Various fixes.

src/cat/cat/cat-func.tex

@@ -57,6 +57,14 @@
     Let $\catc$ be a category with a notion of embedding, then $\catc$ has \textbf{sufficiently many injectives} if every object embeds into an injective object. 
 \end{definition}
 
+\begin{definition}[Extension]
+\label{definition:category-extension-object}
+    Let $\catc$ be a category with a notion of embedding and $X \in \obj(\catc)$, then an \textbf{extension} of $X$ is a pair $(Y, \iota)$ where $Y \in \obj(\catc)$ and $\iota: X \to Y$ is an embedding. 
+    
+    The extension $(Y, \iota)$ is \textbf{essential} if for any $Z \in \obj(\catc)$ and $\phi: Y \To Z$, $\phi \circ \iota: X \to Z$ being an embedding implies that $\phi$ is also an embedding. 
+\end{definition}
+
+
 
 
 

src/dyn/meas/rec.tex

@@ -0,0 +1,67 @@
+\section{Measurable Recurrence}
+\label{section:measure-recurrence}
+
+\begin{theorem}[Poincaré Recurrence]
+\label{theorem:pointcare-recurrence}
+    Let $(\Omega, \cf, \bp)$ be a probability space, $T: \Omega \to \Omega$ be an automorphism, and $A \in \cf$, then $\bracsn{n \in \natp|T^n\omega \in A}$ is infinite for almost every $\omega \in A$. 
+\end{theorem}
+\begin{proof}
+    Let $W = \bracsn{\omega \in A| T^n\omega \not\in A \forall n \in \natp}$. Since $W \subset A$, $T^nW \cap W = \emptyset$ for all $\in \natp$. Thus $\bracsn{T^nW}_0^\infty \subset \cf$ is pairwise disjoint. As $T$ is probability measure preserving,
+    \[
+    1 \ge \bp\paren{\bigsqcup_{n \in \natz}T^nW} = \sum_{n \in \natz}\bp(T^nW) = \sum_{n \in \natz}\bp(W)
+    \]
+
+    so $\bp(W) = 0$. 
+
+    Let $E = \bracsn{\omega \in A|T^n\omega \in A \text{ finitely many times}}$, then for each $\omega \in E$, there exists $n_\omega \in \natp$ such that $T^{n_\omega}\omega \in W$. In other words, $E \subset \bigcup_{k \in \natp} T^{-k}W$, and $E$ is also null. Since $\bracsn{n \in \natp|T^n\omega \in A}$ is infinite for all $\omega \in A \setminus E$, $\bracsn{n \in \natp|T^n\omega \in A}$ is infinite for almost every $\omega \in A$. 
+\end{proof}
+
+\begin{corollary}
+\label{corollary:measure-set-recurrence}
+    Let $(\Omega, \cf, \bp)$ be a probability space, $T: \Omega \to \Omega$ be an automorphism, and $A \in \cf$ with $\bp(A) > 0$, then there exists $n \in \natp$ such that $\bp(A \cap T^{-n}A) > 0$. 
+\end{corollary}
+\begin{proof}
+    By the \hyperref[Poincaré Recurrence Theorem]{theorem:pointcare-recurrence}, $A \subset \bigcup_{n \in \natp}A \cap T^{-n}A$. Since $\bp(A) > 0$, \hyperref[subadditivity]{proposition:measure-properties}, there exists $n \in \natp$ with $\bp(A \cap T^{-n}A) > 0$. 
+\end{proof}
+
+\begin{definition}[Koopman Representation]
+\label{definition:koopman-representation}
+    Let $\Gamma$ be a group acting on a measure space $(X, \cm, \mu)$, then the unitary representation
+    \[
+    \pi: \Gamma \to U(B(L^2(X, \cm, \mu; \complex))) \quad \pi(\gamma)(f) = f \circ \gamma
+    \]
+
+    is the \textbf{Koopman representation} of $\Gamma$. 
+\end{definition}
+
+
+\begin{proposition}
+\label{proposition:koopman-ergodic}
+    Let $\Gamma$ be a group acting on a measure space $(X, \cm, \mu)$ and $\pi: \Gamma \to U(B(L^2(X, \cm, \mu; \complex)))$ be its Koopman representation, then the subspace
+    \[
+    \text{Fix}(\Gamma) = \bracsn{f \in L^2(X, \cm, \mu; \complex)| \pi(\gamma) f = f \forall \gamma \in \Gamma}
+    \]
+
+    is equal to $\complex \one$ if and only if the action of $\Gamma$ is ergodic. 
+\end{proposition}
+
+\begin{definition}[Averaging Operator]
+\label{definition:ergodic-averaging-operator}
+    Let $(X, \cm, \mu)$ be a measure space, $T: X \to X$ be an automorphism, $E$ be a Banach space, $p \in [1, \infty]$, and $N \in \natp$, then the \textbf{$N$-th time averaging operator} is the mapping
+    \[
+    A_N: L^p(X; E) \to L^p(X; E) \quad A_Nf = \frac{1}{N}\sum_{n = 0}^{N-1} f \circ T^n
+    \]
+
+    which is a contraction on $L^p(X; E)$. 
+\end{definition}
+
+\begin{theorem}[von Neumann's Mean Ergodic Theorem]
+\label{theorem:vn-mean-ergodic}
+    Let $H$ be a Hilbert space, $U: H \to H$ be a unitary operator, $P \in B(H)$ be the orthogonal projection onto the fixed points of $U$. For each $N \in \natp$, let
+    \[
+    A_N: H \to H \quad \xi \mapsto \frac{1}{N}\sum_{n = 0}^{N-1} U^n\xi
+    \]
+
+    be the $N$-th time averaging operator. Then, for each $\xi \in H$, $P\xi = \slim_{N \to \infty}A_N\xi$. 
+\end{theorem}
+

src/op/notation.tex

@@ -7,6 +7,7 @@
     $1$ & Identity element of a unital algebra. & \autoref{definition:unital-banach-algebra} \\
     $Z(A)$ & Centre of a Banach algebra. & \autoref{definition:banach-algebra-centre} \\ 
     $G(A)$ & Invertible group of a unital algebra. & \autoref{definition:banach-algebra-invertible} \\
+    $U(A)$ & Unitary group of a unital algebra. & \autoref{definition:banach-algebra-invertible} \\
     $G_0(A)$ & The identity component of $G(A)$. & \autoref{definition:identity-component} \\
     $I(A)$ & The index group of $A$. & \autoref{definition:index-group} \\
     $\sigma_A(x) = \sigma(x)$ & The spectrum of $x$ in $A$. & \autoref{definition:spectrum} \\
@@ -21,12 +22,6 @@
     $\dpn{x, y}{\phi}$ & Defined as $\dpn{y^*x, \phi}{A}$, the pseudo inner product associated to a positive linear functional. & \autoref{definition:cstar-state-pseudo-inner-product} \\ 
     $(H_\phi, \pi_\phi, \xi_\phi)$ & GNS triple associated with $\phi \in S(A)$. & \autoref{definition:gns-triple} \\ 
     $L^2(M, \phi)$ & Hilbert space of the GNS triple associated with a faithful $\phi \in S(A)$. & \autoref{definition:gns-triple} \\ 
-    $U(T)$ & Cayley transform of $T$. & \autoref{definition:cayley-transform-bounded} \\ 
-    $E_{x, y}$ & Section of a spectral measure $E$, where $E_{x, y}(B) = \dpn{E(B)x, y}{H}$. & \autoref{definition:spectral-measure} \\ 
-    $\text{Proj}(A)$ & Projections in $A$. & \autoref{definition:vn-projection-lattice} \\
-    $Z(P)$ & Central support of $P \in \text{Proj}(A)$. & \autoref{definition:central-support-vna} \\ 
-    $P \sim Q$ & $P, Q \in \text{Proj}(A)$ are Murray-von Neumann equivalent & \autoref{definition:murray-von-neumann-equivalent} \\ 
-    $P \preceq Q$ & $P$ is Murray-von Neumann subequivalent to $Q$. & \autoref{definition:murray-von-neumann-subequivalent} \\ 
     
     $M_n(\complex)$ & Algebra of $n \times n$ matrices over $\complex$.  & \autoref{definition:matrix-algebra} \\
     $B(H)$ & Algebra of bounded operators on a Hilbert space. & \autoref{definition:hilbert-endomorphism} \\

src/op/system/inj.tex

@@ -2,7 +2,7 @@
 \label{section:injective-ops}
 
 
-By \hyperref[Arveson's Extension Theorem]{theorem:arveson-extension}, $B(H)$ is injective for any Hilbert space $H$. Further, the \hyperref[Choi-Effros Theorem]{choi-effros} implies that every operator system embeds into $B(H)$, for a suitable Hilbert space $H$. Therefore the category of operator systems, with unital complete order isomorphisms, has sufficiently many injectives. 
+By \hyperref[Arveson's Extension Theorem]{theorem:arveson-extension}, $B(H)$ is injective for any Hilbert space $H$. Further, the \hyperref[Choi-Effros Theorem]{theorem:choi-effros} implies that every operator system embeds into $B(H)$, for a suitable Hilbert space $H$. Therefore the category of operator systems, with unital complete order isomorphisms, has sufficiently many injectives. 
 
 \begin{theorem}
 \label{theorem:opsys-predual}
@@ -17,8 +17,87 @@ By \hyperref[Arveson's Extension Theorem]{theorem:arveson-extension}, $B(H)$ is
 
 \begin{definition}[Injective]
 \label{definition:injective-operator-system}
-    Let 
+    Let $I$ be an operator system, then $I$ is \textbf{injective} if for any pair of operator systems $S, T$ with $S \subset T$ and unital completely positive map $\phi: S \to I$, there exists a unital completely positive map $\Phi: T \to I$ such that the following diagram commutes
+    \[
+    \xymatrix{
+T \ar@{->}[r]^{\Phi} & I \\
+S \ar@{->}[ru]_{\phi} \ar@{^{(}->}[u] & 
+}
+    \]
+\end{definition}
+
+\begin{definition}[Conditional Expectation]
+\label{definition:os-conditional-expectation}
+    Let $S, T$ be operator systems with $S \subset T$ and $\phi: T \to S$ be a unital completely positive map, then $\phi$ is a \textbf{conditional expectation} if $\phi|_{T} = \text{Id}_T$. 
 \end{definition}
 
 
+\begin{proposition}
+\label{proposition:os-injective-ce}
+    Let $I$ be an operator system, then the following are equivalent:
+    \begin{enumerate}
+        \item $I$ is injective. 
+        \item For every complex Hilbert space $H$ and embedding $\iota: I \to B(H)$, there exists a conditional expectation $\phi: B(H) \to \iota(I)$. In other words, $\phi$ is a unital completely positive map such that the following diagram commutes
+        \[
+        \xymatrix{
+        B(H) \ar@{->}[r]^{\phi} & \iota(I) \\
+        I \ar@{=}[ru] \ar@{^{(}->}[u] & 
+        }
+        \]
+    \end{enumerate}
+\end{proposition}
+\begin{proof}
+    (2) $\Rightarrow$ (1): By the \hyperref[Choi-Effros Theorem]{theorem:choi-effros} and \hyperref[Arveson's Extension Theorem]{theorem:arveson-extension}. 
+\end{proof}
+% "obvious"
+
+\begin{example}[Dixmier Algebra]
+    \label{example:sup-not-vna}
+    Let
+    \[
+    A = B^\infty([0, 1])/\bracsn{f \in B^\infty([0, 1])| \bracsn{f \ne 0} \text{ is meagre}}
+    \]
+    
+    then $A$ is an injective, order complete $C^*$-algebra, but $A$ is not a von Neumann algebra. 
+\end{example}
+
+
+\begin{proposition}
+\label{proposition:os-injective-cstar}
+    Let $H$ be a complex Hilbert space, $I \subset B(H)$ be an operator system, and $\phi: B(H) \to I$ be a conditional expectation, then $(x, y) \mapsto \phi(xy)$ is a well-defined product on $I$, making it a $C^*$-algebra. 
+\end{proposition}
+
+
+\begin{proposition}
+\label{proposition:os-injective-cstar}
+    Let $I$ be an injective operator system, then there exists a $C^*$-algebra $A$ such that $I$ is isomorphic to $A$ as operator systems. 
+\end{proposition}
+
+\begin{proposition}
+\label{proposition:os-maximal-essential-extension}
+    Let $S$ be an operator system, then there exists a pair $(\wh S, \iota)$ such that:
+    \begin{enumerate}
+        \item $(\wh S, \iota)$ is an essential extension of $S$. 
+        \item For any proper\footnote{Non-isomorphic.} extension $(T, j)$ of $\wh S$, $(Z, j \circ \iota)$ is not an essential extension of $S$. In other words, every essential extension of $(\wh S, \iota)$ is isomorphic to $(\wh S, \iota)$.
+    \end{enumerate} 
+
+    The pair $(\wh S, \iota)$ is a \textbf{maximal essential extension} of $S$. 
+\end{proposition}
+\begin{proof}
+    Zorn's lemma. 
+\end{proof}
+
+\begin{theorem}
+\label{theorem:maximal-essential-injective}
+    
+\end{theorem}
+
+
+
+% I sure wonder how that will happen 
+
+
+
+
+
 

src/vna/example/tensor.tex

@@ -0,0 +1,16 @@
+\section{Tensor Products of von Neumann Algebras}
+\label{section:vna-tensor}
+
+\begin{definition}[Tensor Product of von Neumann Algebras]
+\label{definition:vna-tensor}
+    Let $H_1, H_2$ be complex Hilbert spaces, $H_1 \wh \otimes_\pi H_2$ be their Hilbert space tensor product, $M_1 \subset B(H_1)$ and $M_2 \subset B(H_2)$ be von Neumann algebras, and
+    \[
+    \iota: M_1 \otimes M_2 \to B(H_1 \wh \otimes_\pi H_2) \quad (x_1 \otimes x_2)(\xi_1 \otimes \xi_2) = (x_1\xi_1) \otimes (x_2\xi_2)
+    \]
+
+    Then, $M_1 \ol{\otimes} M_2 := \iota(M_1 \otimes M_2)'' \subset B(H_1 \wh \otimes_\pi H_2)$ is the \textbf{von Neumann tensor product} of $(M_1, H_1)$ and $(M_2, H_2)$.
+
+    The von Neumann algebra $M_1 \ol{\otimes} M_2$ is unique up to isomorphism, independent of the representations of $M_1$ and $M_2$\footnote{Proof omitted for now.}. 
+\end{definition}
+
+

src/vna/ii1/standard.tex

@@ -0,0 +1,162 @@
+\section{The Standard Representation}
+\label{section:standard-representation}
+
+
+
+\begin{definition}[Standard Representation]
+\label{definition:tracial-vna-standard}
+    Let $(M, \tau)$ be a tracial von Neumann algebra, then the \hyperref[GNS representation]{definition:gns-triple}
+    \[
+    (\pi, L^2(M, \tau), \xi_\tau)
+    \]
+
+    is the \textbf{standard representation} of $(M, \tau)$. For each $x \in M$, identify $\wh x = x\xi_\tau \in L^2(M, \tau)$, so that $M = \wh M \subset L^2(M, \tau)$. 
+\end{definition}
+
+\begin{definition}[Canonical Conjugation]
+\label{definition:tracial-vna-standard-conjugate}
+    Let $(M, \tau)$ be a tracial von Neumann algebra, then the mapping
+    \[
+    J: L^2(M, \tau) \to L^2(M, \tau) \quad \wh x \mapsto \wh{x^*}
+    \]
+
+    is a conjugate linear isometry on $L^2(M, \tau)$, known as the \textbf{canonical conjugation}. 
+\end{definition}
+
+
+\begin{definition}[Left/Right Regular Representations]
+\label{definition:mvn-standard-regular}
+    Let $(M, \tau)$ be a tracial von Neumann algebra and $x \in M$, then
+    \[
+    L_x: L^2(M, \tau) \to L^2(M, \tau) \quad L_x\wh y = \wh{xy}
+    \]
+    and
+
+    \[
+    R_x: L^2(M, \tau) \to L^2(M, \tau) \quad R_x\wh y = \wh{yx}
+    \]
+
+    are the \textbf{left/right multiplication maps} by $x$, respectively. The mapping
+    \[
+    \lambda: M \to B(L^2(M, \tau)) \quad x \mapsto L_x
+    \]
+
+    is the \textbf{left regular representation} of $M$, with $L(M) = \lambda(M)$ being the \textbf{algebra of left multiplications}. Similarly,
+    \[
+    \rho: M^{op} \to B(L^2(M, \tau)) \quad x \mapsto R_x
+    \]
+
+    is the \textbf{right regular representation} of $M^{op}$, with $R(M) = \rho(M)$ being the \textbf{algebra of right multiplications}. 
+\end{definition}
+
+\begin{proposition}
+\label{proposition:vna-standard-conjugation-lr}
+    Let $(M, \tau)$ be a tracial von Neumann algebra, then:
+    \begin{enumerate}
+        \item For each $x \in M$, $JL_xJ = R_{x^*}$. 
+        \item $JL(M)J = R(M)$. 
+    \end{enumerate}
+\end{proposition}
+\begin{proof}
+    (1): For each $x, y \in M$, 
+    \[
+    JL_xJ \wh y = JL_x \wh{y^*} = J \wh{xy^*} = \wh{yx^*} = R_{x^*}y
+    \]
+\end{proof}
+
+\begin{definition}[Multiplication by Vector]
+\label{definition:vna-standard-vector-multiplication}
+    Let $(M, \tau)$ be a tracial von Neumann algebra and $\xi \in L^2(M, \tau)$, then
+    \[
+    L^0_\xi: \wh M \to L^2(M, \tau) \quad L_\xi^0(\wh y) = R_y(\xi) = \xi y
+    \]
+
+    and
+    \[
+    R^0_\xi: \wh M \to L^2(M, \tau) \quad R_\xi^0(\wh y) = L_y(\xi) = y\xi
+    \]
+
+    are densely defined, closable operators. 
+    
+    Let $L_\xi$ and $R_\xi$ be their respective closures, then $L_\xi$ and $R_\xi$ are the \textbf{left/right multiplication operators by $\xi$}. 
+\end{definition}
+\begin{proof}
+    Let $\seq{x_n} \subset M$ and $\eta \in L^2(M, \tau)$ with $\limv{n}\wh{x_n} = 0$ and $\limv{n}L_\xi^0(\wh{x_n}) = \eta$, then for each $y \in M$, 
+    \begin{align*}
+    \dpn{\eta, \wh y}{L^2(M, \tau)} &= \limv{n} \dpn{L_\xi^0(\wh {x_n}), \wh y}{L^2(M, \tau)} = \limv{n} \dpn{R_{x_n}\xi, \wh y}{L^2(M, \tau)} \\
+    &= \limv{n}\dpn{\xi, R_{x_n}^*\wh y}{L^2(M, \tau)} = \limv{n}\dpn{\xi, R_{x_n^*}\wh y}{L^2(M, \tau)} \\
+    &= \limv{n}\dpn{\xi, \wh {yx_n^*}}{L^2(M, \tau)} \\
+    |\dpn{\eta, \wh y}{L^2(M, \tau)}| &\le \limv{n}\norm{\xi}_{L^2(M, \tau)} \normn{\wh {yx_n^*}}_{L^2(M, \tau)} \\
+    &\le  \limv{n}\norm{\xi}_{L^2(M, \tau)} \normn{y}_{M} \normn{\wh {x_n^*}}_{L^2(M, \tau)} = 0
+    \end{align*}
+    
+\end{proof}
+
+\begin{definition}[Convolvers]
+\label{definition:vna-standard-convolver}
+    Let $(M, \tau)$ be a tracial von Neumann algebra and $\xi \in L^2(M, \tau)$, then $\xi$ is a \textbf{left convolver} if $L_\xi$ is bounded, and $\xi$ is a \textbf{right convolver} if $R_\xi$ is bounded. The sets of left and right convolvers are denoted $LC(M)$ and $RC(M)$, respectively. 
+\end{definition}
+
+\begin{lemma}
+\label{lemma:vna-convolver-commute}
+    Let $(M, \tau)$ be a tracial von Neumann algebra, then for each $\xi \in LC(M)$ and $\eta \in RC(M)$, $L_\xi R_\eta = R_\eta L_\xi$.
+\end{lemma}
+\begin{proof}
+    First suppose that $\xi, \eta \in M$. In which case, for each $x, y \in M$, 
+    \[
+        \dpn{L_\xi R_\eta \wh x, \wh y}{L^2(M, \tau)} = \dpn{\wh{\xi x \eta}, \wh y}{L^2(M, \tau)} = \dpn{R_\eta L_\xi \wh x, \wh y}{L^2(M, \tau)}
+    \]
+
+    Now suppose that $\xi \in M$, then for each $\eta \in RC(M)$, by the \hyperref[Cauchy-Schwarz inequality]{proposition:cauchy-schwarz}, 
+    \begin{align*}
+    |\dpn{L_\xi R_\eta \wh x, \wh y}{L^2(M, \tau)}| &\le \norm{L_\xi}_{B(L^2(M, \tau))} \norm{x}_M \norm{y}_M \cdot  \norm{\eta}_{L^2(M, \tau)} \\ 
+    |\dpn{R_\eta L_\xi \wh x, \wh y}{L^2(M, \tau)}| &\le \norm{x}_M \norm{\xi}_M\norm{y}_M \cdot \norm{\eta}_{L^2(M, \tau)}
+    \end{align*}
+    
+
+    so by density of $\wh M$ in $L^2(M, \tau)$, $\dpn{L_\xi R_\eta \wh x, \wh y}{L^2(M, \tau)} = \dpn{R_\eta L_\xi \wh x, \wh y}{L^2(M, \tau)}$ for all $\xi \in M$ and $\eta \in RC(M)$. 
+
+    Finally, suppose that $\xi \in LC(M)$ and $\eta \in RC(M)$, then for each $z \in M$, 
+    \begin{align*}
+        \dpn{L_\xi z, \wh y}{L^2(M, \tau)} &= \dpn{\xi z, \wh y}{L^2(M, \tau)} = \dpn{\xi, \wh y z^*}{L^2(M, \tau)} \\
+        &= \dpn{\xi, \wh y J\wh z}{L^2(M, \tau)} = \dpn{y^*\xi, J\wh z}{L^2(M, \tau)} = \dpn{\wh z, Jy^*\xi}{L^2(M, \tau)}
+    \end{align*}
+
+    so
+    \begin{align*}
+        \dpn{L_\xi R_\eta \wh x, \wh y}{L^2(M, \tau)} &= \dpn{L_\xi (x \eta), \wh y}{L^2(M, \tau)} = \dpn{x \eta, J y^* \xi}{L^2(M, \tau)} \\
+        |\dpn{L_\xi R_\eta \wh x, \wh y}{L^2(M, \tau)}| &\le \norm{x \eta}_{L^2(M, \tau)} \norm{y}_M  \cdot \norm{\xi}_{L^2(M, \tau)}\\
+        |\dpn{R_\eta L_\xi \wh x, \wh y}{L^2(M, \tau)}| &\le \norm{R_\eta}_{B(L^2(M, \tau))} \norm{x}_M \norm{y}_M \cdot \norm{\xi}_{L^2(M, \tau)}
+    \end{align*}
+    
+    By density of $\wh M$ in $L^2(M, \tau)$, $\dpn{L_\xi R_\eta \wh x, \wh y}{L^2(M, \tau)} = \dpn{R_\eta L_\xi \wh x, \wh y}{L^2(M, \tau)}$ for all $\xi \in LC(M)$ and $\eta \in RC(M)$. 
+\end{proof}
+
+\begin{theorem}
+\label{theorem:vna-convolver-identified}
+    Let $(M, \tau)$ be a tracial von Neumann algebra, then:
+    \begin{enumerate}
+        \item $L(M) = LC(M) = R(M)'$. 
+        \item $JL(M)J = R(M) = RC(M) = L(M)'$. 
+        \item In particular, for each $\xi \in L^2(M, \tau)$, $L_\xi \in B(L^2(M, \tau))$ if and only if $\xi \in \wh M$. 
+    \end{enumerate}
+\end{theorem}
+\begin{proof}
+    (1): By definition, $L(M) \subset LC(M)$. By \autoref{lemma:vna-convolver-commute}, $LC(M) \subset RC(M)'$, so
+    \[
+    L(M) \subset LC(M) \subset RC(M)' \subset R(M)'
+    \]
+
+    and
+    \[
+    R(M) \subset RC(M) \subset LC(M)' \subset L(M)'
+    \]
+
+    Using the \hyperref[Bicommutant Theorem]{theorem:bicommutant}, it remains to show that $R(M)' \subset LC(M)$. To this end, let $T \in R(M)'$ and $\xi = T \wh 1$, then for each $x \in M$, 
+    \[
+    T\wh x = TR_x \wh 1 = R_x T \wh 1 = R_x \xi = \xi x = L_\xi^0 \wh x
+    \]
+
+    so $T|_{\wh M} = L_\xi^0$. By density of $\wh M$ in $L^2(M, \tau)$ and continuity of $T$, the closure of $L_\xi^0$ is $T$, hence $T = L_\xi$. 
+\end{proof}
+

src/vna/ii1/trace.tex

@@ -15,11 +15,6 @@
     Let $M$ be a von Neumann algebra and $\tau: M \to \complex$ be a faithful tracial state, then $(M, \tau)$ is a \textbf{tracial von Neumann algebra} if $\tau$ is normal. 
 \end{definition}
 
-\begin{definition}[Standard Representation]
-\label{definition:tracial-vna-standard}
-    Let $(M, \tau)$ be a tracial von Neumann algebra, then its \hyperref[GNS]{definition:gns-triple} representation on $L^2(M, \tau)$ is the \textbf{standard representation} of $(M, \tau)$.  
-\end{definition}
-
 \begin{proposition}
 \label{proposition:cstar-trace-vna}
     Let $M$ be a unital $C^*$-algebra, $\tau: M \to \complex$ be a faithful tracial state, and $(L^2(M, \tau), \pi, 1_M)$ be its \hyperref[GNS]{definition:gns-triple} triple, then:

src/vna/notation.tex

@@ -5,9 +5,37 @@
     \textbf{Notation} & \textbf{Description} & \textbf{Source} \\
     \hline
     
+    
+    $U(T)$ & Cayley transform of $T$. & \autoref{definition:cayley-transform-bounded} \\ 
     $E_{x, y}$ & Section of a spectral measure $E$, where $E_{x, y}(B) = \dpn{E(B)x, y}{H}$. & \autoref{definition:spectral-measure} \\ 
     $\text{Proj}(A)$ & Projections in $A$. & \autoref{definition:vn-projection-lattice} \\
     $Z(P)$ & Central support of $P \in \text{Proj}(A)$. & \autoref{definition:central-support-vna} \\ 
     $P \sim Q$ & $P, Q \in \text{Proj}(A)$ are Murray-von Neumann equivalent & \autoref{definition:murray-von-neumann-equivalent} \\ 
     $P \preceq Q$ & $P$ is Murray-von Neumann subequivalent to $Q$. & \autoref{definition:murray-von-neumann-subequivalent}
+
+    $A'$ & Commutant of $A \subset B(H)$, the operators in $B(H)$ commuting with every element of $A$. & \autoref{definition:commutant} \\
+    $I_E(\phi) = \int_X \phi dE$ & Integral of $\phi$ with respect to a spectral measure $E$. & \autoref{definition:spectral-measure-integral} \\
+    $\phi(T)$ & $L^\infty$-functional calculus of a normal operator $T$, defined as $\int_{\sigma_{B(H)}(T)}\phi dE$. & \autoref{definition:linfty-functional-calculus} \\
+    $A_{\vnI}, A_{\vnII_1}, A_{\vnII_\infty}, A_{\vnIII}$ & Type $\vnI$, $\vnII_1$, $\vnII_\infty$, and $\vnIII$ summands of a von Neumann algebra $A$. & \autoref{theorem:vna-type-decomposition} \\
+ 
+    $\lambda$ & Left regular representation of $G$, and its extension to $L^1(G; \complex)$. & \autoref{definition:group-von-neumann-algebra} \\
+    $\rho$ & Right regular representation of $G$. & \autoref{definition:group-von-neumann-algebra} \\
+    $L(G)$ & Group von Neumann algebra of $G$, defined as $\lambda(G)''$. & \autoref{definition:group-von-neumann-algebra} \\
+    $R(G)$ & Defined as $\rho(G)''$, the von Neumann algebra generated by the right regular representation. & \autoref{definition:group-von-neumann-algebra} \\
+    $L_f$ & Left convolution map $g \mapsto f * g$ by a left convolver $f$. & \autoref{definition:left-convolver} \\
+    $LC(G)$ & Algebra of left convolvers on $G$. & \autoref{definition:left-convolver-algebra} \\
+    $f^*$ & Defined as $f^*(x) = \ol{f(x^{-1})}$ for a left convolver $f$. & \autoref{definition:left-convolver-algebra} \\
+    $u_x$ & Canonical unitary $\lambda(x)$ of $L(G)$, identified with $\delta_x$. & \autoref{definition:group-algebra-fourier-series} \\
+    $T_x$ & Fourier coefficients of $T \in L(G)$, where $T = \sum_{x \in G}T_xu_x$. & \autoref{definition:group-algebra-fourier-series} \\
+    ICC & Infinite conjugacy classes: every non-trivial conjugacy class is infinite. & \autoref{definition:icc} \\
+    $H^{(I)}$ & Defined as $\bigoplus_{i \in I}H$. & \autoref{definition:wreath-product} \\
+    $H \wr_I \Gamma$ & Generalised wreath product $H^{(I)} \rtimes \Gamma$. & \autoref{definition:wreath-product} \\
+    $H \wr \Gamma$ & Wreath product, the case $I = \Gamma$ with $\Gamma$ acting by left translation. & \autoref{definition:wreath-product} \\
+    $\lim_{\longrightarrow}(M_i, \tau_i)$ & Inductive limit of an upward-directed system of tracial von Neumann algebras. & \autoref{definition:inductive-limit-tracial-von-neumann-algebra} \\
+    $M_2(\complex)^{\ol \otimes \infty}$ & The hyperfinite $\vnII_1$ factor, defined as $\lim_{\longrightarrow}(M_{2^n}(\complex), \tau_n)$. & \autoref{definition:hyperfinite-ii1} \\
+ 
+    $Mq$ & Induced von Neumann algebra of $M$ on $qH$ with respect to $q \in \text{Proj}(M')$. & \autoref{definition:induced-vna-commutant} \\
+    $M^t$ & Amplification of a $\vnII_1$ factor $M$, defined as $pM_n(M)p$ with $(\text{Tr}_n \otimes \tau)p = t$. & \autoref{definition:ii1-amplification} \\
+    $\fF(M)$ & Fundamental group of a $\vnII_1$ factor $M$, defined as $\bracsn{t > 0|M^t \iso M}$. & \autoref{definition:ii1-fundamental-group}
+
 \end{tabular}

src/vna/vn/index.tex

@@ -8,5 +8,6 @@
 \input{./commutative.tex}
 \input{./spec.tex}
 \input{./fc.tex}
+\input{./unlimited.tex}
 \input{./projection.tex}
 \input{./type-decomp.tex}

src/vna/vn/spec.tex

@@ -24,10 +24,14 @@
     \begin{enumerate}
         \item For each $x, y \in H$, $\norm{E_{x, y}}_{\text{var}} \le \norm{x}_H \norm{y}_H$. 
         \item For each $x \in H$, $E_{x, x}$ is positive.
+        \item For each $x, y \in H$ and $B \in \cb_X$, 
+        \[
+        |E_{x, y}(B)| \le E_{x, x}(B)^{1/2} E_{y, y}(B)^{1/2} \le \frac{1}{2}(E_{x, x}(B) + E_{y, y}(B))
+        \]
     \end{enumerate}
 
     Let $\mathscr{E} \subset M_R(X; \complex)$ be the closed subspace generated by $\bracsn{E_{x, y}|x, y \in H}$, then
-    \begin{enumerate}[start=2]
+    \begin{enumerate}[start=3]
         \item For any $\mu \in \mathscr{E}$ and $\nu \in M_R(X; \complex)$ with $\nu \ll \mu$, $\nu \in \mathscr{E}$ as well. 
         \item Let 
         \[
@@ -50,7 +54,17 @@
 
     (2): For each $B \in \cb_X$, $E(B)$ is a projection, so $E_{x, x}(B) = \dpn{E(B)x, x}{H} \ge 0$.
 
-    (3): For each $x, y \in H$ and $B, C \in \cb_X$, 
+    (3): For each $B \in \cb_X$, by the \hyperref[Cauchy-Schwarz inequality]{proposition:cauchy-schwarz},
+    \begin{align*}
+        |E_{x, y}(B)| &= |\dpn{E(B)x, y}{H}| = |\dpn{E(B)x, E(B)y}{H}| \\
+        &\le \dpn{E(B)x, E(B)x}{H}^{1/2}\dpn{E(B)y, E(B)y}{H}^{1/2} \\
+        &= E_{x, x}(B)^{1/2} E_{y, y}(B)^{1/2}
+    \end{align*}
+
+    By \hyperref[Young's Inequality]{lemma:young-inequality}, $E_{x, x}(B)^{1/2} E_{y, y}(B)^{1/2} \le \frac{1}{2}(E_{x, x}(B) + E_{y, y}(B))$. 
+    
+
+    (4): For each $x, y \in H$ and $B, C \in \cb_X$, 
     \[
     \int_C \one_B dE_{x, y} = \dpn{E(C \cap B)x, y}{H} = \dpn{E(C)E(B)x, y}{H} = E_{E(B)x, y}(C)
     \]
@@ -59,7 +73,7 @@
     
     Finally, let $\mu \in \mathscr{E}$ and $\nu \in M_R(X; \complex)$ with $\nu \ll \mu$, then by the \hyperref[Radon-Nikodym Theorem]{theorem:lebesgue-radon-nikodym}, there exists $f \in L^1(\mu; \complex)$ such that $d\nu = f d\mu \in \mathscr{E}$. 
 
-    (4): By (3), for any $\mu \in \mathscr{E}$ and $f \in L^1(\mu; \complex)$, $fd\mu \in \mathscr{E}$ as well. By \autoref{proposition:measures-dual-algebra}, there exists a unique weak*-continuous involution and separately weak*-continuous product on $\mathscr{E}^*$ making $\mathscr{E}^*$ a commutative unital $C^*$-algebra, and $J|_{C(X; \complex)}$ a unital *-homomorphism. Since
+    (5): By (4), for any $\mu \in \mathscr{E}$ and $f \in L^1(\mu; \complex)$, $fd\mu \in \mathscr{E}$ as well. By \autoref{proposition:measures-dual-algebra}, there exists a unique weak*-continuous involution and separately weak*-continuous product on $\mathscr{E}^*$ making $\mathscr{E}^*$ a commutative unital $C^*$-algebra, and $J|_{C(X; \complex)}$ a unital *-homomorphism. Since
     \begin{enumerate}[label=(\roman*)]
         \item $J$ is $\sigma(B^\infty(X; \complex), M_R(X; \complex))$-$\sigma(\mathscr{E}^*, \mathscr{E})$ continuous. 
         \item Conjugation on $B^\infty(X; \complex)$ is $\sigma(B^\infty(X; \complex), M_R(X; \complex))$-continuous.
@@ -300,7 +314,7 @@
 \begin{proof}
     Let $E$ be the spectral measure on $\Omega(A)$ associated with $A$, and $\mathscr{E} \subset M_R(\Omega(A); \complex)$ be the closed subspace generated by $\bracsn{E_{x, y}|x, y \in H}$. By \hyperref[Spectral Theorem I]{theorem:spectral-theorem-vn-1}, $A$ is *-isomorphic to $\mathscr{E}^*$.
 
-    (1): By (3) of \autoref{lemma:spectral-measure-properties} and \autoref{theorem:hilbert-measures-dual}, $A$ is *-isomorphic to $[l^\infty(I); L^\infty(\mu_i; \complex)]$, where $\seqi{\mu} \subset \mathscr{E}$ is a maximal mutually singular family. Let $\Omega = \bigsqcup_{i \in I}\Omega(A)$, then $\Omega$ is a LCH space. For each $i \in I$, let $\Omega_i$ denote the $i$-th copy of $\Omega(A)$, then
+    (1): By (4) of \autoref{lemma:spectral-measure-properties} and \autoref{theorem:hilbert-measures-dual}, $A$ is *-isomorphic to $[l^\infty(I); L^\infty(\mu_i; \complex)]$, where $\seqi{\mu} \subset \mathscr{E}$ is a maximal mutually singular family. Let $\Omega = \bigsqcup_{i \in I}\Omega(A)$, then $\Omega$ is a LCH space. For each $i \in I$, let $\Omega_i$ denote the $i$-th copy of $\Omega(A)$, then
     \[
     \mu: \cb_\Omega \to [0, \infty] \quad B \mapsto \sum_{i \in I}\mu_i(B \cap \Omega_i)
     \]

src/vna/vn/unlimited.tex

@@ -0,0 +1,347 @@
+\section{Unbounded Functional Calculus}
+\label{section:unbounded-functional-calculus}
+
+
+
+\begin{definition}[Closed]
+\label{definition:unbounded-closed}
+    Let $H$ be a complex Hilbert space, $T: D(T) \to H$ be a linear operator, then $T$ is \textbf{closed} if its graph
+    \[
+    \Gamma(T) = \bracsn{(x, y) \in D(T) \times H| Tx = y}
+    \]
+
+    is a closed subspace of $H \times H$. 
+
+    The operator $T$ is \textbf{closable} if it admits a closed extension $S \supset T$. 
+\end{definition}
+
+
+\begin{definition}[Adjoint]
+\label{definition:unbounded-adjoint}
+    Let $H$ be a complex Hilbert space, $T: D(T) \to H$ be a densely defined linear operator, 
+    \[
+    D(T^*) = \bracsn{\xi \in H| (\eta \mapsto \dpn{\xi, T\eta}{H}) \text{ is bounded on }D(T)}
+    \]
+
+    and\footnote{$\eta$ sampled from $D(T)$.}
+    \[
+    T^*: D(T^*) \to H \quad \dpn{T^*\xi, \eta}{H} = \dpn{\xi, T\eta}{H}
+    \]
+
+    then $T^*$ is a closed operator, known as the \textbf{adjoint} of $T$. 
+\end{definition}
+\begin{proof}
+    Let $\seq{\xi_n} \subset D(T^*)$ and $\xi, \eta \in H$ such that $\xi_n \to \xi$ and $T^*\xi_n \to \eta$, then for any $\theta \in D(T)$, 
+    \[
+    \dpn{\eta, \theta}{H} = \limv{n}\dpn{T^*\xi_n, \theta}{H} = \limv{n}\dpn{\xi_n, T\theta}{H} = \dpn{\xi, T\theta}{H}
+    \]
+
+    As $D(T)$ is dense, $T \circ \xi \in D(T)^*$, and $\eta = T^*\xi$. Thus the graph of $T^*$ is closed.
+\end{proof}
+
+\begin{definition}[Self-Adjoint]
+\label{definition:unbounded-self-adjoint}
+    Let $H$ be a complex Hilbert space, $T: D(T) \to H$ be a densely defined linear operator, then $T$ is \textbf{self-adjoint} if $D(T) = D(T^*)$\footnote{Written out to emphasise.} and $T = T^*$. 
+\end{definition}
+
+
+
+\begin{lemma}
+\label{lemma:cayley-transform-unbounded}
+    Let $H$ be a complex Hilbert space, $T: D(T) \to H$ be a densely defined, self-adjoint operator, then:
+    \begin{enumerate}
+        \item For any $\xi \in D(T)$,
+        \[
+        \norm{(T \pm i)\xi}_H^2 = \norm{T\xi}_H^2 + \norm{\xi}_H^2 \ge \norm{\xi}_H^2
+        \]
+        
+        In particular, $T \pm i$ is injective. 
+        \item $(T \pm i)(H) = H$.
+    \end{enumerate}
+
+    In other words, $(T \pm i)$ is bijective onto $H$. 
+\end{lemma}
+\begin{proof}
+    (1): For any $\xi \in D(T)$, 
+    \begin{align*}
+        \normn{(T + i)\xi}_H^2 &= \norm{T\xi}_H^2 + \norm{\xi}_H^2 + \dpn{T\xi, i\xi}{H} + \dpn{i\xi, T\xi}{H} \\
+        &= \norm{T\xi}_H^2 + \norm{\xi}_H^2 + i\dpn{T\xi, \xi}{H} -i \dpn{\xi, T\xi}{H} \\
+        &= \norm{T\xi}_H^2 + \norm{\xi}_H^2 \ge \normn{\xi}_H^2 
+    \end{align*}
+
+    (2, closed range): For any $\seq{\xi_n} \subset D(T)$ and $\eta \in H$ with $(T + i)\xi_n \to \eta$, (1) implies that $\seq{\xi_n} \subset D(T)$ and $\seq{T\xi_n}$ are both Cauchy. Let $\xi = \limv{n}\xi_n$, then since $T$ is self-adjoint, it is closed, so $T\xi = \limv{n}T\xi_n \in T(H)$, and $\eta = (T + i)\xi \in (T+i)(H)$. 
+
+    (2, dense range): Let $\eta \in H$, then the following are equivalent:
+    \begin{enumerate}[label=(\alph*)]
+        \item $\eta \perp (T + i)(H)$. 
+        \item $\dpn{T\xi, \eta}{H} = \dpn{\xi, i\eta}{H}$ for all $\xi \in D(T)$. 
+        \item $\eta \in D(T^*)$ with $T^*\eta = i\eta$. 
+        \item $\eta \in D(T)$ with $(T - i)\eta = 0$. 
+    \end{enumerate}
+    
+    By (1), $\eta \perp (T+i)(H)$ if and only if $\eta = 0$. 
+\end{proof}
+
+\begin{definition}[Cayley Transform]
+\label{definition:unbounded-cayley-transform}
+    Let $H$ be a complex Hilbert space, $T: D(T) \to H$ be a densely defined, self-adjoint operator, then
+    \[
+    U(T): H \to H \quad U(T) = (T - i)(T + i)^{-1}
+    \]
+
+    is a unitary operator, known as the \textbf{Cayley transform} of $T$.
+\end{definition}
+\begin{proof}
+    By \autoref{lemma:cayley-transform-unbounded}, $(T+i)^{-1}: H \to D(T)$ and $(T - i): D(T) \to H$ are well-defined linear bijections, so $U(T)$ is a well-defined linear isomorphism. For any $\xi \in D(T)$, (1) of \autoref{lemma:cayley-transform-unbounded} implies that $\norm{(T + i)\xi}_H = \norm{(T - i)\xi}_H$. As $(T+i)^{-1}$ and $(T - i)$ are both bijections, $U(T)$ is an isometry. Therefore $U(T)$ is unitary. 
+\end{proof}
+
+\begin{definition}[Unbounded Integration Against Spectral Measure]
+\label{definition:unbounded-spectral-integral}
+    Let $X$ be a compact Hausdorff space, $H$ be a complex Hilbert space, $E: \cb_X \to B(H)$ be a spectral measure relative to $H$. For each Borel measurable function $f: X \to \complex$, let
+    \[
+    D\paren{\int_X f dE} = \bracs{\xi \in H \bigg | \int_X |f|^2 dE_{\xi, \xi} < \infty}
+    \]
+
+    and
+    \[
+    \int_X f dE: D\paren{\int_X f dE} \to H \quad \angles{\int_X fdE \xi, \eta}_H = \int_X f dE_{\xi, \eta}
+    \]
+
+    Then, 
+    \begin{enumerate}
+        \item $\int_X f dE: D(\int_Xf dE) \to H$ is a densely defined linear operator.
+        \item For any Borel measurable functions $f, g: X \to \complex$, 
+        \[
+        D\paren{\int_X f dE \int_X g dE} = D\paren{\int_X g dE} \cap D\paren{\int_X fg dE}
+        \]
+        
+        and $\int_X f dE \int_X g dE = \int_X fg dE$. 
+        
+        \item For any Borel measurable function $f: X \to \complex$, $(\int_X f dE)^* = \int_X \ol f dE$. In particular, $\int_X f dE$ is always closed. 
+    \end{enumerate}
+\end{definition}
+\begin{proof}
+    (1): For each $n \in \natp$, $\one_{\bracsn{f \le n}}f \in B^\infty(X; \complex)$. Thus for each $\xi \in H$, by a monotone class argument, 
+    \begin{align*}
+        \int_X |f|^2 dE_{E(\bracsn{f \le n})\xi, E(\bracsn{f \le n})\xi} &= \int_{\bracsn{f \le n}}|f|^2 dE_{\xi, \xi} < \infty \\
+        \normn{\xi - E(\bracsn{f \le n})\xi}_H^2 &= \normn{E(\bracsn{f > n})\xi}_H^2
+    \end{align*}
+
+    where by \hyperref[continuity from above]{proposition:measure-properties}, $E(\bracsn{f > n})\xi \to 0$ as $n \to \infty$. Therefore $D(\int_X f dE)$ is dense in $H$. 
+
+    It remains to show that $\int_X f dE$ is defined on the claimed domain. Let $\xi \in D(\int_X f dE)$, $\eta \in H$, and $B \in \cb_X$, then by (1) and (3) of \autoref{lemma:spectral-measure-properties}, 
+    \[
+    \int_X \one_B d|E_{\xi, \eta}| \le E_{\xi, \xi}(B)^{1/2}E_{\eta, \eta}(B)^{1/2}
+    \]
+
+    For any $\phi \in \Sigma(X; \complex)$, by the \hyperref[Cauchy-Schwarz inequality]{proposition:cauchy-schwarz}, 
+    \begin{align*}
+    \int_X |\phi| d|E_{\xi, \eta}| &\le \norm{\eta}_H\sum_{\alpha \in \phi(X)} |\alpha| E_{\xi, \xi}(\bracsn{\phi = \alpha})^{1/2}E_{\eta, \eta}(\bracsn{\phi = \alpha})^{1/2} \\
+    &\le \norm{\eta}_H\braks{\int_X |\phi|^2 dE_{\xi, \xi}}^{1/2}
+    \end{align*}
+    
+
+    By the \hyperref[Monotone Convergence Theorem]{theorem:mct}, 
+    \[
+    \int_X |f| d|E_{\xi, \eta}| \le \norm{\eta}_H \braks{\int_X |f|^2 dE_{\xi, \xi}}^{1/2}
+    \]
+    
+    so $\int_X f dE \xi \in H$ is well-defined. 
+
+    (2): For each $\xi \in D(\int_X g dE)$, by a monotone class argument, 
+    \[
+    \int_X |fg|^2 dE_{\xi, \xi} = \int_X |f|^2 |g|^2 dE_{\xi, \xi} = \int_X |f|^2 dE_{\int_X gdE\xi, \int_X gdE\xi}
+    \]
+
+    so $\int_X g dE \xi \in D(\int_X f dE)$ if and only if $\xi \in D(\int_X fg dE)$, and
+    \[
+    D\paren{\int_X f dE \int_X g dE} = D\paren{\int_X g dE} \cap D\paren{\int_X fg dE}
+    \]
+    
+    
+    Let $\xi \in D(\int_X g dE) \cap D(\int_X fg dE)$ and $\eta \in H$, then
+    \begin{align*}
+        \angles{\int_X f dE \int_X g dE \xi, \eta}_H &= \int_X f dE_{\int_X g dE \xi, \eta} = \int_X fg dE_{\xi, \eta} \\
+        &= \angles{\int_X f  g dE \xi, \eta}_H
+    \end{align*}
+    
+    (3): For any $\xi, \eta \in D(\int_X f dE) = D(\int_X \ol f dE)$, 
+    \begin{align*}
+        \angles{\int_X f dE\xi, \eta}_H &= \int_X f dE_{\xi, \eta} = \ol{\int_X \ol f dE_{\eta, \xi}}  \\
+        &= \ol{\dpn{\int_X \ol f dE\eta, \xi}H} = \dpn{\xi, \int_X \ol f dE \eta}{H}
+    \end{align*}
+    
+    so $\int_X \ol f dE \subset (\int_X f dE)^*$. On the other hand, for each $\xi \in H$ and $\eta \in D[(\int_X f dE)^*]$, by the \hyperref[bounded $L^\infty$ functional calculus]{definition:linfty-functional-calculus} and (2),
+    \begin{align*}
+        \angles{\xi, \int_X \frac{\ol f}{1 + |f|} dE\eta}_H &= \angles{\int_X \frac{f}{1 + |f|} dE\xi, \eta}_H \\
+        &= \angles{\int_X f dE \int_X (1 + |f|)^{-1}dE\xi, \eta}_H \\
+        &=  \angles{ \int_X (1 + |f|)^{-1}dE\xi, \paren{\int_X f dE}^{*}\eta}_H \\
+        &=  \angles{\xi, \int_X (1 + |f|)^{-1}dE\paren{\int_X f dE}^{*}\eta}_H \\
+    \end{align*}
+
+    so
+    \[
+    \int_X \frac{\ol f}{1 + |f|} dE = \int_X (1 + |f|)^{-1}dE\paren{\int_X f dE}^{*}
+    \]
+    
+    By (2) and multiplication of $\int_X (1 + |f|) dE$ from the left, $\int_X \ol f dE = \paren{\int_X f dE}^{*}$.     
+\end{proof}
+
+
+\begin{theorem}[Spectral Theorem I (Unbounded)]
+\label{theorem:spectral-theorem-1-unbounded}
+    Let $H$ be a complex Hilbert space and $T: D(T) \to H$ be a densely defined, self-adjoint operator, then there exists a unique spectral measure $E: \cb_\real \to B(H)$ such that $T = \int_{\real} \lambda E(d\lambda)$. 
+\end{theorem}
+\begin{proof}
+    Let $U(T) = (T - i)(T + i)^{-1} \in U(B(H))$ be the Cayley transform of $T$, then $U(T)$ is unitary. By \hyperref[Spectral Theorem I]{theorem:spectral-theorem-vn-1}, there exists a unique spectral measure $E: \cb_{\partial B_\complex(0, 1)} \to B(H)$ such that $U(T) = \int_{\partial B_\complex(0, 1)} \lambda E(d\lambda)$. Moreover, 
+    \[
+    (I - U) = (T+i)(T+i)^{-1} - (T - i)(T+i)^{-1} = 2i(T + i)^{-1}
+    \]
+
+    is injective, so $1$ is not an eigenvalue of $U$, and $E(\bracsn{1}) = 0$. Let
+    \[
+    \phi: \partial B_\complex(0, 1) \setminus \bracsn{1} \to \real \quad \lambda \mapsto i\frac{1 + \lambda}{1 - \lambda}
+    \]
+
+    then the pushforward $F = \phi_* E$ defined by $F(B) = E(\phi^{-1}(B))$ is also a spectral measure, with
+    \begin{align*}
+    \int_\real \lambda F(d\lambda) &= \int_{\partial B_\complex(0, 1) \setminus \bracsn{1}} i\frac{1 + \lambda}{1 - \lambda} E(d\lambda) = i(1 + U)(1 - U)^{-1} \\
+    &= i[(T + i)(T+i)^{-1} + (T-i)(T+i)^{-1}] \cdot (2i(T + i)^{-1})^{-1} \\
+    &= \frac{1}{2}[(T + i)(T+i)^{-1}(T+i) + (T-i)(T+i)^{-1}(T+i)] = T
+    \end{align*}
+    
+\end{proof}
+
+\begin{lemma}
+\label{lemma:closed-unbonuded-adjoint}
+    Let $H$ be a complex Hilbert space, $T: D(T) \to H$ be a densely defined linear operator, and
+    \[
+    \Gamma(T) = \bracsn{(\xi, T\xi)|\xi \in D(T)} \subset H \oplus H
+    \]
+
+    be the graph of $T$, then:
+    \begin{enumerate}
+        \item $\Gamma(T)^\perp = \bracsn{(-T^*\eta, \eta)| \eta \in D(T^*)}$. 
+        \item If $T$ is closed, then the mapping $I + T^*T: D(T^*T) \to H$ is surjective. 
+    \end{enumerate}
+\end{lemma}
+\begin{proof}
+    (1): Let $\Gamma(T) = \bracsn{(\xi, T\xi)| \xi \in D(T)}$ be the graph of $T$, then for any $(\xi, \eta) \in H \oplus H$, the following are equivalent:
+    \begin{enumerate}[label=(\roman*)]
+        \item $(\xi, \eta) \perp \Gamma(T)$. 
+        \item For each $\theta \in D(T)$, $\dpn{\eta, T\theta}{H}  = -\dpn{\xi, \theta}{H}$.
+        \item $\eta \in D(T^*)$, with $\dpn{T^*\eta, \theta}{H} = -\dpn{\xi, \theta}{H}$ for all $\theta \in H$. 
+        \item $\eta \in D(T^*)$ with $\xi = -T^*\eta$. 
+    \end{enumerate}
+
+    so $\Gamma(T)^\perp = \bracsn{(-T^*\eta, \eta)| \eta \in D(T^*)}$.
+
+    (2): Given that $T$ is closed, $H \oplus H = \Gamma(T) \oplus \Gamma(T)^\perp$. In particular, for each $\theta \in H$, there exists $\xi \in D(T)$ and $\eta \in D(T^*)$ such that
+    \[
+    (\theta, 0) = (\xi, T\xi) + (-T^*\eta, \eta)
+    \]
+
+    In which case, $T\xi = -\eta \in D(T^*)$, so $\xi \in D(T^*T)$ with
+    \[
+    \theta = \xi - T^*\eta = \xi + T^*T\xi
+    \]
+
+    so $I + T^*T: D(T^*T) \to H$ is surjective. 
+\end{proof}
+
+\begin{theorem}[von Neumann]
+\label{theorem:von-neumann-closed-operator}
+    Let $H$ be a complex Hilbert space, $T: D(T) \to H$ be a closed, densely defined linear operator, then $T^*T: D(T^*T) \to H$ is a densely defined, self-adjoint, and positive operator. 
+\end{theorem}
+\begin{proof}
+    For each $\xi, \eta \in D(T^*T)$, by definition of the adjoint,
+    \[
+    \dpn{T^*T\xi, \eta}{H} = \dpn{T\xi, T\eta}{H} = \dpn{\xi, T^*T\eta}{H}
+    \] 
+
+    (Densely Defined): Let $\eta \in D(T^*T)^\perp$. By (2) of \autoref{lemma:closed-unbonuded-adjoint}, there exists $\xi \in D(T^*T)$ such that $\eta = \xi + T^*T\xi$. In which case, 
+    \[
+    0 = \dpn{\xi, \eta}{H} = \dpn{\xi, \xi}{H} + \dpn{\xi, T^*T\xi}{H} = \norm{\xi}_H^2 + \norm{T\xi}_H^2
+    \]
+
+    so $T\xi = \xi = 0$, and $\eta = 0$ as well. 
+
+    ($T^*T \subset (T^*T)^*$): For each $\xi, \eta \in D(T^*T)$, $\dpn{T^*T\xi, \eta}{H} = \dpn{\xi, T^*T\eta}{H}$, so $T^*T \subset (T^*T)^*$. 
+
+    ($T^*T \supset (T^*T)^*$): Let $\xi \in D((T^*T)^*)$, then by (2) of \autoref{lemma:closed-unbonuded-adjoint}, there exists $\eta \in D(T^*T)$ such that $\xi + (T^*T)^*\xi = \eta + T^*T\eta$. In which case, for each $\theta \in D(T^*T)$, 
+    \begin{align*}
+        \dpn{(I+T^*T)^*\xi, \theta}{H} &= \dpn{(I+T^*T)\eta, \theta}{H} \\
+        \dpn{\xi, (I + T^*T)\theta}{H} &= \dpn{\eta, (I+T^*T)^*\theta}{H}
+    \end{align*}
+    
+    As the above holds for all $\theta \in D(T^*T)$, (2) of \autoref{lemma:closed-unbonuded-adjoint} implies that $\dpn{\xi, \theta}{H} = \dpn{\eta, \theta}{H}$ for all $\theta \in H$. Therefore $\xi = \eta \in D(T^*T)$, and $T^*T \supset (T^*T)^*$. 
+\end{proof}
+
+\begin{definition}[Absolute Value]
+\label{definition:unbounded-absolute-value}
+    Let $H$ be a complex Hilbert space, $T: D(T) \to H$ be a closed, densely defined linear operator, and $|T| = (T^*T)^{1/2}: D(|T|) \to H$, then:
+    \begin{enumerate}
+        \item $|T|$ is a densely defined, self-adjoint, and positive operator. 
+        \item Equip $D(T)$ with the inner product $\dpn{\xi, \eta}{D(T)} = \dpn{\xi, \eta}{H} + \dpn{T\xi, T\eta}{H}$, then $D(T)$ is a Hilbert space with $D(T^*T)$ being dense in $D(T)$. 
+        \item Equip $D(|T|)$ with the inner product $\dpn{\xi, \eta}{D(|T|)} = \dpn{\xi, \eta}{H} + \dpn{|T|\xi, |T|\eta}{H}$, then $D(|T|)$ is a Hilbert space with $D(T^*T)$ being dense in $D(|T|)$. 
+        \item $D(T) = D(|T|)$. 
+        \item For each $\xi \in D(T) = D(|T|)$, $\normn{T\xi}_H = \norm{|T|\xi}_H$. In particular, $\ker T = \ker |T|$. 
+    \end{enumerate}
+
+    The operator $|T|$ is the \textbf{absolute value} of $T$. 
+\end{definition}
+\begin{proof}
+    (1): By \hyperref[von Neumann's theorem]{theorem:von-neumann-closed-operator}, $T^*T: D(T^*T) \to H$ is a densely defined, self-adjoint, and positive operator. Thus $|T| = (T^*T)^{1/2}$ is well-defined, self-adjoint, and positive via the \hyperref[spectral theorem]{theorem:spectral-theorem-1-unbounded}. 
+
+    (2): By definition, $D(T)$ is isomorphic to the graph of $T$, which is a Hilbert space given that $T$ is closed. 
+
+    Let $\xi \in D(T)$ with $\xi \perp D(T^*T)$, then for each $\eta \in D(T^*T)$, 
+    \[
+    0 = \dpn{\xi, \eta}{H} + \dpn{T\xi, T\eta}{H} = \dpn{\xi, (I + T^*T)\eta}{H}
+    \]
+
+    By (2) of \autoref{lemma:closed-unbonuded-adjoint}, $(I+T^*T): D(T^*T) \to H$ is surjective, so $\xi = 0$, and $D(T^*T)$ is dense in $D(T)$. 
+
+    (3): By (3) of \autoref{definition:unbounded-spectral-integral}, $|T|$ is self-adjoint, and in particular closed.  Since $D(T)$ is isomorphic to the graph of $|T|$, $D(|T|)$ is a Hilbert space. Similarly, for any $\xi \in D(|T|)$ with $\xi \perp D(T^*T)$ and $\eta \in D(T^*T)$, by the \hyperref[Borel functional calculus]{definition:unbounded-spectral-integral}\footnote{The fact that the spectral measure of a positive element is supported in $[0, \infty)$ is needed here.},
+    \[
+    0 = \dpn{\xi, \eta}{H} + \dpn{|T|\xi, |T|\eta}{H} = \dpn{\xi, (I + T^*T)\eta}{H}
+    \]
+
+    By (2) of \autoref{lemma:closed-unbonuded-adjoint}, $\xi = 0$, and $D(T^*T)$ is dense in $D(|T|)$. 
+
+    (4): For any $\xi \in D(T^*T)$, $\norm{\xi}_{D(T)} = \norm{\xi}_{D(|T|)}$. By (2) and (3), $D(T)$ and $D(|T|)$ are both the completions of $D(T^*T)$ under the same norm, so, $D(T) = D(|T|)$. 
+
+    (5): For each $\xi \in D(T^*T)$, 
+    \[
+    \normn{T\xi}_H^2 = \dpn{T\xi, T\xi}{H} = \dpn{\xi, T^*T\xi}{H} = \dpn{|T|\xi, |T|\xi}{H} = \norm{|T|\xi}_H^2
+    \]
+
+    By the density provided in (2) and (3), $\normn{T\xi}_H^2 = \norm{|T|\xi}_H^2$ for all $\xi \in D(T)$. 
+\end{proof}
+
+
+\begin{theorem}[Polar Decomposition (Unbounded)]
+\label{theorem:unbounded-polar-decomposition}
+    Let $H$ be a complex Hilbert space, $T: D(T) \to H$ be a closed, densely defined linear operator, then there exists a unique partial isometry $U \in B(H)$ with $\ker U = \ker T$ such that $T = U|T|$, where $|T|$ is the \hyperref[absolute value]{definition:unbounded-absolute-value} of $T$. 
+\end{theorem}
+\begin{proof}
+    By \autoref{definition:unbounded-absolute-value}, $|T|: D(T) \to H$ is a densely defined, self-adjoint, and positive operator with $\normn{T\xi}_H = \norm{|T|\xi}_H$ for all $\xi \in D(T)$ and $\ker(T) = \ker(|T|)$. Let
+    \[
+    U: |T|(D(T)) \to T(D(T)) \quad |T|\xi \mapsto T\xi
+    \]
+
+    then $U$ extends into a well-defined isometry $\ol{|T|(D(T))} \to \ol{T(D(T))}$. Further extend $U$ to $H$ by setting $U|_{|T|(D(T))^\perp} = 0$, then $U: H \to \ol{T(D(T))} \subset H$ is a partial isometry with $T = U|T|$. As $|T|$ is self-adjoint, 
+    \[
+    \ker(U) = |T|(D(T))^\perp = \ker(|T|) = \ker(T)
+    \]
+
+    Finally, let $V: H \to H$ be a partial isometry with $T = V|T|$ and $\ker(V) = \ker(T)$, then for each $\xi \in D(T)$, $V|T|\xi = T\xi = U|T|\xi$, so $U = V$ on $|T|(D(T))$. By continuity, $U = V$ on $\ol{|T|(D(T))}$. As
+    \[
+    \ker(V) = \ker(T) = \ker(U) = |T|(D(T))^\perp
+    \]
+
+    and $H = \ker(V) \oplus \ol{|T|(D(T))}$, $U = V$. 
+
+
+\end{proof}
+