Far Eastern Mathematical Journal

To content of the issue


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


Abstract
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)

References

[1] 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.
[2] P. Ara, M. Mathieu, Local multipliers of $C^*$-algebras, Springer-Verlag, New York-Heidelberg-Berlin, 2002.
[3] R. Banning, M. Mathieu, “Commutativity preserving mappings on semiprime rings”, Comm. Algebra, 25 (1997), 247-265.
[4] M. Bresar, “Commuting traces of biadditive mappings, commutativity-preserving mappings and Lie mappings", Trans. Amer. Math. Soc., 335 (1993), 525-546.
[5] H. G. Dales, Banach algebras and automatic continuity, London Math. Soc. Monographs, Oxford Press., Oxford., 2000.
[6] I. N. Herstein, “Lie and Jordan structures in simple, associative rings”, Bull. Amer. Math. Soc., 67 (1961), 517-531.
[7] B. E. Johnson, “Symmetric amenability and the nonexistence of Lie and Jordan derivations”, Math. Proc. Cambridge Philos. Soc., 120 (1996), 455-473.
[8] W. S. Martindale III, “Lie derivations of primitive rings”, Michigan Math. J., 11 (1964), 183-187.
[9] M. Mathieu, A. R. Villena, “The structure of Lie derivations on $C^*$-algebras”, J. Func. Anal., 202 (2003), 504-525.
[10] C. R. Miers, “Lie derivations of von Neumann algebras”, J. Duke Math., 40 (1973), 403-409.
[11] S. Sakai, $C^*$-algebras and $W^*$-algebras, Springer-Verlag, New York-Heidelberg-Berlin, 1971.
[12] I. Segal, “A non-commutative extension of abstract integration”, Ann. of Math., 57 (1953), 401-457.
[13] A. R. Villena, “Lie derivations on Banach algebras”, J. Algebra, 226 (2000), 390-409.
[14] I. M. Zhuraev, “The structure of Lie derivations of unbounded operator algebras of type I. (Russian)”, Uzbek. Mat. Zh., 1 (2011), 71-77.

To content of the issue