From fbf94061cb5afce69923b5a31e96b4c988759525 Mon Sep 17 00:00:00 2001
From: Bokuan Li <bokuan.li@mail.mcgill.ca>
Date: Wed, 26 Aug 2026 19:26:51 -0400
Subject: [PATCH] Typo fixes in projection section.

---
 src/op/vn/projection.tex | 10 +++++-----
 1 file changed, 5 insertions(+), 5 deletions(-)

diff --git a/src/op/vn/projection.tex b/src/op/vn/projection.tex
index 7868584..5705ca0 100644
--- a/src/op/vn/projection.tex
+++ b/src/op/vn/projection.tex
@@ -134,7 +134,7 @@
         \item There exists a partial isometry $V \in A$ from $P(H)$ to $Q(H)$. 
     \end{enumerate}
     
-    If the above holds, then $P$ and $Q$ are \textbf{Murrey-von Neumann equivalent}, denoted $P \sim Q$. The relation $\sim$ is an equivalence relation on $\text{Proj}(A)$. 
+    If the above holds, then $P$ and $Q$ are \textbf{Murray-von Neumann equivalent}, denoted $P \sim Q$. The relation $\sim$ is an equivalence relation on $\text{Proj}(A)$. 
 \end{definition}
 \begin{proof}
     (1) $\Rightarrow$ (2): Let $V \in A$ with $P = V^*V$ and $Q = VV^*$. By \autoref{proposition:partial-isometry-characterisation}, $V$ is a partial isometry with initial space $\ker(P)^\perp$, and $V^*$ is a partial isometry with initial space $\ker(Q)^\perp$. Therefore $V$ is a partial isometry from $P(H)$ to $Q(H)$.  
@@ -142,9 +142,9 @@
     (2) $\Rightarrow$ (1): By \autoref{proposition:partial-isometry-characterisation}, $P = V^*V$ is a projection onto $\ker(V)^\perp$, and $Q = VV^*$ is a projection onto $\ker(V^*)^\perp = V(H)$.
 \end{proof}
 
-\begin{definition}[Murrey-von Neumann Subequivalent]
+\begin{definition}[Murray-von Neumann Subequivalent]
 \label{definition:murray-von-neumann-subequivalent}
-    Let $H$ be a complex Hilbert space, $A \subset B(H)$ be a von Neumann algebra, and $P, Q \in \text{Proj}(A)$, then $P$ is \textbf{Murrey-von Neumann subequivalent} to $Q$, denoted $P \preceq Q$, if there exists $R \in \text{Proj}(A)$ such that $P \sim R$ and $R \le Q$. 
+    Let $H$ be a complex Hilbert space, $A \subset B(H)$ be a von Neumann algebra, and $P, Q \in \text{Proj}(A)$, then $P$ is \textbf{Murray-von Neumann subequivalent} to $Q$, denoted $P \preceq Q$, if there exists $R \in \text{Proj}(A)$ such that $P \sim R$ and $R \le Q$. 
 \end{definition}
 
 \begin{lemma}
@@ -259,7 +259,7 @@
 
 \begin{theorem}[The Comparability Theorem]
 \label{theorem:vna-comparability}
-    Let $H$ be a complex Hilbert space, $A \subset H$ be a von Neumann algebra, and $P, Q \in \text{Proj}(A)$, then there exists a central projection $R$ such that $RP \preceq RQ$ and $(1 - R)Q \preceq (1 - R)P$. 
+    Let $H$ be a complex Hilbert space, $A \subset B(H)$ be a von Neumann algebra, and $P, Q \in \text{Proj}(A)$, then there exists a central projection $R$ such that $RP \preceq RQ$ and $(I - R)Q \preceq (I - R)P$. 
 \end{theorem}
 \begin{proof}[Proof, {{\cite[Theorem 25.4]{Zhu}}}. ]
     By Zorn's lemma, there exists maximal families $\seqi{P}, \seqi{Q} \subset \text{Proj}(A)$ such that:
@@ -289,7 +289,7 @@
     Let $H$ be a complex Hilbert space and $A \subset B(H)$ be a factor, then for any $P, Q \in \text{Proj}(A)$, either $P \prec Q$, $P \sim Q$, or $Q \prec P$. 
 \end{corollary}
 \begin{proof}
-    By the \hyperref[comparability theorem]{theorem:vna-comparability}, there exists a central projection $R$ such that $PR \preceq QR$ and $Q(I - R) \preceq P(I - R)$. As $A$ is a factor, either $R = 0$ or $R = I$. In which case, $P \preqeq Q$ or $Q \preqeq P$. 
+    By the \hyperref[comparability theorem]{theorem:vna-comparability}, there exists a central projection $R$ such that $PR \preceq QR$ and $Q(I - R) \preceq P(I - R)$. As $A$ is a factor, either $R = 0$ or $R = I$. In which case, $P \preceq Q$ or $Q \preceq P$. 
 \end{proof}