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/Part 8: Von Neumann Algebras/Chapter 44: Tracial von Neumann Algebras

44.2 The Standard Representation

Definition 44.2.1 (Standard Representation).label Let $(M, \tau)$ be a tracial von Neumann algebra, then the GNS representation

\[(\pi, L^{2}(M, \tau), \xi_{\tau})\]

is the standard representation of $(M, \tau)$. For each $x \in M$, identify $\wh x = x\xi_{\tau} \in L^{2}(M, \tau)$, so that $M = \wh M \subset L^{2}(M, \tau)$.

Definition 44.2.2 (Canonical Conjugation).label 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 canonical conjugation.

Definition 44.2.3 (Left/Right Regular Representations).label 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 left/right multiplication maps by $x$, respectively. The mapping

\[\lambda: M \to B(L^{2}(M, \tau)) \quad x \mapsto L_{x}\]

is the left regular representation of $M$, with $L(M) = \lambda(M)$ being the algebra of left multiplications. Similarly,

\[\rho: M^{op}\to B(L^{2}(M, \tau)) \quad x \mapsto R_{x}\]

is the right regular representation of $M^{op}$, with $R(M) = \rho(M)$ being the algebra of right multiplications.

Proposition 44.2.4.label Let $(M, \tau)$ be a tracial von Neumann algebra, then:

  1. (1)

    For each $x \in M$, $JL_{x}J = R_{x^*}$.

  2. (2)

    $JL(M)J = R(M)$.

Proof. (1): For each $x, y \in M$,

\[JL_{x}J \wh y = JL_{x} \wh{y^*}= J \wh{xy^*}= \wh{yx^*}= R_{x^*}y\]

$\square$

Definition 44.2.5 (Multiplication by Vector).label 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 left/right multiplication operators by $\xi$.

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*}

$\square$

Definition 44.2.6 (Convolvers).label Let $(M, \tau)$ be a tracial von Neumann algebra and $\xi \in L^{2}(M, \tau)$, then $\xi$ is a left convolver if $L_{\xi}$ is bounded, and $\xi$ is a right convolver if $R_{\xi}$ is bounded. The sets of left and right convolvers are denoted $LC(M)$ and $RC(M)$, respectively.

Lemma 44.2.7.label Let $(M, \tau)$ be a tracial von Neumann algebra, then for each $\xi \in LC(M)$ and $\eta \in RC(M)$, $L_{\xi} R_{\eta} = R_{\eta} L_{\xi}$.

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 Cauchy-Schwarz inequality,

\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 $\eta \in RC(M)$.$\square$

Theorem 44.2.8.label Let $(M, \tau)$ be a tracial von Neumann algebra, then:

  1. (1)

    $L(M) = LC(M) = R(M)'$.

  2. (2)

    $JL(M)J = R(M) = RC(M) = L(M)'$.

  3. (3)

    In particular, for each $\xi \in L^{2}(M, \tau)$, $L_{\xi} \in B(L^{2}(M, \tau))$ if and only if $\xi \in \wh M$.

Proof. (1): By definition, $L(M) \subset LC(M)$. By Lemma 44.2.7, $LC(M) \subset RC(M)'$, 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 Bicommutant Theorem, 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 $T = L_{\xi}$.$\square$

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