163 lines
6.9 KiB
TeX
163 lines
6.9 KiB
TeX
\section{The Standard Representation}
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\label{section:standard-representation}
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\begin{definition}[Standard Representation]
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\label{definition:tracial-vna-standard}
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Let $(M, \tau)$ be a tracial von Neumann algebra, then the \hyperref[GNS representation]{definition:gns-triple}
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\[
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(\pi, L^2(M, \tau), \xi_\tau)
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\]
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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 $M = \wh M \subset L^2(M, \tau)$.
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\end{definition}
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\begin{definition}[Canonical Conjugation]
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\label{definition:tracial-vna-standard-conjugate}
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Let $(M, \tau)$ be a tracial von Neumann algebra, then the mapping
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\[
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J: L^2(M, \tau) \to L^2(M, \tau) \quad \wh x \mapsto \wh{x^*}
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\]
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is a conjugate linear isometry on $L^2(M, \tau)$, known as the \textbf{canonical conjugation}.
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\end{definition}
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\begin{definition}[Left/Right Regular Representations]
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\label{definition:mvn-standard-regular}
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Let $(M, \tau)$ be a tracial von Neumann algebra and $x \in M$, then
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\[
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L_x: L^2(M, \tau) \to L^2(M, \tau) \quad L_x\wh y = \wh{xy}
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\]
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and
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\[
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R_x: L^2(M, \tau) \to L^2(M, \tau) \quad R_x\wh y = \wh{yx}
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\]
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are the \textbf{left/right multiplication maps} by $x$, respectively. The mapping
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\[
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\lambda: M \to B(L^2(M, \tau)) \quad x \mapsto L_x
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\]
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is the \textbf{left regular representation} of $M$, with $L(M) = \lambda(M)$ being the \textbf{algebra of left multiplications}. Similarly,
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\[
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\rho: M^{op} \to B(L^2(M, \tau)) \quad x \mapsto R_x
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\]
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is the \textbf{right regular representation} of $M^{op}$, with $R(M) = \rho(M)$ being the \textbf{algebra of right multiplications}.
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\end{definition}
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\begin{proposition}
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\label{proposition:vna-standard-conjugation-lr}
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Let $(M, \tau)$ be a tracial von Neumann algebra, then:
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\begin{enumerate}
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\item For each $x \in M$, $JL_xJ = R_{x^*}$.
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\item $JL(M)J = R(M)$.
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\end{enumerate}
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\end{proposition}
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\begin{proof}
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(1): For each $x, y \in M$,
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\[
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JL_xJ \wh y = JL_x \wh{y^*} = J \wh{xy^*} = \wh{yx^*} = R_{x^*}y
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\]
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\end{proof}
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\begin{definition}[Multiplication by Vector]
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\label{definition:vna-standard-vector-multiplication}
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Let $(M, \tau)$ be a tracial von Neumann algebra and $\xi \in L^2(M, \tau)$, then
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\[
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L^0_\xi: \wh M \to L^2(M, \tau) \quad L_\xi^0(\wh y) = R_y(\xi) = \xi y
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\]
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and
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\[
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R^0_\xi: \wh M \to L^2(M, \tau) \quad R_\xi^0(\wh y) = L_y(\xi) = y\xi
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\]
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are densely defined, closable operators.
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Let $L_\xi$ and $R_\xi$ be their respective closures, then $L_\xi$ and $R_\xi$ are the \textbf{left/right multiplication operators by $\xi$}.
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\end{definition}
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\begin{proof}
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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 $y \in M$,
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\begin{align*}
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\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)} \\
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&= \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)} \\
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&= \limv{n}\dpn{\xi, \wh {yx_n^*}}{L^2(M, \tau)} \\
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|\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)} \\
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&\le \limv{n}\norm{\xi}_{L^2(M, \tau)} \normn{y}_{M} \normn{\wh {x_n^*}}_{L^2(M, \tau)} = 0
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\end{align*}
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\end{proof}
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\begin{definition}[Convolvers]
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\label{definition:vna-standard-convolver}
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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.
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\end{definition}
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\begin{lemma}
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\label{lemma:vna-convolver-commute}
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Let $(M, \tau)$ be a tracial von Neumann algebra, then for each $\xi \in LC(M)$ and $\eta \in RC(M)$, $L_\xi R_\eta = R_\eta L_\xi$.
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\end{lemma}
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\begin{proof}
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First suppose that $\xi, \eta \in M$. In which case, for each $x, y \in M$,
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\[
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\dpn{L_\xi R_\eta \wh x, \wh y}{L^2(M, \tau)} = \dpn{\wh{\xi x \eta}, \wh y}{L^2(M, \tau)} = \dpn{R_\eta L_\xi \wh x, \wh y}{L^2(M, \tau)}
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\]
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Now suppose that $\xi \in M$, then for each $\eta \in RC(M)$, by the \hyperref[Cauchy-Schwarz inequality]{proposition:cauchy-schwarz},
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\begin{align*}
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|\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)} \\
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|\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)}
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\end{align*}
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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 $\eta \in RC(M)$.
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Finally, suppose that $\xi \in LC(M)$ and $\eta \in RC(M)$, then for each $z \in M$,
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\begin{align*}
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\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)} \\
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&= \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)}
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\end{align*}
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so
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\begin{align*}
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\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)} \\
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|\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)}\\
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|\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)}
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\end{align*}
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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 $\eta \in RC(M)$.
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\end{proof}
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\begin{theorem}
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\label{theorem:vna-convolver-identified}
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Let $(M, \tau)$ be a tracial von Neumann algebra, then:
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\begin{enumerate}
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\item $L(M) = LC(M) = R(M)'$.
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\item $JL(M)J = R(M) = RC(M) = L(M)'$.
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\item In particular, for each $\xi \in L^2(M, \tau)$, $L_\xi \in B(L^2(M, \tau))$ if and only if $\xi \in \wh M$.
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\end{enumerate}
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\end{theorem}
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\begin{proof}
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(1): By definition, $L(M) \subset LC(M)$. By \autoref{lemma:vna-convolver-commute}, $LC(M) \subset RC(M)'$, so
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\[
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L(M) \subset LC(M) \subset RC(M)' \subset R(M)'
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\]
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and
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\[
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R(M) \subset RC(M) \subset LC(M)' \subset L(M)'
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\]
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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 $x \in M$,
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\[
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T\wh x = TR_x \wh 1 = R_x T \wh 1 = R_x \xi = \xi x = L_\xi^0 \wh x
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\]
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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 $T = L_\xi$.
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\end{proof}
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