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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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/

[1] Dixmier J., Les algèbres d'opérateurs dans l'espace Hilbertien (algèbres de von Neumann), 2$^e$ édition, Gauthier-Villars, Paris, 1969

[2] Haagerup U., “Normal weights on $W^*$-algebras”, J. Funct. Anal., 19:3 (1975), 302–317 | DOI | MR | Zbl

[3] Pedersen G. K., Takesaki M., “The Radon–Nikodym theorem for von Neumann algebras”, Acta Math., 130:1–2 (1973), 53–87 | DOI | MR | Zbl

[4] Upmeier H., “Automorphism groups of Jordan $C^*$-algebras”, Math. Z., 176:1 (1981), 21–34 | DOI | MR | Zbl

[5] Ayupov Sh. A., Klassifikatsiya i predstavlenie uporyadochennykh iordanovykh algebr, FAN, Tashkent, 1986

[6] Petz D., Zemánek J., “Characterizations of the trace”, Linear Algebra Appl., 111 (1988), 43–52 | DOI | MR | Zbl

[7] Komb F., “Vesa i uslovnye ozhidaniya na algebrakh fon Neimana”, Matem. Sb. perevodov, 18:6 (1974), 80–113

[8] Pedersen G. K., “The trace in semi-finite von Neumann algebras”, Math. Scand., 37:1 (1975), 142–144 | MR | Zbl

[9] Sherstnev A. N., “Kazhdyi gladkii ves yavlyaetsya $l$-vesom”, Izv. vuzov. Matem., 1977, no. 8, 88–91 | Zbl

[10] Trunov N. V., Sherstnev A. N., “Vvedenie v teoriyu nekommutativnogo integrirovaniya”, Itogi nauki i tekhniki. Sovremennye problemy matematiki. Noveishie dostizheniya, 27, VINITI, M., 1985, 167–190 | MR

[11] Gardner L. T., “An inequality characterizes the trace”, Canad. J. Math., 31:6 (1979), 1322–1328 | MR | Zbl

[12] Trunov N. V., Lokalno konechnye vesa na algebrakh Neimana, Dep. VINITI 10 yanvarya 1979 g., No. 101-79, Kazanskii un-t, Kazan, 1978

[13] Trunov N. V., “K teorii normalnykh vesov na algebrakh Neimana”, Izv. vuzov. Matem., 1982, no. 8, 61–70 | MR | Zbl

[14] Trunov N. V., “Lokalno konechnye vesa na algebrakh Neimana i kvadratichnye formy”, Konstruktivnaya teoriya funktsii i funktsion. analiz, 5, Izd-vo Kazanskogo un-ta, Kazan, 1985, 80–94 | MR