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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 $(x_a)$.

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    $\begingroup$ Maybe I missing something... But what do you intend to do about "infinite values"? E.g. $\tau$ does extend to $L_0(M)_+$, but it may take the value $\infty$. So what are you hoping to do with general CP maps? $\endgroup$
    – Matthew Daws
    Commented Sep 21, 2017 at 8:27
  • $\begingroup$ It seems to me that a normal cp map should extend to a map between extended positive parts of $M$. Would that be helpful? $\endgroup$
    – Mateusz Wasilewski
    Commented Oct 19, 2017 at 15:33
  • $\begingroup$ @Wasilwski. What do you mean by the extended positive part of $M$? $\endgroup$
    – A beginner mathmatician
    Commented Jul 29, 2019 at 13:00

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