Added a proof for Kadison-Schwarz.

src/op/system/completely.tex

@@ -259,7 +259,7 @@
     \phi(x)^*\phi(x) \le \norm{\phi(1_A)}_B \phi(x^*x)
     \]
 \end{proposition}
-\begin{proof}
+\begin{proof}[Proof, general case. ]
     Since
     \[
     \begin{bmatrix}
@@ -290,6 +290,14 @@
 
     By \autoref{lemma:norm-matrix-test}, $\phi(x)^*\phi(x) \le \norm{\phi(1_A)}_B\phi(x^*x)$. 
 \end{proof}
+\begin{proof}[Proof, completely positive case. ]
+    Using the \hyperref[Gelfand-Naimark Theorem]{theorem:gns}, let $H$ be a complex Hilbert space such that $B \subset B(H)$. Assume in addition that $\phi$ is completely positive. Let $(K, V, \pi)$ be a \hyperref[Stinespring representation]{theorem:stinespring-dilation} for $\phi$, then for each $a \in A$ and $\xi \in H$, 
+    \begin{align*}
+        \dpn{\phi(a^*)\phi(a)\xi, \xi}{H} &= \dpn{V^*\pi(a^*)VV^*\pi(a)V\xi, \xi}{H} = \dpn{\pi(a^*)VV^*\pi(a)V\xi, V\xi}{K} \\
+        &\le \norm{VV^*}_{B(K)}\dpn{\pi(a^*)\pi(a)V\xi, V\xi}{K} \\
+        &\le \norm{V}_{B(K)}^2 \dpn{\pi(a^*a)V\xi, V\xi}{K} = \norm{\phi(1_A)}_{B(H)}\dpn{\phi(a^*a)\xi, \xi}{K}
+    \end{align*}
+\end{proof}