Geodesic
Generalized notion of a trace for von Neumann algebras
Matematičeskie trudy, Tome 13 (2010) no. 1, pp. 146-155 
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/
@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},
     year = {2010},
     volume = {13},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MT_2010_13_1_a5/}
}
TY  - JOUR
AU  - B. S. Zakirov
TI  - Generalized notion of a trace for von Neumann algebras
JO  - Matematičeskie trudy
PY  - 2010
SP  - 146
EP  - 155
VL  - 13
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/MT_2010_13_1_a5/
LA  - ru
ID  - MT_2010_13_1_a5
ER  - 
%0 Journal Article
%A B. S. Zakirov
%T Generalized notion of a trace for von Neumann algebras
%J Matematičeskie trudy
%D 2010
%P 146-155
%V 13
%N 1
%U http://geodesic.mathdoc.fr/item/MT_2010_13_1_a5/
%G ru
%F MT_2010_13_1_a5

Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Ganiev I. G., Chilin V. I., “Izmerimye rassloeniya *-algebr izmerimykh operatorov”, Uzbekskii mat. zhurn., 2001, no. 1, 8–14 | MR

[2] Muratov M. A., Chilin V. I., Algebry izmerimykh i lokalno izmerimykh operatorov, Tr. In-ta matematiki NAN Ukrainy, 69, Izd-vo In-ta matematiki NAN Ukrainy, Kiev, 2007

[3] Brown L. G., Kosaki H., “Jensen's inequality in semi-finite von Neumann algebras”, J. Operator Theory, 23:1 (1990), 3–19 | MR | Zbl

[4] Fack T., Kosaki H., “Generalized $s$-numbers of $\tau$-measurable operators”, Pacific. J. Math., 123:2 (1986), 269–300 | MR | Zbl

[5] Segal I. E., “A non-commutative extension of abstract integration”, Ann. of Math. (2), 57 (1953), 401–457 | DOI | MR | Zbl

[6] Strătilă Ş., Zsidó L., Lectures on von Neumann Algebras, Abacus Press, Bucharest, 1979 | MR | Zbl

[7] Takesaki M., Theory of Operator Algebras, v. I, Springer-Verlag, New York–Heidelberg, 1979 | MR

[8] Yeadon F. J., “Convergence of measurable operators”, Proc. Cambridge Philos. Soc., 74 (1973), 257–268 | DOI | MR | Zbl