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*}
$\square$
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