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In this section we prove the existence of a conditional expectation in noncommutative $L_p(M,\Phi)$-spaces associated with center-valued traces, and give representation such operators as measurable bundle of expectation operators in usual noncommutative $L_p$-spaces associated with numerical traces.

Let $M$ be any finite von Neumann algebra, with unit element $\id$, and $Z$ be some subalgebra of $Z(M)$. Then one can identify $Z=L_\infty(\Omega,\Sigma,m)$ and $S(Z) = L^0(\Omega,\Sigma,m)$. Let $\Phi$ be a strictly normal trace on $M$ with values in $L^0(\Omega)$, associated with $Z$. Let $p\geq 1$ and $L_p(M,\Phi)$ be the noncommutative $L_p$-space, associated with von Neumann algebra $M$ and the trace $\Phi$.

\[Exp1\] Let $M$ and $\Phi$ be as above. Let $N$ be a von Neumann subalgebra of $M$, and the restriction $\Phi$ to $N$ also denoted by $\Phi$. Then there exists a unique map $\mathcal{E}:
M\rightarrow N$ satisfying the following conditions:

1.  $\mathcal{E}$ is a normal positive contraction from $M$ onto $N$;

2.  $\mathcal{E}(axb)=a\mathcal{E}(x)b$ for any $x\in M$ and $a,b\in N;$

3.  $\Phi(\mathcal{E}(x))=\Phi(x)$ for any $x\in M$.

Without loss of generality, we may assume $\Phi(x)\in L^\infty(\Omega)$ (otherwise, we put $\Phi_0(x) =
 \Phi(x)(\id + \Phi(\id))^{-1}$.) Let $\nu$ be a faithful normal trace on $L^\infty(\Omega)$. Then $\tau(x)=\nu(\Phi(x))$ is also a faithful normal 