$x \in L^p({\mathcal{L}}G)$ with $2 \leq p \leq \infty$, then $D(L^2G,\psi) \subset {\mathrm{dom}}(x)$.

3.  Let $1 < p \leq \infty$. For any $\xi,\eta \in D(L^2G,\psi)$ there is a bounded linear map $\omega_{\xi,\eta}^p: L^p({\mathcal{L}}G) \to \C$ satisfying $\omega_{\xi,\eta}^p(x) = \langle |x|^{1/2} \xi,|x|^{1/2} u^* \eta \rangle$ if $x= u |x|$ is the polar decomposition of $x \in L^p({\mathcal{L}}G)$.

As a direct consequence, using the inclusion $L^2 F   \subset D(L^2G,\psi)$ and the closed graph theorem we get the following.

\[prop=def\_of\_PFxPF\] Let $F \subset G$ be a relatively compact Borel subset with positive measure, and $P_F:L^2G  \to L^2 F$ the orthogonal projection.

1.  If $p \geq 2$, $L^2F \subset {\mathrm{dom}}(x)$ for every $x \in L^p({\mathcal{L}}G)$, and $x \in L^p({\mathcal{L}}G) \to x P_F \in B(L^2 G)$ is a linear bounded map.

2.  If $1 \leq p \leq \infty$ there is a bounded linear map $L^p({\mathcal{L}}G) \to B(L^2 F)$, that maps an element $x =u |x| \in L^p({\mathcal{L}}G)$ to $ (|x|^{1/2}u^*P_F)^* |x|^{1/2} P_F$. We abusively denote this map by $x \mapsto P_F x P_F$.

Completely bounded maps between von Neumann algebras
----------------------------------------------------

Let ${\mathcal{M}}\subset B({\mathcal{H}})$ a von Neumann algebra, $\psi$ a weight on ${\mathcal{M}}'$ and $L^p({\mathcal{M}},\psi)$ the spatial non-commutative $L^p$-space. The commutant of $M_n \otimes {\mathcal{M}}$ 