From d000e08aae3043bc555dea4f098c18efc69a3b98 Mon Sep 17 00:00:00 2001
From: Bokuan Li <bokuan.li@mail.mcgill.ca>
Date: Mon, 19 Oct 2026 14:30:38 -0400
Subject: [PATCH] Housekeeping.

---
 src/op/system/dilation.tex | 11 ++++++++---
 src/op/system/index.tex    |  2 +-
 2 files changed, 9 insertions(+), 4 deletions(-)

diff --git a/src/op/system/dilation.tex b/src/op/system/dilation.tex
index ecbab47..3b58bdf 100644
--- a/src/op/system/dilation.tex
+++ b/src/op/system/dilation.tex
@@ -290,7 +290,7 @@
 
 \begin{theorem}[Stinespring's Dilation Theorem]
 \label{theorem:stinespring-dilation}
-    Let $A$ be a unital $C^*$-algebra, $H$ be a complex Hilbert space, and $\phi: A \to B(H)$ be a completely positive map, then there exists a triple $(K, V, \pi)$, where:
+    Let $A$ be a unital $C^*$-algebra, $H$ be a complex Hilbert space, and $\phi: A \to B(H)$ be a completely positive map, then there exists a triple $(\pi, V, K)$, where:
     \begin{enumerate}[label=(\alph*)]
         \item $K$ is a complex Hilbert space. 
         \item $V: H \to K$ is a continuous linear map.
@@ -309,7 +309,7 @@
         \item Under the identification that $H = V(H) \subset K$, $(K, \pi(a))$ is a dilation of $\phi(a)$ for all $a \in A$. 
     \end{enumerate}
     
-    The triple $(K, V, \pi)$ is a \textbf{Stinespring representation} of $\phi$. If $\bracsn{\pi(a)VH|a \in A}$ is dense in $K$, then $(K, V, \pi)$ is \textbf{minimal}.
+    The triple $(\pi, V, K)$ is a \textbf{Stinespring representation} of $\phi$. If $\bracsn{\pi(a)VH|a \in A}$ is dense in $K$, then $(\pi, V, K)$ is \textbf{minimal}.
 \end{theorem}
 \begin{proof}[Proof, {{\cite[Theorem 4.1]{Paulsen}}}. ]
     (a): Let $A \otimes H$ and define the Hermitian form
@@ -392,6 +392,11 @@
     \]
 
     so $V$ is an isometry. 
-    
 \end{proof}
 
+\begin{proposition}[Multiplicative Domain of Stinespring Representations]
+\label{proposition:stinespring-multiplicative-domain}
+    Let $A$ be a unital $C^*$-algebra, $H$ be a complex Hilbert space, $\phi: A \to B(H)$ be a unital completely positive map, and $()
+\end{proposition}
+
+
diff --git a/src/op/system/index.tex b/src/op/system/index.tex
index b2aa376..fed8176 100644
--- a/src/op/system/index.tex
+++ b/src/op/system/index.tex
@@ -1,7 +1,6 @@
 \chapter{Operator Systems}
 \label{chap:opertaor-systems}
 
-\input{./dilation.tex}
 \input{./os.tex}
 \input{./two.tex}
 \input{./completely.tex}
@@ -9,3 +8,4 @@
 \input{./module.tex}
 \input{./numerical-radius.tex}
 \input{./schur.tex}
+\input{./dilation.tex}