Revision

src/op/example/hii1.tex

@@ -77,9 +77,9 @@
         &= \dpn{UL_x1_{M_0}, \pi(zy^*)\xi}{K} = \dpn{L_x1_{M_0}, U^*\pi(zy^*)\xi}{H}
     \end{align*}
 
-    so $x \mapsto \dpn{\psi_0(x)\pi(y)\xi, \pi(z)\xi}{K}$ is a weak-operator continuous functional on $M_0$. Since $\pi(N)\xi$ is dense in $K$, by the \hyperref[Arzelà-Ascoli Theorem]{theorem:arzela-ascoli}, the weak operator topology on $\ol{B_{\pi(N)}(0, 1)}$ is induced by maps of the form $x \mapsto \dpn{\pi(x)\pi(y)\xi, \pi(z)\xi}{K}$ with $y, z \in N$. As such, $\psi_0: M_0 \to N$ is weak-operator continuous when restricted to bounded sets. 
+    so $x \mapsto \dpn{\phi_0(x)\pi(y)\xi, \pi(z)\xi}{K}$ is a weak-operator continuous functional on $M_0$. Since $\pi(N)\xi$ is dense in $K$, by the \hyperref[Arzelà-Ascoli Theorem]{theorem:arzela-ascoli}, the weak operator topology on $\ol{B_{\pi(N)}(0, 1)}$ is induced by maps of the form $x \mapsto \dpn{\pi(x)\pi(y)\xi, \pi(z)\xi}{K}$ with $y, z \in N$. As such, $\phi_0: M_0 \to N$ is weak-operator continuous when restricted to bounded sets. 
 
-    By the \hyperref[Kaplansky Density Theorem]{theorem:kaplansky-density} and \autoref{theorem:uniform-continuous-extension}, $\psi_0$ extends into a *-homomorphism $\phi: M \to N$. Since $(N, \sigma)$ is a tracial von Neumann algebra, $\psi$ is also trace-preserving. Finally, if $\Phi: M \to N$ is another such map, then it is also weak-operator continuous on bounded sets by \autoref{lemma:tracial-von-neumann-gns}. Therefore $\Phi = \phi$ by uniqueness of continuous extensions. 
+    By the \hyperref[Kaplansky Density Theorem]{theorem:kaplansky-density} and \autoref{theorem:uniform-continuous-extension}, $\phi_0$ extends into a *-homomorphism $\phi: M \to N$. Since $(N, \sigma)$ is a tracial von Neumann algebra, $\phi$ is also trace-preserving. Finally, if $\Phi: M \to N$ is another such map, then it is also weak-operator continuous on bounded sets by \autoref{lemma:tracial-von-neumann-gns}. Therefore $\Phi = \phi$ by uniqueness of continuous extensions. 
 \end{proof}
 
 \begin{definition}[Hyperfinite $\vnII_1$ Factor]
@@ -114,7 +114,7 @@
 
     where
     \begin{align*}
-    \norm{u(x - x_\alpha) u^*}_H^2 &= \tau(u(x - x_\alpha)^*u^*u(x - x_\alpha)u^*) \\
+    \norm{u(x - x_\alpha) u^*\xi}_H^2 &= \tau(u(x - x_\alpha)^*u^*u(x - x_\alpha)u^*) \\
     &= \tau((x - x_\alpha)^*(x - x_\alpha)uu^*uu^*) \\
     &= \tau((x - x_\alpha)^*(x - x_\alpha)) = \norm{(x - x_\alpha)\xi}_H^2
     \end{align*}

src/op/example/lg.tex

@@ -103,7 +103,7 @@
     (2): Let $x, y \in G$, then
     \begin{align*}
         \dpn{L_f\delta_x, \delta_y}{l^2(G; \complex)} &= (f * \delta_x)(y) = f(yx^{-1}) \\ 
-    \dpn{\delta_x, L_{f^*}\delta_y}{l^2(G;\complex)} &= \ol{(f^* * \delta_y)(x)} = \ol{f^*(y^{-1}x)} = f(yx^{-1})
+    \dpn{\delta_x, L_{f^*}\delta_y}{l^2(G;\complex)} &= \ol{(f^* * \delta_y)(x)} = \ol{f^*(xy^{-1})} = f(yx^{-1})
     \end{align*}
     
 
@@ -168,7 +168,7 @@
 \begin{proof}[Proof, {{\cite[Proposition 1.3.9]{AnantharamanPopa}}}. ]
     (2) $\Rightarrow$ (1): Let $T \in Z(L(G))$, then for each $x \in G$, 
     \[
-    T\delta_e = \lambda(x)T\lambda(x^{-1}) = \lambda(x)T\rho(x) = \lambda(x)\rho(x)(T\delta_e)
+    T\delta_e = \lambda(x)T\lambda(x^{-1})\delta_e = \lambda(x)T\rho(x)\delta_e = \lambda(x)\rho(x)(T\delta_e)
     \]
 
     so $T\delta_e$ is invariant under conjugation, and hence constant on conjugacy classes of $G$.