{*}x)\end{bmatrix} = \phi_{2}\begin{bmatrix}1&x \\ x^{*}&x^{*}x\end{bmatrix} \ge 0\]
By Lemma 41.3.6, $\phi(x)^{*}\phi(x) \le \phi(x^{*}x)$.$\square$
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