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Let $\mathcal M$ be a noncommutative probability space i.e. there is a trace $\tau$ on $\mathcal M$ such that it is normal, faithful and $\tau(1)=1.$ Let $\tau_1$ be another normal trace on $\mathcal M$. Furthermore, assume that for any $x\geq 0$ in $\mathcal M,$ $\tau(x)=0$ implies $\tau_1(x)=0.$ In this situation what would be a kind of Radon-Nikodym derivative of $\tau_1$ with respect to $\tau$? (if it exists!)?

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    $\begingroup$ OK, now your revised question is DIFFERENT from your original one, because in the original version you only had semifinite densely-defined traces, whereas now at least one of the traces is finite. Do you also require $\tau_1$ to be a finite trace? or are you still asking about the semifinite case $\endgroup$
    – Yemon Choi
    Aug 17, 2019 at 23:40
  • $\begingroup$ "What would be a kind of Radon-Nikodym derivative?" I don't understand what's being asked. $\endgroup$
    – Nik Weaver
    Sep 3, 2019 at 13:13

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