housekeeping

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/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}
+
+