I came across the following form of Schwarz inequality for completely positive maps in Arveson's paper:
Let $\delta:\mathcal{A}\to\mathcal{B}$ be a unital completely positive linear map between two $C^*$-algebras, then \begin{equation}\delta(A)^*\delta(A)\le \delta(A^*A),\end{equation}
which is a crucial part in proving proposition 3.
I do not know this inequality and thus searched on wiki, something related is
(Kadison-Schwarz) If $\phi$ is a unital positive map, then \begin{equation}\phi(a^*a)\ge\phi(a^*)\phi(a)\end{equation} for all normal elements $a$.
However, I cannot find a proof of Kadison-Schwarz. Also, since Kadison-Schwarz works only for normal elements, there seems to be a gap between Kadison-Schwarz and Schwarz.
I wonder where I can find a proof to the first inequality. Thanks!