Edited, standard rep.

src/vna/ii1/standard.tex

@@ -10,17 +10,17 @@
     (\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 $M = \wh M \subset L^2(M, \tau)$. 
+    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 Conjuation]
+\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 $L^2(M, \tau)$, konwn as the \textbf{canonical conjugation}. 
+    is a conjugate linear isometry on $L^2(M, \tau)$, known as the \textbf{canonical conjugation}. 
 \end{definition}
 
 
@@ -73,12 +73,12 @@
 
     and
     \[
-    R^0_\xi: \wh M \to L^2(M, \tau) \quad R_0^\xi(\wh y) = L_y(\xi) = y\xi
+    R^0_\xi: \wh M \to L^2(M, \tau) \quad R_\xi^0(\wh y) = L_y(\xi) = y\xi
     \]
 
     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 opeartors by $\xi$}. 
+    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 $y \in M$, 
@@ -87,34 +87,34 @@
     &= \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)} \\
-    &=  \limv{n}\norm{\xi}_{L^2(M, \tau)} \normn{y}_{M} \normn{\wh {x_n^*}}_{L^2(M, \tau)} = 0
+    &\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 set of left and right convolvers are denoted $LC(M)$ and $RC(M)$, respectively. 
+    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 $\eta \in RC(M)$, $L_\xi R_\eta = R_\xi L_\eta$.
+    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 $x, y \in M$, 
     \[
-        \dpn{L_\xi R_\eta \wh x, \wh y}{L^2(M, \tau)} = \dpn{\wh{\xi x \eta}, \wh y}{L^2(M, \tau)} = \dpn{R_\eta L_\xi \wh x, \wh y}{L^2(m, \tau)}
+        \dpn{L_\xi R_\eta \wh x, \wh y}{L^2(M, \tau)} = \dpn{\wh{\xi x \eta}, \wh y}{L^2(M, \tau)} = \dpn{R_\eta L_\xi \wh x, \wh y}{L^2(M, \tau)}
     \]
 
     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)}
+    |\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 $\eta \in RC(M)$. 
+    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*}
@@ -129,7 +129,7 @@
         |\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 $\eta \in RC(M)$. 
+    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}
@@ -157,6 +157,6 @@
     T\wh x = TR_x \wh 1 = R_x T \wh 1 = R_x \xi = \xi x = L_\xi^0 \wh x
     \]
 
-    so $T = L_\xi^0$. By density of $\wh M$ in $L^2(M, \tau)$ and continuity of $T$, the closure of $L_\xi^0$ is $T$. Hence $T = L_\xi^0$. 
+    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}