Lie derivations on the algebra of measurable operators affiliated with a type I finite von Neumann algebra
Ilkhom M. Juraev
2013, issue 1, Ñ. 43-51
|Let $M$ be a type I finite von Neumann algebra and let $S(M)$ be the algebra of all measurable operators affiliated with M. We prove that every Lie derivation on $S(M)$ has standard form, that is, it is decomposed into the sum of a derivation and a center-valued trace.|
Keywords: von Neumann algebra, measurable operator, type I von Neumann algebra, derivation, inner derivation, Lie derivation, center-valued trace
Download the article (PDF-file)
| S. Albeverio, Sh. A. Ayupov, K. K. Kudaybergenov, “Structure of derivations on various algebras of measurable operators for type I von Neumann algebras”, J. Func. Anal., 256 (2009), 2917-2943.|
 P. Ara, M. Mathieu, Local multipliers of $C^*$-algebras, Springer-Verlag, New York-Heidelberg-Berlin, 2002.
 R. Banning, M. Mathieu, “Commutativity preserving mappings on semiprime rings”, Comm. Algebra, 25 (1997), 247-265.
 M. Bresar, “Commuting traces of biadditive mappings, commutativity-preserving mappings and Lie mappings", Trans. Amer. Math. Soc., 335 (1993), 525-546.
 H. G. Dales, Banach algebras and automatic continuity, London Math. Soc. Monographs, Oxford Press., Oxford., 2000.
 I. N. Herstein, “Lie and Jordan structures in simple, associative rings”, Bull. Amer. Math. Soc., 67 (1961), 517-531.
 B. E. Johnson, “Symmetric amenability and the nonexistence of Lie and Jordan derivations”, Math. Proc. Cambridge Philos. Soc., 120 (1996), 455-473.
 W. S. Martindale III, “Lie derivations of primitive rings”, Michigan Math. J., 11 (1964), 183-187.
 M. Mathieu, A. R. Villena, “The structure of Lie derivations on $C^*$-algebras”, J. Func. Anal., 202 (2003), 504-525.
 C. R. Miers, “Lie derivations of von Neumann algebras”, J. Duke Math., 40 (1973), 403-409.
 S. Sakai, $C^*$-algebras and $W^*$-algebras, Springer-Verlag, New York-Heidelberg-Berlin, 1971.
 I. Segal, “A non-commutative extension of abstract integration”, Ann. of Math., 57 (1953), 401-457.
 A. R. Villena, “Lie derivations on Banach algebras”, J. Algebra, 226 (2000), 390-409.
 I. M. Zhuraev, “The structure of Lie derivations of unbounded operator algebras of type I. (Russian)”, Uzbek. Mat. Zh., 1 (2011), 71-77.