Suppose that $M$ is a von Neumann algebra with a finite, normal, faithful trace $\tau$. Let $T\colon M\to M$ be a completely positive, normal map.
Can $T$ be extended to a `positively linear map' on $L_0(M)_+$, the positive part of the *-algebra of $\tau$-measurable operators affiliated with $M$?
The idea is obvious. We express a positive element from $L_0(M)$ as the limit of an increasing net of positive elements $(x_a)$ in $M$. In this case we ought to define
$Tx = \sup Tx_a$,
as the net $(Tx_a)$ is increasing due to positivity of $T$. This definition readily works for $T=1_M\cdot \tau$, however in full generality there is an issue with the choice of $(x_a)$. I don't think it is obvious that $Tx$ does not depend on $(x_a)$.