From f165b4fd2d273bbc579486250ac1c146b7e40a0e Mon Sep 17 00:00:00 2001
From: Bokuan Li <bokuan.li@mail.mcgill.ca>
Date: Thu, 27 Aug 2026 20:30:50 -0400
Subject: [PATCH] Added citation for the type 1 proof.

---
 refs.bib                  | 10 ++++++++++
 src/op/vn/type-decomp.tex |  2 +-
 2 files changed, 11 insertions(+), 1 deletion(-)

diff --git a/refs.bib b/refs.bib
index e2379b0..2bd01af 100644
--- a/refs.bib
+++ b/refs.bib
@@ -283,3 +283,13 @@
  volume = {2},
  year = {1951}
 }
+
+
+@MISC {TownesType1,
+    TITLE = {Classification of Type 1 factors},
+    AUTHOR = {Leslie Townes (https://math.stackexchange.com/users/18076/leslie-townes)},
+    HOWPUBLISHED = {Mathematics Stack Exchange},
+    NOTE = {URL:https://math.stackexchange.com/q/150258 (version: 2012-05-27)},
+    EPRINT = {https://math.stackexchange.com/q/150258},
+    URL = {https://math.stackexchange.com/q/150258}
+}
\ No newline at end of file
diff --git a/src/op/vn/type-decomp.tex b/src/op/vn/type-decomp.tex
index 4addd5f..f8e10c4 100644
--- a/src/op/vn/type-decomp.tex
+++ b/src/op/vn/type-decomp.tex
@@ -234,7 +234,7 @@
         \item There exists a complex Hilbert space $K$ and a *-isomorphism $\pi: A \to B(K)$. 
     \end{enumerate}
 \end{theorem}
-\begin{proof}
+\begin{proof}[Proof, {{\cite{TownesType1}}}. ]
     (1) $\Rightarrow$ (2): Since $A$ is of type $\vnI$, $A$ admits a non-zero abelian projection $Q \in \text{Proj}(A)$. By \autoref{lemma:abelian-minimal-factor}, $Q$ is minimal. 
 
     (2) $\Rightarrow$ (3): Let $P \in \text{Proj}(A)$ and $Q \in \text{Proj}(A)$ be a minimal projection. By the \hyperref[comparability theorem]{corollary:vna-factor-comparability}, either $P \preceq Q$ or $Q \preceq P$.