\( \xi \)-Lie derivations on algebras of locally measurable operators. (Russian. English summary) Zbl 1468.46068

Zhuraev, Ilkhom Mukhitdinovich

Vladikavkaz. Mat. Zh. 18, No. 1, 26-35 (2016).

Summary: We study \(\xi \)-Lie derivations on algebras of locally measurable operators \(LS(M)\), where \(M\) is a von Neumann algebra without central summands of type \(I_1\).

MSC:

46L51	Noncommutative measure and integration
47B47	Commutators, derivations, elementary operators, etc.

Keywords:

von Neumann algebra; locally measurable operator; derivation; Lie derivations; \(\xi\)-Lie derivations; center valued trace

References:

[1]	Albeverio S., Ayupov Sh. A., Kudaybergenov K. K., “Structure of derivations on various algebras of measurable operators for type I von Neumann algebras”, #J. Func. Anal., #256 (2009), 2917-2943 · Zbl 1175.46054
[2]	Ayupov Sh. A., Kudaybergenov K. K., “Additive derivations algebras of measurable operators”, #J. Operator Theory, #67:2 (2012), 101-116 · Zbl 1263.46057
[3]	Zhuraev I. M., “Ctruktura lievykh differentsirovanii algebr izmerimykh operatorov”, #Vladikavk. mat. zhurn., #14:3 (2012), 58-62 · Zbl 1326.47107
[4]	Kadison R. V., Ringrose J. R., #Fundamentals of the Theory of Operator Algebras, v. I, Elementary Theory, Acad. Press, N.Y. etc., 1983, 398 pp. · Zbl 0518.46046
[5]	Kusraev A. G., “Automorphisms and derivations on a universally complete complex \(f\)-algebra”, #Siberian Math. J., #47:1 (2006), 77-85 · Zbl 1113.46043
[6]	Mathieu M., Villena A. R., “The structure of Lie derivations on \(C^*\)-algebras”, #J. Func. Anal., #202 (2003), 504-525 · Zbl 1032.46086
[7]	Muratov M. A., Chilin V. I., #Algebry izmerimykh i lokalno izmerimykh operatorov, Pratsi In-ty matemematiki NAN Ukraini, #69, Kiiv, 2007, 390 pp.
[8]	Robert Miers C., “Lie derivations of von Neumann algebras”, #Duke Math. J., #40 (1973), 403-409 · Zbl 0264.46064
[9]	Sakai S., #\(C^*\)-Algebras and \(W^*\)-Algebras, Springer-Verlag, N.Y.-Heidelberg-Berlin, 1971 · Zbl 0219.46042
[10]	Segal I., “A non-commutative extention of abstract integration”, #Ann. Math., #57 (1953), 401-457 · Zbl 0051.34201
[11]	Chilin V. I., Zhuraev I. M., “Kommutatory lokalno izmerimykh operatorov, prisoedinennykh k algebre fon Neimana tipa I”, #Materialy Respublikanskoi nauch. konf., v. 2, Urgench, 2012, 122-124
[12]	Chilin V. I., Zhuraev I. M., “Additivnye lievo differentsirovaniya na algebrakh lokalno izmerimykh operatorov”, #Materialy Respublikanskoi nauch. konf., Tashkent, 2013, 256-258