ADP related house keeping.

src/op/c-star/gns.tex

@@ -70,6 +70,8 @@
     \end{enumerate}
     
     The representation $(H_\phi, \pi_\phi)$ is the \textbf{cyclic representation of $A$ induced by $\phi$}, and the triple $(H_\phi, \pi_\phi, \xi_\phi)$ is the \textbf{Gelfand-Naimark-Segal (GNS) triple associated with $\phi$}. 
+
+    If $\phi$ is faithful, then $L^2(M, \phi)$ also denotes $H_\phi$. 
 \end{definition}
 \begin{proof}[Proof, {{\cite[Proposition 14.2]{Zhu}}}. ]
     (2): Fix $x \in A$, then for each $y_1, y_2 \in A$ with $y_1 - y_2 \in N_\phi$, $x(y_1 - y_2) \in N_\phi$ by \autoref{lemma:cstar-state-kernel}, so $\pi_\phi^0(x)$ is well-defined on $A/N_\phi$. 

src/op/system/index.tex

@@ -12,4 +12,5 @@
 \input{./os-mn.tex}
 \input{./arveson.tex}
 \input{./aos.tex}
+\input{./tensor.tex}
 

src/op/vn/fc.tex

@@ -41,6 +41,10 @@
     Let $H$ be a complex Hilbert space and $A \subset B(H)$ be a von Neumann algebra, then:
     \begin{enumerate}
         \item For each normal operator $T \in A$ and $f \in B^\infty(\sigma_{B(H)}(T); \complex)$ with $f(0) = 0$, $f(T) \in A$. 
+        \item For each $T \in A$ with $0 \le T \le 1$, there exists projections $\seq{P_n} \subset A$ such that
+        \[
+        T = \sum_{n = 1}^\infty \frac{P_n}{2^n}
+        \]
         \item The linear span of projections in $A$ is norm dense in $A$. 
     \end{enumerate}
 \end{theorem}
@@ -52,6 +56,20 @@
     If $B$ is arbitrary, then $\one_{B \setminus B_\complex(0, r)} \to \one_B$ in the weak* topology of $C(\sigma_{B(H)}(T); \complex)^{**}$ as $r \downto 0$. As $\one_{B \setminus B_{\complex}(0, r)}(T) \in A$ for all $r > 0$, $\one_B(T) \in A$ as well. 
 
     By linearity, $g(T) \in A$ for all $g \in \Sigma(\sigma_{B(H)}(T); \complex)$ with $g(0) = 0$. By \autoref{lemma:separable-metric-space-approx-identity}, $\bracsn{g \in \Sigma(\sigma_{B(H)}(T); \complex)|g(0) = 0}$ is uniformly dense in $\bracsn{f \in B^\infty(\sigma_{B(H)(T)}; \complex)|f(0) = 0}$. Therefore $f(T) \in A$ for all $f \in B^\infty(\sigma_{B(H)}(T); \complex)$ with $f(0) = 0$. 
+
+    
+
+    (2): Let $f = \text{Id}: [0, 1] \to [0, 1]$, then
+    \[
+    f = \sum_{n = 1}^\infty \frac{\one_{\bracsn{x \in [0, 1]| \lceil 2^nx \rceil = 0 \mod 2}}}{2^n}
+    \]
+
+    Let $E: \cb_{\sigma_{A}(T)} \to \text{Proj}(A)$ be the resolution of the identity for $f$, then by the \hyperref[$L^\infty$ functional calculus]{definition:linfty-functional-calculus}, 
+    \begin{align*}
+    T &= f(T) = \sum_{n = 1}^\infty\frac{\one_{\bracsn{x \in [0, 1]| \lceil 2^nx \rceil = 0 \mod 2}}(T)}{2^n} \\
+    &= \sum_{n = 1}^\infty 2^{-n}E(\bracsn{x \in [0, 1]| \lceil 2^nx \rceil = 0 \mod 2})
+    \end{align*}
+    
 \end{proof}
 
 \begin{theorem}
@@ -77,4 +95,21 @@
     (2): In particular, as $e^{t\phi}(\partial B(0, 1)) \subset \partial B(0, 1)$, the spectrum of $e^{t\phi}$ as an element in the domain of the $L^\infty$ functional calculus, is contained in $\partial B(0, 1)$. Thus if $T$ is unitary, then the path $t \mapsto e^{t\phi(T)}$ lies in the unitary group of $A$ by \autoref{corollary:spectrum-characterisation-iff}.
 \end{proof}
 
+\begin{proposition}
+\label{proposition:vna-two-sided-ideal}
+    Let $H$ be a complex Hilbert space, $M \subset B(H)$ be a von Neumann algebra, and $I \subset M$ be a two-sided ideal, then:
+    \begin{enumerate}
+        \item $I$ is self-adjoint. 
+        \item Let $x \in I$ with $x > 0$, $E: \cb_{[0, \infty)} \to \text{Proj}(M)$ be its resolution of the identity, and $t > 0$, then $E([t, \infty)) \in I$. 
+        \item If $I$ is weak-operator closed, then there exists a unique projection $z \in Z(M)$ with $I = Mz$. 
+    \end{enumerate}
+\end{proposition}
+\begin{proof}[Proof, {{\cite[Proposition 2.4.15]{AnantharamanPopa}}}. ]
+    (1): Let $x = u|x|$ be the \hyperref[polar decomposition]{theorem:hilbert-polar-decomposition} of $x$, then $|x| = u^*x \in I$, and $x^* = |x|u^* \in I$. 
+
+    (2): Let $f: [0, \infty) \to \real$ be defined by $f(s) = s^{-1}\one_{\bracsn{s \ge t}}(s)$, then by the \hyperref[Borel functional calculus]{definition:linfty-functional-calculus}, $E([t, \infty)) = xf(x) \in I$. 
+
+    (3): By (5) of \autoref{theorem:existence-of-projections-vna}. 
+\end{proof}
+
 

src/op/vn/index.tex

@@ -4,6 +4,7 @@
 \input{./topologies.tex}
 \input{./cayley.tex}
 \input{./vn.tex}
+\input{./weak.tex}
 \input{./commutative.tex}
 \input{./spec.tex}
 \input{./fc.tex}

src/op/vn/type-decomp.tex

@@ -28,6 +28,12 @@
     (2) $\Rightarrow$ (1): Given that there exist no projections strictly between $0$ and $P$, the only non-zero projection in $PAP$ is $P$ itself. By \autoref{theorem:vn-projection-norm-dense}, the linear span of projections in $PAP$ is norm-dense in $PAP$. Therefore $PAP = \complex P$. 
 \end{proof}
 
+\begin{definition}[Diffuse]
+\label{definition:diffuse-vna}
+    Let $H$ be a complex Hilbert space, $M \subset B(H)$ be a von Neumann algebra, then $M$ is \textbf{diffuse} if it admits no minimal projections. 
+\end{definition}
+
+
 \begin{lemma}
 \label{lemma:projection-types-gymnastics}
     Let $H$ be a complex Hilbert space, $A \subset B(H)$ be a von Neumann algebra, and $P, Q \in \text{Proj}(A)$. 

src/op/vn/vn.tex

@@ -5,7 +5,7 @@
 \label{theorem:existence-of-projections-vna}
     Let $H$ be a complex Hilbert space and $A \subset B(H)$ be a strong-operator closed $C^*$-subalgebra, then:
     \begin{enumerate}
-        \item For any bounded directed family $\cf \subset A_{sa}$, $\sup(\cf) = \sotlim_{T \in \cf}T \in A_{sa}$. 
+        \item For any bounded, upward-directed family $\cf \subset A_{sa}$, $\sup(\cf) = \sotlim_{T \in \cf}T \in A_{sa}$. 
         \item For any directed family of projections $\mathcal{P} \subset A_{sa}$, $\sup(\mathcal{P}) \in A$ is the projection onto $\ol{\bigcup_{P \in \mathcal{P}}P(H)}$. 
     \end{enumerate}