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. 
@article{MT_2010_13_1_a5,
     author = {B. S. Zakirov},
     title = {Generalized notion of a~trace for von {Neumann} algebras},
     journal = {Matemati\v{c}eskie trudy},
     pages = {146--155},
     publisher = {mathdoc},
     volume = {13},
     number = {1},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MT_2010_13_1_a5/}
}
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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/