From 1263ac993917ecf1aa30bf00b5d1772e00ac4716 Mon Sep 17 00:00:00 2001 From: Bokuan Li Date: Mon, 19 Oct 2026 14:26:13 -0400 Subject: [PATCH] Added a proof for Kadison-Schwarz. --- src/op/system/completely.tex | 10 +++++++++- 1 file changed, 9 insertions(+), 1 deletion(-) diff --git a/src/op/system/completely.tex b/src/op/system/completely.tex index 28fc0c1..52e4800 100644 --- a/src/op/system/completely.tex +++ b/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}