a faithful, normal and finite trace on $M(\w)$ (see [@Dik Proposition 6.8.3]). Now let us consider a non-commutative $L_1$-space $L_1(M(\w),\t_w)$, where $\t_\w(\cdot)=(1+\Phi(\id))(\w)\m_\w(\cdot)$. By $i_\w:\G_\w\to M(\w)$ one denotes the canonical embedding, and $j_\w:(M(\w),\t_w)\to L_1(M(\w),\t_\w)$ denotes the natural embedding. Then $\g_\w=j_\w\circ i_\w\circ\pi_\w$ is a linear mapping from $M$ to $L_1(M(\w),\t_\w)$.

Let us define $$\ell(x)(\omega)=\gamma_\omega(x)$$ for any $x\in M$.

\(a) Since any element $x\in M$ is identified with the element $\widehat{\gamma_\omega(x)}$, then one has $\ell(x)\in x$ (see [@GaC]).

\(b) The linearity of $\ell$ is obvious.

\(c) Let $x\in M$. Then $$\begin{aligned}
\|\ell(x)(\omega)\|_{L_1(M(\omega),\tau_\omega)}&=&\|\gamma_\omega(x)\|_{L_1(M(\omega),\tau_\omega)}=\tau_\omega(|\pi_\omega(x)|)\\
&=&\tau_\omega(\pi_\omega(|x|))=\rho(\Phi(|x|))(\omega)\\
&=&\rho(\|x\|_1)(\omega)\end{aligned}$$ for all $\omega\in\Omega$.

\(d) Let $\chi_A\in L^\infty(\Omega)$ and $x\in M$, then $\chi_A\cdot x\in M$. By $\tilde{\Sigma}$ we denote a complete Boolean algebra of equivalent classes w.r.t. a.e. equality, of sets from $\Sigma$. The lifting $\rho :
 L^\infty(\Omega)\rightarrow\mathcal{ L}^\infty(\Omega)$ induces a lifting $\tilde{\rho} : \tilde{\Sigma}\rightarrow\Sigma$ such that $p(\chi_A)=\chi_{\tilde{p}(A)}$. Due to $$\begin{aligned}
 \|\pi_\omega(\chi_A x)\|_{L_1(M(\omega),\tau_\omega)} &=&  \rho(\|\chi_A \cdot
 x\|_1)(\omega)=\rho(\chi_A)(\omega)\cdot
 p(\|x\|_1)(\omega)\\
 &=& \rho(\chi_A)(\omega)\cdot\|\pi_\omega(x)\|_{L_1(M(\omega),\tau_\omega)}\\
 &=&
 \chi_{\tilde{p}(A)}(\omega)\|\pi_\omega(x)\|_{L_1(M(\omega),\tau_\omega)},
 \end{aligned}$$ we obtain $\pi_\omega(\chi_A\cdot x)=0$, if $\omega\in\tilde{\rho}(A)$. Let $\omega\in\tilde{\rho}(A)$, then $$\begin{aligned}
 \|\pi_\omega(\chi_A\cdot
 x)-\pi_\omega(x)\|_{L_1(M(\omega),\tau_\omega)}&=& \|\pi_\omega(\chi_{\Omega\setminus
 A}\cdot x)\|_{L_1(M(\omega),\tau_\omega)}\\
 & =&\rho(\|\chi_{\Omega\setminus A}\cdot
 x\|_1)(\omega)= \rho(\chi_{\Omega\setminus A})(\omega)\cdot
 p(\|x\|_1)(\omega)\\
 &=& \chi_{\tilde{\rho}(\Omega\setminus
 A)}(\omega)\cdot \|\pi_\omega(x)\|_{L_1(M(\omega),\tau_\omega)}=0.
 \end{aligned}$$ Therefore, we have $\pi_\omega(\chi_A \cdot x)= \chi_{\tilde{\rho}(A)}(\omega)
 \cdot\pi_\omega(x)= p(\chi_A)(\omega)\cdot\pi_\omega(x)$ for all $\omega\in\Omega$.

Let $\lambda=\sum\limits_{i=1}^n r_i\chi_{A_i}\in
 L^\infty(\Omega)$ be a simple function. 