Editing.

src/op/vn/projection.tex

@@ -0,0 +1,34 @@
+\section{The Projection Lattice}
+\label{section:vn-projection-lattice}
+
+\begin{definition}[Projection Lattice]
+\label{definition:vn-projection-lattice}
+    Let $H$ be a complex Hilbert space and $A \subset B(H)$ be a von Neumann algebra, then the set of all projections $\text{Proj}(A)$ in $A$ is the \textbf{projection lattice} of $A$.
+\end{definition}
+
+\begin{proposition}
+\label{proposition:vn-projection-lattice-complete}
+    Let $H$ be a complex Hilbert space and $A \subset B(H)$ be a von Neumann algebra, then $\text{Proj}(A)$ is order complete. 
+\end{proposition}
+\begin{proof}
+    By \autoref{theorem:existence-of-projections-vna}. 
+\end{proof}
+
+
+\begin{definition}[Murray-von Neumann Equivalent]
+\label{definition:murrey-von-neuman-equivalent}
+    Let $H$ be a complex Hilbert space, $A \subset B(H)$ be a von Neumann algebra, and $P, Q \in \text{Proj}(A)$, then the following are equivalent:
+    \begin{enumerate}
+        \item There exists $V \in A$ such that $P = V^*V$ and $Q = VV^*$. 
+        \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 $\text{Proj}(A)$. 
+\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 $Q(H)$.  
+
+    (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}
+
+

src/op/vn/spec.tex

@@ -292,7 +292,7 @@
 \label{corollary:commutative-von-neumann-linfty}
     Let $H$ be a complex Hilbert space and $A \subset B(H)$ be a commutative von Neumann algebra with $I \in A$, then:
     \begin{enumerate}
-        \item There exists a LCH space $\Omega$ and a decomposable Radon measure $\mu$ on $\Omega$ such that $A$ is *-isomorphic to $L^\infty(\Omega; \complex)$. 
+        \item There exists a LCH space $\Omega$ and a decomposable Radon measure $\mu$ on $\Omega$ such that $A$ is *-isomorphic to $L^\infty(\mu; \complex)$. 
         \item If $A$ admits a cyclic vector, then $\Omega$ may be taken to be compact. 
         \item If $H$ is separable, then $\Omega$ may be taken to be compact. 
     \end{enumerate}
@@ -309,7 +309,7 @@
 
     (2): By \hyperref[Spectral Theorem II]{theorem:spectral-theorem-vn-2}, there exists a single positive Radon measure $\mu \in \mathscr{E}$ on $\Omega(A)$ such that $\mathscr{E}$ is absolutely continuous with respect to it. Therefore the index set in (1) can be taken to be a singleton. 
 
-    (3): If $H$ is separable, then so is $\mathscr{E}$. As such, there exists a single positive Radon measure on $\mu \in \mathscr{E}$ on $\Omega(A)$ such that $\mathscr{E}$ is absolutely continuous with respect to it. Therefore the index set in (1) can be taken to be a singleton. 
+    (3): If $H$ is separable, then so is $\mathscr{E}$. As such, there exists a single positive Radon measure $\mu \in \mathscr{E}$ on $\Omega(A)$ such that $\mathscr{E}$ is absolutely continuous with respect to it. Therefore the index set in (1) can be taken to be a singleton. 
 \end{proof}