From 1f3c20a235cfedd837f75a16429d66b044d387e9 Mon Sep 17 00:00:00 2001 From: Bokuan Li Date: Sun, 1 Nov 2026 23:38:40 -0500 Subject: [PATCH] Revisions. --- src/op/example/hii1.tex | 16 ++++++++-------- src/op/example/lg.tex | 26 +++++++++++++------------- 2 files changed, 21 insertions(+), 21 deletions(-) diff --git a/src/op/example/hii1.tex b/src/op/example/hii1.tex index a80c6b8..39dd01a 100644 --- a/src/op/example/hii1.tex +++ b/src/op/example/hii1.tex @@ -26,10 +26,10 @@ \item $\braks{\bigcup_{i \in I}\phi^i(M_i)}'' = M$, where the bicommutant is taken in $M$. \end{enumerate} - The pair $((M, \tau), \bracsn{\phi^i}_{i \in I}) = \lim_{\longleftarrow}(M_i, \tau_i)$ is the \textbf{inductive limit} of $(\bracsn{(M_i, \tau_i)}_{i \in I}, \bracsn{\phi^i_j|i, j \in I, i \le j})$. + The pair $((M, \tau), \bracsn{\phi^i}_{i \in I}) = \lim_{\longrightarrow}(M_i, \tau_i)$ is the \textbf{inductive limit} of $(\bracsn{(M_i, \tau_i)}_{i \in I}, \bracsn{\phi^i_j|i, j \in I, i \le j})$. \end{definition} \begin{proof} - Let $(M_0, \bracsn{\phi^i|i \in I})$ be the inductive limit of $(\bracsn{M_i}_{i \in i}, \bracsn{\phi^i_j|i \le j})$ as vector spaces. Since $\bracsn{\phi^i_j|i, j \in I, i \le j}$ are trace-preserving and $\bracsn{\tau_i}_{i \in I}$ are faithful, $\bracsn{\phi^i_j|i, j \in I, i \le j}$ are also unital isometric embeddings. As such, $\bracsn{M_i}_{i \in I}$ may be identified as subspaces of $M_0$, the norms on $\bracsn{M_i}_{i \in I}$ extend to $M_0$, and $\phi^i: M_i \to M_0$ are isometric embeddings for all $i \in I$. + Let $(M_0, \bracsn{\phi^i|i \in I})$ be the inductive limit of $(\bracsn{M_i}_{i \in I}, \bracsn{\phi^i_j|i \le j})$ as vector spaces. Since $\bracsn{\phi^i_j|i, j \in I, i \le j}$ are trace-preserving and $\bracsn{\tau_i}_{i \in I}$ are faithful, $\bracsn{\phi^i_j|i, j \in I, i \le j}$ are also unital isometric embeddings. As such, $\bracsn{M_i}_{i \in I}$ may be identified as subspaces of $M_0$, the norms on $\bracsn{M_i}_{i \in I}$ extend to $M_0$, and $\phi^i: M_i \to M_0$ are isometric embeddings for all $i \in I$. For each $i, j \in I$ with $i \le j$, $\phi^i_j: M_i \to M_j$ is trace-preserving. By universal property of the inductive limit, there exists a linear functional $\tau: M_0 \to \complex$ such that $\tau \circ \phi^i = \tau_i$ for all $i \in I$. Similarly, $\bracsn{\phi^i_j|i, j \in I, i \le j}$ are *-homomorphisms, the products and involutions from $\bracsn{M_i}_{i \in I}$ extend to $M_0$, and $\phi^i: M_i \to M_0$ is a unital trace-preserving *-homomorphism for all $i \in I$. In particular, $1_{M_0} = \phi^i(1_{M_i})$ for all $i \in I$. @@ -62,7 +62,7 @@ \[ \xymatrix{ M_i \ar@{->}[d]_{\phi^i} \ar@{->}[rd]^{\psi^i} & \\ - M_0 \ar@{->}[r]_{\psi_0} & N + M_0 \ar@{->}[r]_{\phi_0} & N } \] @@ -71,15 +71,15 @@ U: H \to K \quad \dpn{UL_x1_{M_0}, \pi(y)\xi}{K} = \sigma(y^*\phi_0(x)) \] - then for any $x \in M$, $\norm{UL_x1_{M_0}}_K^2 = \sigma(\phi_0(x^*x)) = \tau(x^*x) = \norm{L_x1_{M_0}}_H^2$. Thus $U$ extends into an isometry from $H$ to $K$. Moreover, for each $x \in M_0$ and $y, z \in N$, + then for any $x \in M_0$, $\norm{UL_x1_{M_0}}_K^2 = \sigma(\phi_0(x^*x)) = \tau(x^*x) = \norm{L_x1_{M_0}}_H^2$. Thus $U$ extends into an isometry from $H$ to $K$. Moreover, for each $x \in M_0$ and $y, z \in N$, \begin{align*} - \dpn{(\pi \circ \psi_0)(x)\pi(y)\xi, \pi(z)\xi}{K} &= \sigma(z^*\phi_0(x)y) = \sigma(yz^*\phi_0(x)) = \sigma((zy^*)^*\phi_0(x)) \\ + \dpn{(\pi \circ \phi_0)(x)\pi(y)\xi, \pi(z)\xi}{K} &= \sigma(z^*\phi_0(x)y) = \sigma(yz^*\phi_0(x)) = \sigma((zy^*)^*\phi_0(x)) \\ &= \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. - By the \hyperref[Kaplanski 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}, $\psi_0$ 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. \end{proof} \begin{definition}[Hyperfinite $\vnII_1$ Factor] @@ -121,8 +121,8 @@ By the \hyperref[Cauchy-Schwarz inequality]{proposition:cauchy-schwarz}, \begin{align*} - \normn{(x - \lambda_\alpha)\xi}_H^2 &= \int_{U(M_{2^n}(\complex))} \dpn{u(x - \lambda_\alpha)u^*\xi, (x - \lambda_\alpha)\xi}{H} \mu(du) \\ - &\le \norm{(x - \lambda_\alpha)\xi}_H\braks{\int_{U(M_{2^n}(\complex))}\normn{u(x - x_\alpha) u^*\xi}_H^2 \mu(du)} \\ + \normn{(x - \lambda_\alpha)\xi}_H^2 &= \int_{U(M_{2^n}(\complex))} \dpn{u(x - x_\alpha)u^*\xi, (x - \lambda_\alpha)\xi}{H} \mu(du) \\ + &\le \norm{(x - \lambda_\alpha)\xi}_H\braks{\int_{U(M_{2^n}(\complex))}\normn{u(x - x_\alpha) u^*\xi}_H^2 \mu(du)}^{1/2} \\ &= \normn{(x - \lambda_\alpha)\xi}_H \norm{(x - x_\alpha)\xi}_H \end{align*} diff --git a/src/op/example/lg.tex b/src/op/example/lg.tex index 6aa22f0..d4b7a9a 100644 --- a/src/op/example/lg.tex +++ b/src/op/example/lg.tex @@ -39,7 +39,7 @@ \begin{lemma} \label{lemma:left-convolver-cc} - Let $G$ be a locally compact group and $f \in C_c(G; \complex)$, $f$ is a lefy convolver. + Let $G$ be a locally compact group and $f \in C_c(G; \complex)$, $f$ is a left convolver. \end{lemma} \begin{proof} By \hyperref[Young's Inequality]{theorem:young-fourier}, for each $g \in L^2(G; \complex)$, $\norm{f * g}_{L^2(G; \complex)} \le \norm{f}_{L^1(G; \complex)}\norm{g}_{L^2(G; \complex)}$. @@ -48,7 +48,7 @@ \begin{lemma} \label{lemma:left-convolver} - Let $G$ be a locally compacy group and $f: G \to \complex$ be a left convolver, then: + Let $G$ be a locally compact group and $f: G \to \complex$ be a left convolver, then: \begin{enumerate} \item $L_f \in B(L^2(G; \complex))$. \item $f \in L_{\text{loc}}^1(G; \complex)$. @@ -64,8 +64,8 @@ In which case, for almost every $x \in G$, \begin{align*} - \limv{N} f * (g_N - g)(x) &= \int_G f(y)(g_N - g)(y^{-1}x)dy \\ - &\le \int_G|f(y)|\sum_{n \ge N}|(g_n - g)(y^{-1}x)| dy + f * (g_N - g)(x) &= \int_G f(y)(g_N - g)(y^{-1}x)dy \\ + |f * (g_N - g)(x)| &\le \int_G|f(y)|\sum_{n \ge N}|(g_n - g)(y^{-1}x)| dy \end{align*} where $\int_G|f(y)|\sum_{n \in \natp}|(g_n - g)(y^{-1}x)| dy < \infty$ by (ii). As such, by the \hyperref[Dominated Convergence Theorem]{theorem:dct}, $f * g_N \to f * g$ almost everywhere. By (i), $f * g = h$ almost everywhere. Therefore the \hyperref[Closed Graph Theorem]{theorem:closed-graph} implies that $L_f$ is continuous. @@ -87,7 +87,7 @@ \begin{definition}[Space of Left Convolvers] \label{definition:left-convolver-algebra} - Let $G$ be a group, $LC(G) := \bracsn{L_f| G \to \complex\text{ is a left convolver}} \subset B(l^2(G; \complex))$, and $f, g: G \to \complex$ be left convolvers, then: + Let $G$ be a group, $LC(G) := \bracsn{L_f| f: G \to \complex\text{ is a left convolver}} \subset B(l^2(G; \complex))$, and $f, g: G \to \complex$ be left convolvers, then: \begin{enumerate} \item $f \in l^2(G; \complex)$. \item $L_f^* = L_{f^*}$, where $f^*(x) = \ol{f(x^{-1})}$ for all $x \in G$. @@ -97,13 +97,13 @@ The set $LC(G)$ is the \textbf{algebra of left convolvers} on $G$. \end{definition} -\begin{proof}[Proof, {{\cite[Propositon 1.3.5]{AnantharamanPopa}}}. ] +\begin{proof}[Proof, {{\cite[Proposition 1.3.5]{AnantharamanPopa}}}. ] (1): $f * \delta_e = f \in l^2(G; \complex)$. (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(x^{-1}y) \\ - \dpn{\delta_x, L_{f^*}\delta_y}{l^2(G;\complex)} &= \ol{(f^* * \delta_y)(x)} = f^*(y^{-1}x) + \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}) \end{align*} @@ -114,7 +114,7 @@ (4): Let $T \in \ol{LC(G)}^{\text{\small SOT}}$ and $\angles{f_\alpha}_{\alpha \in A}$ be left convolvers such that $T = \sotlim_{\alpha \in A}L_{f_\alpha}$. Let $f = T\delta_e \in l^2(G; \complex)$, then \[ - f = T\delta_e = \paren{\sotlim_{\alpha \in A}L_{f_\alpha}}\delta_e = l^2\text{-}\lim_{\alpha \in A}f_\alpha * e + f = T\delta_e = \paren{\sotlim_{\alpha \in A}L_{f_\alpha}}\delta_e = l^2\text{-}\lim_{\alpha \in A}f_\alpha * \delta_e \] For any $g \in l^2(G; \complex)$, @@ -132,7 +132,7 @@ \begin{proof}[Proof, {{\cite[Theorem 1.3.6]{AnantharamanPopa}}}. ] ($\subset$): For any $x, y, z \in G$ and $f \in l^2(G; \complex)$, \[ - (\lambda(x)\rho(y)f)(z) = (\rho(h)f)(x^{-1}z) = f(x^{-1}zy) = (\lambda(x)f)(zy) = (\rho(y)\lambda(x)f)(z) + (\lambda(x)\rho(y)f)(z) = (\rho(y)f)(x^{-1}z) = f(x^{-1}zy) = (\lambda(x)f)(zy) = (\rho(y)\lambda(x)f)(z) \] so $\lambda(G) \subset R(G)'$. By \autoref{definition:left-convolver-algebra}, $L(G) \subset LC(G) \subset R(G)'$. @@ -147,7 +147,7 @@ \begin{definition} \label{definition:group-algebra-fourier-series} - Let $G$ be a group. For each $x \in G$, let $u_x = \lambda(x)$ and identify it with $\delta_x$, then $\lambda(G) = \bracsn{u_x|x \in G}$ is the \textbf{canonical unitaries} of $L(G)$. + Let $G$ be a group. For each $x \in G$, let $u_x = \lambda(x)$ and identify it with $\delta_x$, then $\lambda(G) = \bracsn{u_x|x \in G}$ are the \textbf{canonical unitaries} of $L(G)$. For each $f \in l^2(G)$, $f = \sum_{x \in G}f(x)u_x$. In particular, for each $T \in L(G) = LC(G) \subset l^2(G; \complex)$, there exists $\bracsn{T_x}_{x \in G} \subset \complex$ such that $T$ may be written as $T = \sum_{x \in G}T_x u_x$\footnote{This sum does not converge in the strong-operator or the weak-operator topology. }. In which case, the scalars $\bracsn{T_x}_{x \in G}$ are the \textbf{Fourier coefficients} of $T$. \end{definition} @@ -187,7 +187,7 @@ \label{proposition:wreath-product-icc} Let $H$ be a non-trivial group, $\Gamma$ be a group acting on a set $I$, and $\Gamma_f \subset \Gamma$ be the subgroup of elements whose conjugacy class is finite. If \begin{enumerate}[label=(\alph*)] - \item The orbits of $\Gamma \curvearrowleft I$ are finite. + \item The orbits of $\Gamma \curvearrowleft I$ are infinite. \item The restricted action $\Gamma_f \curvearrowright I$ is faithful. \end{enumerate} @@ -198,7 +198,7 @@ Let $g = (\xi, \gamma) \in H \wr_I \Gamma$. First suppose that $\xi \ne e_{H^{(I)}}$, then the support of $\xi$ is non-empty. By (a), the support of $\xi$ has infinite orbit under $\Gamma$, so $g^G \supset g^\Gamma$ is infinite. - Now suppose that $\xi = e_{H^{(I)}}$ and $\gamma \ne e_{\Gamma}$. If $\gamma \in \Gamma \setminus \Gamma_f$, then the conjugacy class of $\gamma$ is infinite, and the conjugacy class of $g$ is also infinite. Now suppose that $\gamma \in \Gamma_f$. By (b), there exists $i \in I$ with $\gamma i_0 \ne i_0$. As $H$ is non-trivial, there exists $\xi_0 \in H^{(I)} \setminus \bracsn{e_{H^{(I)}}}$ with support $\bracsn{i_0}$. In which case, if $g_0 = (\xi_0, e_{\Gamma})$, then + Now suppose that $\xi = e_{H^{(I)}}$ and $\gamma \ne e_{\Gamma}$. If $\gamma \in \Gamma \setminus \Gamma_f$, then the conjugacy class of $\gamma$ is infinite, and the conjugacy class of $g$ is also infinite. Now suppose that $\gamma \in \Gamma_f$. By (b), there exists $i_0 \in I$ with $\gamma i_0 \ne i_0$. As $H$ is non-trivial, there exists $\xi_0 \in H^{(I)} \setminus \bracsn{e_{H^{(I)}}}$ with support $\bracsn{i_0}$. In which case, if $g_0 = (\xi_0, e_{\Gamma})$, then \[ g_0^{-1}gg_0 = (\xi_0^{-1}\gamma\xi_0, \gamma) \]