From 7e649d64f8bbedda0f563e525b114246ca54aad6 Mon Sep 17 00:00:00 2001 From: Bokuan Li Date: Sun, 8 Nov 2026 19:29:34 -0500 Subject: [PATCH] Standard form, first draft. --- src/vna/ii1/standard.tex | 162 +++++++++++++++++++++++++++++++++++++++ 1 file changed, 162 insertions(+) create mode 100644 src/vna/ii1/standard.tex diff --git a/src/vna/ii1/standard.tex b/src/vna/ii1/standard.tex new file mode 100644 index 0000000..9a1f787 --- /dev/null +++ b/src/vna/ii1/standard.tex @@ -0,0 +1,162 @@ +\section{The Standard Representation} +\label{section:standard-representation} + + + +\begin{definition}[Standard Representation] +\label{definition:tracial-vna-standard} + Let $(M, \tau)$ be a tracial von Neumann algebra, then the \hyperref[GNS representation]{definition:gns-triple} + \[ + (\pi, L^2(M, \tau), \xi_\tau) + \] + + 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 $M = \wh M \subset L^2(M, \tau)$. +\end{definition} + +\begin{definition}[Canonical Conjuation] +\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}. +\end{definition} + + +\begin{definition}[Left/Right Regular Representations] +\label{definition:mvn-standard-regular} + Let $(M, \tau)$ be a tracial von Neumann algebra and $x \in M$, then + \[ + L_x: L^2(M, \tau) \to L^2(M, \tau) \quad L_x\wh y = \wh{xy} + \] + and + + \[ + R_x: L^2(M, \tau) \to L^2(M, \tau) \quad R_x\wh y = \wh{yx} + \] + + are the \textbf{left/right multiplication maps} by $x$, respectively. The mapping + \[ + \lambda: M \to B(L^2(M, \tau)) \quad x \mapsto L_x + \] + + is the \textbf{left regular representation} of $M$, with $L(M) = \lambda(M)$ being the \textbf{algebra of left multiplications}. Similarly, + \[ + \rho: M^{op} \to B(L^2(M, \tau)) \quad x \mapsto R_x + \] + + is the \textbf{right regular representation} of $M^{op}$, with $R(M) = \rho(M)$ being the \textbf{algebra of right multiplications}. +\end{definition} + +\begin{proposition} +\label{proposition:vna-standard-conjugation-lr} + Let $(M, \tau)$ be a tracial von Neumann algebra, then: + \begin{enumerate} + \item For each $x \in M$, $JL_xJ = R_{x^*}$. + \item $JL(M)J = R(M)$. + \end{enumerate} +\end{proposition} +\begin{proof} + (1): For each $x, y \in M$, + \[ + JL_xJ \wh y = JL_x \wh{y^*} = J \wh{xy^*} = \wh{yx^*} = R_{x^*}y + \] +\end{proof} + +\begin{definition}[Multiplication by Vector] +\label{definition:vna-standard-vector-multiplication} + Let $(M, \tau)$ be a tracial von Neumann algebra and $\xi \in L^2(M, \tau)$, then + \[ + L^0_\xi: \wh M \to L^2(M, \tau) \quad L_\xi^0(\wh y) = R_y(\xi) = \xi y + \] + + and + \[ + R^0_\xi: \wh M \to L^2(M, \tau) \quad R_0^\xi(\wh y) = L_y(\xi) = y\xi + \] + + 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 $\xi$}. +\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 $y \in M$, + \begin{align*} + \dpn{\eta, \wh y}{L^2(M, \tau)} &= \limv{n} \dpn{L_\xi^0(\wh {x_n}), \wh y}{L^2(M, \tau)} = \limv{n} \dpn{R_{x_n}\xi, \wh y}{L^2(M, \tau)} \\ + &= \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 + \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. +\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 $\eta \in RC(M)$, $L_\xi R_\eta = R_\xi L_\eta$. +\end{lemma} +\begin{proof} + First suppose that $\xi, \eta \in M$. In which case, for each $x, y \in M$, + \[ + \dpn{L_\xi R_\eta \wh x, \wh y}{L^2(M, \tau)} = \dpn{\wh{\xi x \eta}, \wh y}{L^2(M, \tau)} = \dpn{R_\eta L_\xi \wh x, \wh y}{L^2(m, \tau)} + \] + + 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)} + \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 $\eta \in RC(M)$. + + Finally, suppose that $\xi \in LC(M)$ and $\eta \in RC(M)$, then for each $z \in M$, + \begin{align*} + \dpn{L_\xi z, \wh y}{L^2(M, \tau)} &= \dpn{\xi z, \wh y}{L^2(M, \tau)} = \dpn{\xi, \wh y z^*}{L^2(M, \tau)} \\ + &= \dpn{\xi, \wh y J\wh z}{L^2(M, \tau)} = \dpn{y^*\xi, J\wh z}{L^2(M, \tau)} = \dpn{\wh z, Jy^*\xi}{L^2(M, \tau)} + \end{align*} + + so + \begin{align*} + \dpn{L_\xi R_\eta \wh x, \wh y}{L^2(M, \tau)} &= \dpn{L_\xi (x \eta), \wh y}{L^2(M, \tau)} = \dpn{x \eta, J y^* \xi}{L^2(M, \tau)} \\ + |\dpn{L_\xi R_\eta \wh x, \wh y}{L^2(M, \tau)}| &\le \norm{x \eta}_{L^2(M, \tau)} \norm{y}_M \cdot \norm{\xi}_{L^2(M, \tau)}\\ + |\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 $\eta \in RC(M)$. +\end{proof} + +\begin{theorem} +\label{theorem:vna-convolver-identified} + Let $(M, \tau)$ be a tracial von Neumann algebra, then: + \begin{enumerate} + \item $L(M) = LC(M) = R(M)'$. + \item $JL(M)J = R(M) = RC(M) = L(M)'$. + \item In particular, for each $\xi \in L^2(M, \tau)$, $L_\xi \in B(L^2(M, \tau))$ if and only if $\xi \in \wh M$. + \end{enumerate} +\end{theorem} +\begin{proof} + (1): By definition, $L(M) \subset LC(M)$. By \autoref{lemma:vna-convolver-commute}, $LC(M) \subset RC(M)'$, so + \[ + L(M) \subset LC(M) \subset RC(M)' \subset R(M)' + \] + + and + \[ + R(M) \subset RC(M) \subset LC(M)' \subset L(M)' + \] + + Using the \hyperref[Bicommutant Theorem]{theorem:bicommutant}, it remains to show that $R(M)' \subset LC(M)$. To this end, let $T \in R(M)'$ and $\xi = T \wh 1$, then for each $x \in M$, + \[ + T\wh x = TR_x \wh 1 = R_x T \wh 1 = R_x \xi = \xi x = L_\xi^0 \wh x + \] + + 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 $T = L_\xi^0$. +\end{proof} +