numerical trace on $M$. Hence, we can define a faithful normal numerical trace $\tau$ on $M$. Then the existence of $\mathcal{E}$ and properties (i) and (ii) follow from [@P Proposition 2.1].

Let us show (iii). Let $x\in M$. Then by Proposition 2.1. (iii) [@P] we get $\tau(\mathcal{E}(x))=\tau(x).$ By definition of $\tau$ one has $\nu(\Phi((\mathcal{E}(x))=\nu(\Phi(x))$ for any $x\in M$.

Let $\Phi((\mathcal{E}(x))-\Phi(x)=v|\Phi((\mathcal{E}(x))-\Phi(x)|$ be a polar decomposition of $\Phi((\mathcal{E}(x))-\Phi(x)$, for some partial isometry $v\in L^\infty(\Omega)$. Then one finds $$0=\nu(\Phi((\mathcal{E}(vx))-\nu(\Phi(vx))=\nu(v(\Phi((\mathcal{E}(x))-\Phi(x))=\nu(|\Phi(\mathcal{E}(x))-\Phi(x)|).$$

The faithfulness of $\nu$ implies that $\Phi(\mathcal{E}(x))=\Phi(x)$ for any $x\in M$. Similarly, we can show $\Phi(\mathcal{E}(x)y)=\Phi(xy)$ for any $x\in M, y\in N$.

\[Exp2\] The operator $\mathcal{E}$ (from Proposition \[Exp1\]) can be extended to a positive $L_p(M,\Phi)$–contraction from $L_p(M,\Phi)$ onto $L_p(N,\Phi)$, $1\leq p<\infty$. Moreover, if $a,b\in N$ and $x\in L_p(M,\Phi)$ then $$\mathcal{E}(axb)=a\mathcal{E}(x)b.$$

Let $x\in M, y\in N$. Then $|\Phi(\mathcal{\mathcal{E}}(x)y)|=|\Phi(xy)|\leq \Phi(|xy|)
\leq \|y\|_N\|x\|_1$. Hence, according Proposition \[1.1\] (i) we have $$\|\mathcal{E}(x)\|_1=\sup\{|\Phi(\mathcal{E}(x)y)|: \|y\|_M\leq
1\}\leq \|x\|_1.$$

Let $p>1$. By Proposition \[1.1\] (ii) one gets $$|\Phi(\mathcal{\mathcal{E}}(x)y)|=|\Phi(xy)|\leq \Phi(|xy|)\leq \|y\|_q\|x\|_p,$$ where $\frac{1}{p}+\frac{1}{q}=1.$ Then one finds $$\|\mathcal{E}(x)\|_p=\sup\{|\Phi(\mathcal{E}(x)y)|: \|y\|_q\leq
\mathbf{1}\}\leq\|x\|_p.$$

Hence $\|\mathcal{E}(x)\|_p\leq\|x\|_p$, for any $x\in M$, i.e. $\mathcal{E}$ is an $L_p(M,\Phi)$–contraction. Since $M$ is $(bo)$-dense in $L_p(M,\Phi)$, so we can continue $\mathcal{E}$ until $L_p(M,\Phi)$–contraction to $L_p(M,\Phi)$, which is denoted by the same symbol 