\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 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 $L^2(M, \tau)$, known 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
    \[
    R^0_\xi: \wh M \to L^2(M, \tau) \quad R_\xi^0(\wh y) = L_y(\xi) = y\xi
    \]

    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 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 $y \in M$, 
    \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)} \\
    &\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 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 $\eta \in RC(M)$, $L_\xi R_\eta = R_\eta L_\xi$.
\end{lemma}
\begin{proof}
    First suppose that $\xi, \eta \in M$. In which case, for each $x, y \in M$, 
    \[
        \dpn{L_\xi R_\eta \wh x, \wh y}{L^2(M, \tau)} = \dpn{\wh{\xi x \eta}, \wh y}{L^2(M, \tau)} = \dpn{R_\eta L_\xi \wh x, \wh y}{L^2(M, \tau)}
    \]

    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 $T|_{\wh M} = L_\xi^0$. 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}