Geodesic
Generalized notion of a~trace for von Neumann algebras
Matematičeskie trudy, Tome 13 (2010) no. 1, pp. 146-155.

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On a von Neumann algebra $M$, we consider traces with values in the algebra $L^0$ of measurable complex-valued functions. We show that every faithful normal $L^0$-valued trace on $M$ generates an $L^0$-valued metric on the algebra of measurable operators that are affiliated with $M$. Moreover, convergence in this metric coincides with local convergence in measure. 
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B. S. Zakirov. Generalized notion of a~trace for von Neumann algebras. Matematičeskie trudy, Tome 13 (2010) no. 1, pp. 146-155. http://geodesic.mathdoc.fr/item/MT_2010_13_1_a5/

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