Then $$\begin{aligned}
\pi_\omega(\lambda
 x)&=& \pi_\omega\bigg(\sum\limits_{i=1}^n r_i \chi_{A_i} x\bigg)=
 \sum\limits_{i=1}^n \pi_\omega(r_i \chi_{A_i} x)\\
 &=&
 \sum\limits_{i=1}^n r_i \rho(\chi_{A_i})(\omega)\pi_\omega(x) =
\rho(\lambda)(\omega) \pi_\omega(u).
 \end{aligned}$$

The density argument implies that for any $\lambda\in
L^\infty(\Omega)$ there exists a sequence of simple functions $\{\lambda_n\}$ such that $\|\lambda_n-\lambda\|_{L^\infty(\Omega)}\rightarrow0$ as $n\rightarrow\infty$. From $$\begin{aligned}
 \|\pi_\omega(\lambda_n x)-\pi_\omega(\lambda
 x)\|_{L_1(M(\omega),\tau_\omega)}&=& \|\pi_\omega((\lambda_n - \lambda)x)\|_{L_1(M(\omega),\tau_\omega)} =
\rho(\|(\lambda_n-\lambda)x\|_1)(\omega)\\[2mm]
 &=& \rho(|\lambda_n -
 \lambda|)(\omega)\cdot \rho(\|x\|_1)(\omega)\\[2mm]
 &\leq&
 \|\rho(\lambda_n - \lambda)\|_{\mathcal{ L}^\infty(\Omega)}\cdot
 \|\rho(\|x\|_1)\|_{\mathcal{ L}^\infty(\Omega)}\\[2mm]
 &=&
 \|\lambda_n-\lambda\|_{L^\infty(\Omega)}\cdot \|\|x\|_1 \|_{L^\infty(\Omega)}
 \end{aligned}$$ one gets $\pi_\omega(\lambda
 x)=\lim\limits_{n\rightarrow\infty}\pi_\omega(\lambda_n x)$ for all $\omega\in\Omega$. So, $$\pi_\omega(\lambda
 x)=\lim\limits_{n\to\infty}\pi_\omega(\lambda_n x)=
 \lim\limits_{n\rightarrow\infty} \rho(\lambda_n)(\omega)\pi_\omega(x)=
\rho(\lambda)(\omega)\pi_\omega(u)$$ for all $\omega\in\Omega$.

Hence, $$i_\omega
 (\pi_\omega(\lambda x)=
 i_\omega(\rho(\lambda)(\omega)\pi_\omega(x))=
 \rho(\lambda)(\omega)i_\omega(\pi_\omega(x))= \rho(\lambda)(\omega)
 i_\omega (\pi_\omega(x))$$ and $j_\omega(\lambda x))=\rho(\lambda)(\omega)
 j_\omega (x)$ for all $\omega\in\Omega$. These mean that $\ell(\lambda x)=\rho(\lambda)\ell(x)$ for any $x\in M$ and $\lambda\in
 L^\infty(\Omega)$.

\(e) According to $\gamma_\omega(x^*)=\gamma_\omega(x)^*$ for any $x\in M$ we get $\ell(x^*)=\ell(x)^*$.

\(f) From $\gamma_\omega(xy)=\gamma_\omega(x)\gamma_\omega(y)$ for any $x,y\in
M$ it follows that $\ell(xy)=\ell(x)\ell(y)$ for any $x,y\in M$.

\(g) By the construction of $\gamma_\omega$ the set $\{\gamma_\omega(x): x\in M\}$ is dense in $L_1(M(\omega),\tau_\omega)$ for any $\omega\in\Omega$. Therefore, the set $\{\ell(x)(\omega) : x\in M\}$ is dense in $L_1(M(\omega),\tau_\omega)$ for all $\omega\in\Omega$.

The proof is complete.

The defined map $\ell$ in Theorem \[1.3\] is called *a noncommutative vector-valued lifting associated with the lifting $\rho$*.

Conditional expectations in $L_p(M,\Phi)$
