Geodesic
Characterization of Normal Traces on Von Neumann Algebras by Inequalities for the Modulus
Matematičeskie zametki, Tome 72 (2002) no. 3, pp. 448-454

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It is proved that if a normal semifinite weight $\varphi $ on a von Neumann algebra $\mathscr M$ satisfies the inequality $\varphi (|a_1+a_2|)\le \varphi (|a_1|)+\varphi (|a_2|)$ for any selfadjoint operators $a_1,a_2$ in $\mathscr M$ , then this weight is a trace. Several similar characterizations of traces among the normal semifinite weights are proved. In particular, Gardner's result on the characterization of traces by the inequality $|\varphi (a)|\le \varphi (|a|)$ is refined and reinforced. 
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     author = {A. I. Stolyarov and O. E. Tikhonov and A. N. Sherstnev},
     title = {Characterization of {Normal} {Traces} on {Von} {Neumann} {Algebras} by {Inequalities} for the {Modulus}},
     journal = {Matemati\v{c}eskie zametki},
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     publisher = {mathdoc},
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     number = {3},
     year = {2002},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2002_72_3_a11/}
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A. I. Stolyarov; O. E. Tikhonov; A. N. Sherstnev. Characterization of Normal Traces on Von Neumann Algebras by Inequalities for the Modulus. Matematičeskie zametki, Tome 72 (2002) no. 3, pp. 448-454. http://geodesic.mathdoc.fr/item/MZM_2002_72_3_a11/