From 53d4bfe34d2a3083db9814dd4784675714ce03f2 Mon Sep 17 00:00:00 2001 From: Bokuan Li Date: Fri, 6 Nov 2026 16:01:46 -0500 Subject: [PATCH] Corrections. --- src/op/ii1/trace.tex | 24 ++++++++++++------------ src/op/vn/weak.tex | 30 +++++++++++++----------------- 2 files changed, 25 insertions(+), 29 deletions(-) diff --git a/src/op/ii1/trace.tex b/src/op/ii1/trace.tex index bd6ec6e..143e7a3 100644 --- a/src/op/ii1/trace.tex +++ b/src/op/ii1/trace.tex @@ -5,8 +5,8 @@ \label{definition:trace} Let $H$ be a complex Hilbert space, $M \subset B(H)$ be a von Neumann algebra, and $\tau: M_+ \to [0, \infty]$, then $\tau$ is a \textbf{trace} on $M$ if: \begin{enumerate} - \item For each $x, y \in M$ and $\lambda \ge 0$, $\tau(\lambda x + y) = \lambda \tau(x) + \tau(y)$. - \item For each $x \in M$ and $u \in U(M)$, $\tau(uxu^*) = \tau(x)$. + \item For each $x, y \in M_+$ and $\lambda \ge 0$, $\tau(\lambda x + y) = \lambda \tau(x) + \tau(y)$. + \item For each $x \in M_+$ and $u \in U(M)$, $\tau(uxu^*) = \tau(x)$. \end{enumerate} \end{definition} @@ -32,20 +32,20 @@ (1): For any $x, y \in \ol{B_M(0, 1)}$, \begin{align*} \norm{x - y}_{L^2(M, \tau)}^2 &= \tau((x - y)^*(x - y)) = \dpn{\pi(x - y)^*\pi(x - y)1_M, 1_M}{L^2(M, \tau)} \\ - &= \dpn{\pi(x - y)1_M, \pi(x - y)1_M}{L^2(M, \tau)} = \norm{\pi(x - y)1_M}_{L^2(M, \tau)} + &= \dpn{\pi(x - y)1_M, \pi(x - y)1_M}{L^2(M, \tau)} = \norm{\pi(x - y)1_M}_{L^2(M, \tau)}^2 \end{align*} so the uniform structure of pointwise convergence is stronger than the uniformity induced by the $L^2(M, \tau)$ norm. On the other hand, by the \hyperref[Arzelà-Ascoli Theorem]{theorem:arzela-ascoli}, the uniform structure of pointwise convergence on $\ol{B_M(0, 1)}$ coincides with the uniform structure of pointwise convergence on any dense subset of $L^2(M, \tau)$. In particular, this includes $M \subset L^2(M, \tau)$. For each $x, y \in \ol{B_M(0, 1)}$ and $t \in M$, since $\tau$ is tracial, \begin{align*} - \norm{(x - y)t}_{L^2(M, \tau)}^2 &= \tau(t(x - y)^*(x - y)t) \le \tau(t^*t(x - y)^*(x - y)) \\ - &\le \norm{t}_M \tau((x - y)^*(x - y)) = \norm{t}_M^2 \norm{x - y}_{L^2(M, \tau)}^2 + \norm{(x - y)t}_{L^2(M, \tau)}^2 &= \tau(t^*(x - y)^*(x - y)t) \le \tau(t^*t(x - y)^*(x - y)) \\ + &\le \norm{t}_M^2 \tau((x - y)^*(x - y)) = \norm{t}_M^2 \norm{x - y}_{L^2(M, \tau)}^2 \end{align*} so the uniformity induced by the $L^2(M, \tau)$ norm is stronger than the uniform structure of pointwise convergence. - (2): Using (1), it is sufficient to show that $\ol{B_M(0, 1)} \subset B(L^2(M, \tau))$ is complete if and only if it is strong-operator closed. By \autoref{proposition:complete-closed}, $\ol{B_M(0, 1)}$ is complete if and only if it is closed in $H^H$ with respect to the product topology. Finally, the \hyperref[Arzelà-Ascoli Theorem]{theorem:arzela-ascoli} implies that $\ol{B_M(0, 1)}$ is closed in $H^H$ if and only if it is strong-operator closed in $B(L^2(M, \tau))$. + (2): Using (1), it is sufficient to show that $\ol{B_M(0, 1)} \subset B(L^2(M, \tau))$ is complete if and only if it is strong-operator closed. By \autoref{proposition:complete-closed}, $\ol{B_M(0, 1)}$ is complete if and only if it is closed in $L^2(M, \tau)^{L^2(M, \tau)}$ with respect to the product topology. Finally, the \hyperref[Arzelà-Ascoli Theorem]{theorem:arzela-ascoli} implies that $\ol{B_M(0, 1)}$ is closed in $H^H$ if and only if it is strong-operator closed in $B(L^2(M, \tau))$. \end{proof} \begin{proposition} @@ -63,7 +63,7 @@ (2) $\Rightarrow$ (3): By \autoref{proposition:cstar-trace-vna}. - (3) $\Rightarrow$ (4): Since the \hyperref[GNS representation]{definition:gns-triple} is cyclic, $M$ is dense in $L^2(M, \tau)$. + (3) $\Rightarrow$ (4): By \autoref{lemma:tracial-von-neumann-gns}, the standard representation $(L^2(M, \tau), \pi, 1_M)$ is faithful and normal. As $M$ is dense in $L^2(M, \tau)$, $\ol{B_M(0, 1)}$ being separable in $L^2(M, \tau)$ implies that $L^2(M, \tau)$ is separable. \end{proof} \begin{proposition} @@ -87,10 +87,10 @@ \begin{lemma} \label{lemma:diffuse-factor-subsplit} - Let $M$ be a diffuse factor, $p \in \text{Proj}(M)$ be a non-zero projection, then there exists non-zero projections $p_1, p_2 \in \text{Proj}(M)$ with $p_1 \sim p_2$ and $p_1 + p_2 \le p$. + Let $M$ be a diffuse factor, $p \in \text{Proj}(M)$ be a non-zero projection, then there exist non-zero projections $p_1, p_2 \in \text{Proj}(M)$ with $p_1 \sim p_2$ and $p_1 + p_2 \le p$. \end{lemma} \begin{proof}[Proof, {{\cite[Lemma 4.1.1]{AnantharamanPopa}}}. ] - Since $p$ is not minimal, there exists a projection $q \le p$ with $0 \ne q \ne p$. As $M$ is a factor, $Z(q)Z(p - q) = 1_M$. By \autoref{proposition:central-support-mvn}, there exists non-zero projections $p_1, p_2 \in \text{Proj}(M)$ with $p_1 \sim p_2$, $p_1 \le q$, and $p_2 \le p - q$. + Since $p$ is not minimal, there exists a projection $q \le p$ with $0 \ne q \ne p$. As $M$ is a factor, $Z(q)Z(p - q) = 1_M$. By \autoref{proposition:central-support-mvn}, there exist non-zero projections $p_1, p_2 \in \text{Proj}(M)$ with $p_1 \sim p_2$, $p_1 \le q$, and $p_2 \le p - q$. \end{proof} \begin{lemma} @@ -105,7 +105,7 @@ and $\mathcal{M} \subset \mathcal{F}$ be a maximal family such that $\bracsn{q, r|(q,r) \in \mathcal{M}}$ are all mutually orthogonal. Let $q = \sum_{(q, r) \in \mathcal{M}}q$ and $r = \sum_{(q, r) \in \mathcal{M}}r$, then by \autoref{lemma:mvn-equivalent-direct-sum}, $q \sim r$, $q \perp r$, and $q + r \le p$. - If $q + r \ne p$, then by \autoref{lemma:diffuse-factor-subsplit} applied to $p - (q + r)$, there exists non-zero projections $(q', r') \in \text{Proj}(M)$ with $q' \sim r$, $q' \perp r'$, and $q + r \le p - (q + r)$, which contradicts the maximality of $\mathcal{M}$. + If $q + r \ne p$, then by \autoref{lemma:diffuse-factor-subsplit} applied to $p - (q + r)$, there exists non-zero projections $q', r' \in \text{Proj}(M)$ with $q' \sim r'$, $q' \perp r'$, and $q' + r' \le p - (q + r)$, which contradicts the maximality of $\mathcal{M}$. \end{proof} \begin{corollary} @@ -159,7 +159,7 @@ Let $(M, \tau)$ be a tracial factor, then $M$ admits no non-trivial two-sided ideal. \end{proposition} \begin{proof}[Proof, {{\cite[Proposition 4.1.5]{AnantharamanPopa}}}. ] - Let $I \subset M$ be a non-zero two-sided ideal. By \autoref{proposition:vna-two-sided-ideal}, $I$ admits a non-zero projection $e \in I$. Let $\mathcal{P} \subset \text{Proj}(P)$ be a maximal family of mutually orthogonal projections equivalent to $e$. Since $\tau$ is faithful, $\tau(p) > 0$ for all $p \in \mathcal{P}$, so $\mathcal{P}$ is finite. + Let $I \subset M$ be a non-zero two-sided ideal. By \autoref{proposition:vna-two-sided-ideal}, $I$ admits a non-zero projection $e \in I$. Let $\mathcal{P} \subset \text{Proj}(M)$ be a maximal family of mutually orthogonal projections equivalent to $e$. Since $\tau$ is faithful, $\tau(p) > 0$ for all $p \in \mathcal{P}$, so $\mathcal{P}$ is finite. Let $q = 1_M - \sum_{p \in \mathcal{P}}p$, then by the \hyperref[comparability theorem]{corollary:vna-factor-comparability}, either $q \preceq e$ or $e \prec q$. If $e \prec q$, then there exists a subprojection $p \le q$ with $e \sim p$, which contradicts the maximality of $\mathcal{P}$. Thus $q \preceq e$. @@ -171,7 +171,7 @@ Let $M$ be a $\vnII_1$ factor with tracial normal state $\tau: M \to \complex$, then the mapping $p \mapsto \tau(p)$ induces a bijection from the equivalence classes of $\text{Proj}(M)$ to $[0, 1]$. \end{proposition} \begin{proof}[Proof, {{\cite[Proposition 4.1.6]{AnantharamanPopa}}}. ] - (Injective): Let $p, q \in \text{Proj}(M)$ with $\tau(p) = \tau(q)$. By the \hyperref[comparability theorem]{corollary:vna-factor-comparability}, either $p \prec q$, $p \sim q$, or $q \prec p$. Since $\tau$ is faithful, $p \prec q$ implies that $\tau(p) < \tau(q)$, and $q \prec p$ implies that $\tau(q) < \tau(p)$. Therefore the only possiblity is $p \sim q$. + (Injective): Let $p, q \in \text{Proj}(M)$ with $\tau(p) = \tau(q)$. By the \hyperref[comparability theorem]{corollary:vna-factor-comparability}, either $p \prec q$, $p \sim q$, or $q \prec p$. Since $\tau$ is faithful, $p \prec q$ implies that $\tau(p) < \tau(q)$, and $q \prec p$ implies that $\tau(q) < \tau(p)$. Therefore the only possibility is $p \sim q$. (Surjective): By \autoref{lemma:diffuse-factor-split}, there exists a family $\bracsn{p_r|r \in \mathbb D \cap [0, 1]} \subset \text{Proj}(M)$ such that: \begin{enumerate}[label=(\roman*)] diff --git a/src/op/vn/weak.tex b/src/op/vn/weak.tex index 88e4672..1fd91ee 100644 --- a/src/op/vn/weak.tex +++ b/src/op/vn/weak.tex @@ -3,7 +3,7 @@ \begin{lemma}[Jordan Decomposition] \label{lemma:weak-operator-jordan} - Let $H$ be a complex Hilbert space, $M \subset H$ be a von Nuemann algebra, and $\omega: M \to \complex$ be a weak-operator continuous linear functional, then there exists positive weak-operator continuous functionals $\bracsn{\omega_k}_1^4$ such that $\omega = \sum_{k = 0}^3 i^k \omega_k$. + Let $H$ be a complex Hilbert space, $M \subset B(H)$ be a von Neumann algebra, and $\omega: M \to \complex$ be a weak-operator continuous linear functional, then there exists positive weak-operator continuous functionals $\bracsn{\omega_k}_0^3$ such that $\omega = \sum_{k = 0}^3 i^k \omega_k$. \end{lemma} \begin{proof} Assume without loss of generality that there exists $\xi, \eta \in H$ such that $\dpn{x, \omega}{M} = \dpn{x\xi, \eta}{H}$ for all $x \in M$. In which case, by \hyperref[polarisation]{proposition:polarisation-complex}, @@ -27,10 +27,10 @@ so the mapping $(x, y) \mapsto \omega(y^*x)$ is a bounded sesquilinear form on $M\xi$. By the \hyperref[Riesz Representation Theorem]{theorem:riesz-hilbert}, there exists $z \in B(\ol{M\xi})$ such that $\omega(y^*x) = \dpn{zx\xi, y\xi}{H}$ for all $x, y \in M$. In particular, $0 \le \omega(x^*x) = \dpn{zx\xi, x\xi}{H}$ for all $x \in M$, so $z \ge 0$. Moreover, for any $x, y, t \in M$, \[ - \dpn{ztx\xi, zy\xi}{H} = \omega(y^*tx) = \omega((t^*y)^*x) = \dpn{zx\xi, t^*y\xi}{H} = \dpn{tzx\xi, y\xi}{H} + \dpn{ztx\xi, y\xi}{H} = \omega(y^*tx) = \omega((t^*y)^*x) = \dpn{zx\xi, t^*y\xi}{H} = \dpn{tzx\xi, y\xi}{H} \] - so $t \in M'_+$. Finally, let $p: H \to \ol{M\xi}$ be the orthogonal projection onto $\ol{M\xi}$, and $x' = (zp)^{1/2} \in M'$, then for each $x \in M$, + so $z \in M'_+$. Finally, let $p: H \to \ol{M\xi}$ be the orthogonal projection onto $\ol{M\xi}$, and $x' = (zp)^{1/2} \in M'$, then for each $x \in M$, \[ \omega(x) = \dpn{zx\xi, \xi}{H} = \dpn{zpx\xi, \xi}{H} = \dpn{x(zp)^{1/2}\xi, (zp)^{1/2}\xi}{H} \] @@ -39,7 +39,7 @@ \begin{lemma} \label{lemma:completely-additive-projection-choose} - Let $H$ be a complex Hilbert space, $M \subset B(H)$ be a von Neumann algebra, and $\varphi: M \to \complex$ be a positive linear functional. If for any mutually orthogonal projections $\mathcal{P} \subset M$, $\varphi\paren{\sum_{p \in \mathcal{P}}p} = \sum_{p \in P}\varphi(p)$, then there exists a non-zero projection $p \in M$ and $\xi \in pH$ such that $\varphi(x) = \dpn{x\xi, \xi}{H}$ for all $x \in pMp$. + Let $H$ be a complex Hilbert space, $M \subset B(H)$ be a von Neumann algebra, and $\varphi: M \to \complex$ be a positive linear functional. If for any mutually orthogonal projections $\mathcal{P} \subset M$, $\varphi\paren{\sum_{p \in \mathcal{P}}p} = \sum_{p \in \mathcal{P}}\varphi(p)$, then there exists a non-zero projection $p \in M$ and $\xi \in pH$ such that $\varphi(x) = \dpn{x\xi, \xi}{H}$ for all $x \in pMp$. \end{lemma} \begin{proof}[Proof, {{\cite[Lemma 2.5.6.]{AnantharamanPopa}}}. ] Let $\eta \in H$ with $\varphi(1_M) < \dpn{\eta, \eta}{H}$. Let $\mathcal{P} \subset M$ be a maximal family of mutually orthogonal projections with $\varphi(p) \ge \dpn{p\eta, \eta}{H}$ for all $p \in \mathcal{P}$. By assumption, @@ -53,7 +53,7 @@ Let $x \in (pMp)_+$ with $0 \le x \le 1$. By \autoref{theorem:vn-projection-norm-dense}, there exists projections $\seq{q_n} \subset pMp$ such that $x = \sum_{n = 1}^\infty \frac{q_n}{2^n}$. In which case\footnote{Here, only continuity of $\varphi$ is used.}, \[ - \dpn{x\eta, \eta}{H} = \sum_{n = 1}^\infty 2^{-n}\dpn{q_n\eta, \eta}{H} \ge \sum_{n = 1}^\infty \frac{\varphi(q^n)}{2^n} = \varphi(x) + \dpn{x\eta, \eta}{H} = \sum_{n = 1}^\infty 2^{-n}\dpn{q_n\eta, \eta}{H} \ge \sum_{n = 1}^\infty \frac{\varphi(q_n)}{2^n} = \varphi(x) \] Therefore $\dpn{x\eta, \eta}{H} \ge \varphi(x)$ for all $x \in (pMp)_+$. By the \hyperref[Radon-Nikodym theorem for positive functionals]{lemma:vna-radon-nikodym}, there exists $\xi \in pH$ with $\varphi(x) = \dpn{x\xi, \xi}{H}$ for all $x \in pMp$. @@ -66,8 +66,8 @@ Let $H$ be a complex Hilbert space, $M \subset B(H)$ be a von Neumann algebra, and $\omega: M \to \complex$ be a positive linear functional, then the following are equivalent: \begin{enumerate} \item For any bounded, upward-directed family $A \subset M$, $\omega(\sup(A)) = \sup(\omega(A))$. - \item For any mutually orthogonal projections $\mathcal{P} \subset M$, $\omega\paren{\sum_{p \in \mathcal{P}}p} = \sum_{p \in P}\omega(p)$. - \item There exists $\seq{\xi_n} \in l^2(\natp; H)$ such that $\omega(x) = \sum_{n \in \natp}\dpn{x\xi_n, \xi_n}{H}$ for all $x \in M$. + \item For any mutually orthogonal projections $\mathcal{P} \subset M$, $\omega\paren{\sum_{p \in \mathcal{P}}p} = \sum_{p \in \mathcal{P}}\omega(p)$. + \item $\omega$ is the norm limit of a sequence of weak-operator continuous positive linear functionals. \item $\omega$ is weak-operator continuous on bounded subsets of $M$. \item $\omega$ is strong-operator continuous on bounded subsets of $M$. @@ -90,7 +90,7 @@ so $\omega_F$ is weak-operator continuous. Finally, for any $x \in M$, \begin{align*} - |\omega(x) - \omega_F(x)| &= |\omega(x) - \omega(xq_F)| + |\omega(xq_F) - \omega(q_Fxq_F)| \\ + |\omega(x) - \omega_F(x)| &\le |\omega(x) - \omega(xq_F)| + |\omega(xq_F) - \omega(q_Fxq_F)| \\ &= |\omega(x(1_M - q_F))| + |\omega((1 - q_F)xq_F)| \\ &\le 2\norm{x}_M\omega(1)^{1/2} \cdot \omega(1 - q_F)^{1/2} \end{align*} @@ -123,11 +123,11 @@ If the above holds, then $\phi$ is \textbf{normal}. \end{definition} \begin{proof}[Proof, {{\cite[Proposition 2.5.8]{AnantharamanPopa}}}. ] - (2) $\Rightarrow$ (3): Let $\omega \in M^*$ be a weak-operator continuous functional. By \autoref{lemma:weak-operator-jordan}, there exists positive weak-operator continuous functionals $\bracsn{\omega_k}_1^4$ such that $\omega = \sum_{k = 0}^3 i^k \omega_k$. By assumption, $\omega_k \circ \phi$ is weak-operator continuous on bounded subsets of $M$. In which case, $\omega \circ \phi = \sum_{k = 0}^3 i^k \omega_k \circ \phi$ is weak-operator continuous on bounded subsets of $M$. Therefore $\phi$ is weak-operator continuous on bounded subset of $M$. + (2) $\Rightarrow$ (3): Let $\omega \in N^*$ be a weak-operator continuous functional. By \autoref{lemma:weak-operator-jordan}, there exists positive weak-operator continuous functionals $\bracsn{\omega_k}_0^3$ such that $\omega = \sum_{k = 0}^3 i^k \omega_k$. By assumption, $\omega_k \circ \phi$ is weak-operator continuous on bounded subsets of $M$. In which case, $\omega \circ \phi = \sum_{k = 0}^3 i^k \omega_k \circ \phi$ is weak-operator continuous on bounded subsets of $M$. Therefore $\phi$ is weak-operator continuous on bounded subsets of $M$. (3) $\Rightarrow$ (1), (4) $\Rightarrow$ (1): By (1) of \autoref{theorem:existence-of-projections-vna}. - (3) $\Rightarrow$ (4)\footnote{Central here is the observation that $x_\alpha \to 0$ in the weak operator topology i.f.f. $x_\alpha^*x \to 0$ in the strong operator topology. }: Suppose that $M$ acts on the Hilbert space $H$, and $N$ acts on the Hilbert space $K$. Let $\angles{x_\alpha}_{\alpha \in A} \subset M$ be a bounded net converging to $x \in M$, then for each $\xi \in H$, + (3) $\Rightarrow$ (4)\footnote{Central here is the observation that $x_\alpha \to 0$ in the strong operator topology i.f.f. $x_\alpha^*x_\alpha \to 0$ in the weak operator topology. }: Suppose that $M$ acts on the Hilbert space $H$, and $N$ acts on the Hilbert space $K$. Let $\angles{x_\alpha}_{\alpha \in A} \subset M$ be a bounded net converging to $x \in M$ in the strong operator topology, then for each $\xi \in H$, \[ \lim_{\alpha \in A}\dpn{(x - x_\alpha)^*(x - x_\alpha)\xi, \xi}{H} = \lim_{\alpha \in A}\normn{(x - x_\alpha)\xi}_H^2 = 0 \] @@ -139,7 +139,7 @@ as well. In particular, for any $\xi \in K$, \[ - \lim_{\alpha \in A}\norm{\phi(x - x_\alpha)}_K^2 = \lim_{\alpha \in A}\dpn{\phi(x - x_\alpha)^*\phi(x - x_\alpha)\xi, \xi}{K} = 0 + \lim_{\alpha \in A}\norm{\phi(x - x_\alpha)\xi}_K^2 = \lim_{\alpha \in A}\dpn{\phi(x - x_\alpha)^*\phi(x - x_\alpha)\xi, \xi}{K} = 0 \] Therefore $\phi$ restricted to bounded subsets of $M$ is strong-operator continuous. @@ -147,7 +147,7 @@ \begin{proposition} \label{proposition:vna-isomorphism-normal} - Let $M, N$ be von Neumann algebras, then every isomorphism $\phi: M \to N$ is normal. In particular, the weak and strong operator topologies on $M$, restricted to bounded subsets, is independent of its representation. + Let $M, N$ be von Neumann algebras, then every isomorphism $\phi: M \to N$ is normal. In particular, the weak and strong operator topologies on $M$, restricted to bounded subsets, are independent of its representation. \end{proposition} \begin{proof} As $\phi$ is an isomorphism, it is an order isomorphism. By (1) of \autoref{definition:normal-linear-map}, $\phi$ is normal. By (3) and (4) of \autoref{definition:normal-linear-map}, its restriction to bounded subsets of $M$ is both weak-operator and strong-operator continuous. @@ -176,7 +176,7 @@ \ol{B_{\pi(M)}(0, 1)}^{B(H)} \supset \pi(\ol{B_M(0, 1)}^M) \supset \ol{B_{\pi(M)}(0, 1)}^{\text{\small WOT}} \supset \ol{B_N(0, 1)}^{B(H)} \] - and $\ol{B_{\pi(M)}(0, 1)}^{B(H)} = \ol{B_{N}(0, 1)}^{B(H)}$, so $\pi(M) = \ol{\pi(M)}^{B(H)} = \pi(N)$ by \autoref{theorem:cstar-quotient}. + and $\ol{B_{\pi(M)}(0, 1)}^{B(H)} = \ol{B_{N}(0, 1)}^{B(H)}$, so $\pi(M) = \ol{\pi(M)}^{B(H)} = N$ by \autoref{theorem:cstar-quotient}. \end{proof} @@ -185,10 +185,6 @@ \begin{lemma} \label{lemma:tracial-von-neumann-gns} Let $M$ be a von Neumann algebra, $\varphi \in S(M)$ be a state, and $(H, \pi, \xi)$ be the \hyperref[GNS triple]{definition:gns-triple} associated with $\varphi$, then $\pi$ is normal if and only if $\varphi$ is normal. In which case, $\pi(M) \subset B(H)$ is a von Neumann algebra. - \begin{enumerate} - \item $\pi|_{\ol{B_M(0, 1)}}: \ol{B_M(0, 1)} \to \ol{B_{\pi(M)}(0, 1)}$ is weak-operator continuous. - \item $\pi(M) \subset B(H)$ is a von Neumann algebra. - \end{enumerate} \end{lemma} \begin{proof} (1): Suppose that $\varphi$ is normal. Since $\pi(M)\xi$ is dense in $H$, the \hyperref[Arzelà-Ascoli Theorem]{theorem:arzela-ascoli} implies that the weak operator topology on $\ol{B_{\pi(M)}(0, 1)}$ coincides with the topology of pointwise convergence on $\bracsn{\eta \otimes \theta|\eta, \theta \in \pi(M)\xi}$. For each $x, y, z \in M$,