Abstract
Let \({\mathcal {A}}\) be a factor von Neumann algebra with dim\({\mathcal {A}}\ge 2\). We prove that a map \(\phi : {\mathcal {A}}\rightarrow {\mathcal {A}}\) satisfies \(\phi ([A, B]_{\diamond })=[\phi (A), B]_{\diamond }+[A, \phi (B)]_{\diamond }\) for all \(A, B\in {\mathcal {A}}\) if and only if \(\phi \) is an additive \(*\)-derivation, where \([A, B]_{\diamond }=AB^{*}-BA^{*}\).
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References
Brešar, M.: Commuting traces of biadditive mappings, commutativity-preserving mappings and Lie mappings. Trans. Am. Math. Soc. 335(2), 525–546 (1993)
Chen, L., Zhang, J.: Nonlinear Lie derivations on upper triangular matrices. Linear Multilinear Algebra 56(6), 725–730 (2008)
Cheung, W.S.: Lie derivations of triangular algebras. Linear Multilinear Algebra 51(3), 299–310 (2003)
Jing, W., Lu, F.: Lie derivable mappings on prime rings. Linear Multilinear Algebra 60(2), 167–180 (2012)
Jing, W.: Nonlinear \(\ast \)-Lie derivations of standard operator algebras. Quaest. Math. 39(8), 1037–1046 (2016)
Lu, F., Liu, B.: Lie derivable maps on \(B(X)\). J. Math. Anal. Appl. 372(2), 369–376 (2010)
Lu, F., Liu, B.: Lie derivations of reflexive algebras. Integr. Equ. Oper. Theory 64(2), 261–271 (2009)
Martindale 3rd, W.S.: Lie derivations of primitive rings. Michigan Math. J. 11(2), 183–187 (1964)
Mathieu, M., Villena, A.R.: The structure of Lie derivations on \(C^{\ast }\)-algebras. J. Funct. Anal. 202(2), 504–525 (2003)
Yu, W., Zhang, J.: Nonlinear Lie derivations of triangular algebras. Linear Algebra Appl. 432(11), 2953–2960 (2010)
Yu, W., Zhang, J.: Nonlinear \(\ast \)-Lie derivations on factor von Neumann algebras. Linear Algebra Appl. 437(8), 1979–1991 (2012)
Zhang, J.: Lie derivations on nest subalgebras of von Neumann algebras. Acta. Math. Sin. 46(4), 657–664 (2003)
Acknowledgements
The authors wish to thank the anonymous referee for valuable comments and suggestions which have considerably improved the presentation of the paper. This work was supported by National Natural Science Foundation of China [Grant no. 11471199] and Shangluo University Key Disciplines Project, Discipline name: Mathematics.
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Communicated by Thomas Schick.
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Kong, L., Zhang, J. Nonlinear Bi-skew Lie Derivations on Factor von Neumann Algebras. Bull. Iran. Math. Soc. 47, 1097–1106 (2021). https://doi.org/10.1007/s41980-020-00430-5
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DOI: https://doi.org/10.1007/s41980-020-00430-5