Korean J. Math. Vol. 27 No. 4 (2019) pp.969-976
DOI: https://doi.org/10.11568/kjm.2019.27.4.969

Nonlinear $\xi$-Lie-$\ast$-derivations on von Neumann algebras

Main Article Content

Aili Yang


Let $\mathscr{B(H)}$ be the algebra of all bounded linear operators on a complex Hilbert space $\mathscr{H}$ and $\mathscr{M}\subseteq\mathscr{B(H)}$ be a von Neumann algebra without central abelian projections. Let $\xi$ be a non-zero scalar. In this paper, it is proved that a mapping $\varphi:\mathscr{M}\rightarrow\mathscr{B(H)}$ satisfies $\varphi([A,B]^{\xi}_{\ast})=[\varphi(A),B]^{\xi}_{\ast}+[A,\varphi(B)]^{\xi}_{\ast}$ for all $A,B\in\mathscr{M}$ if and only if $\varphi$ is an additive $\ast$-derivation and $\varphi(\xi A)=\xi\varphi(A)$ for all $A\in\mathscr{M}.$

Article Details

Supporting Agencies

College of Science Xi’an University of Science and Technology Xi’an 710054 P. R. China


[1] B.E. Johnson, Symmetric amenability and the nonexistence of Lie and Jordan derivations, Math. Proc. Cambridge Philos. Soc. 120 (1996), 455–473. Google Scholar

[2] M. Mathieu and A.R. Villena, The structure of Lie derivations on C∗-algebras, J. Funct. Anal. 202 (2003), 504–525. Google Scholar

[3] J.-H. Zhang, Lie derivations on nest subalgebras of von Neumann algebras, Acta Math. Sinica 46 (2003), 657–664. Google Scholar

[4] W.S. Cheung, Lie derivation of triangular algebras, Linear and Multilinear Algebra 51 (2003), 299–310. Google Scholar

[5] X.F. Qi and J.C. Hou, Additive Lie (ξ-Lie) derivations and generalized Lie (ξ- Lie) derivations on nest algebras, Linear Algebra Appl. 431 (2009), 843–854. Google Scholar

[6] M. Breˇsar, Commuting traces of biadditive mappings, commutativity preserving mappings and Lie mappings, Trans. Amer. Math. Soc. 335 (1993), 525–546. Google Scholar

[7] Lin Chen and J. H. Zhang, Nonlinear Lie derivation on upper triangular matrix Google Scholar

[8] algebras, Linear and Multilinear Algebra 56 (6) (2008)725–730. Google Scholar

[9] W. Y. Yu and J. H. Zhang, Nonlinear Lie derivations of triangular algebras, Google Scholar

[10] Linear Algebra Appl. 432 (2010), 2953–2960. Google Scholar

[11] Miers CR, Lie isomorphisms of operator algebras, Pacific J. Math. 1971;38, 717–735. Google Scholar