$H(\mathbf{1})(\omega)=
     \mathbf{1}_\omega$ for almost all $\omega$.\

4.  the section $(H(x)(\omega))^*$ is measurable for all $x\in L_p(M,\Phi)$.\

5.  the section $H(x)(\omega)\cdot H(y)(\omega)$ is measurable for all $x,y\in M$.

The existence of vector-valued lifting
======================================

In this section we establishes the existence of the lifting in a non-commutative setting. Note that in the case of $C^*$-algebras, the existence of the lifting has been given in [@GC1] (see also [@GM3]).

Let $M$ be a von Neumann algebra. Then it can be identify with a linear subspace of $L^\infty(\Omega,X)$ by the isomorphism $H$, since if $x\in M$, then one has $$\|H(x)\|_{L_0(\Omega,X)}=\|x\|_1=\Phi(|x|)\in L^\infty(\Omega)$$

\[1.3\] There exists a mapping $\ell: M(\subset
L^\infty(\Omega,X))\rightarrow \mathcal{L}^\infty(\Omega,X)$ with following properties

1.  for every $x\in M $ one has $\ell(x)\in x,\ {\rm dom}\ \ell (x)=\Omega$;\

2.  if $x_1, x_2\in M$ and $\lambda_1, \lambda_2\in
    \mathbb{R}$, then $\ell(\lambda_1 x_1+ \lambda_2 x_2)=\lambda_1\ell(x_1)+\lambda_2\ell(x_2)$;\

3.  $\|\ell(x)(\omega)\|_{L_p(M(\omega),\tau_\omega)}=p(\|x\|_p)(\omega)$ for all $x\in M$ and for all $\omega\in\Omega$;\

4.  if $x\in M,\ \lambda\in
     L^\infty(\Omega)$, then $\ell(ex)=p(e)\ell(x)$;\

5.  if $x\in M$, then $\ell(x^*)=\ell(x)^*$;\

6.  if $x,y\in M$, then $\ell(xy)=\ell(x)\ell(y)$;\

7.  the set $\{\ell(x)(\omega) : x\in M\}$ is dense in $L_p(M(\omega),\tau_\omega)$ for all $\omega\in\Omega$.

Following [@GaC] for every $x\in M$ we define $$\Phi_0(x)=\Phi(x)(1+\Phi(\id))^{-1}.$$ One can see that $\Phi_0$ is an $L^{\infty}(\Omega)$-valued faithful 