Abstract
Let \(\mathcal {R}\) be a 2-torsion free unital prime \(*\)-ring containing a nontrivial symmetric idempotent. We prove that if a map \(\phi : \mathcal {R}\rightarrow \mathcal {R}\) satisfies \(\phi ([A, B]_{*})=[\phi (A), B]_{*}+[A, \phi (B)]_{*}\) for all \(A, B\in \mathcal {R}\), then \(\phi \) is an additive \(*\)-derivation, where \([A, B]_{*}=AB-BA^{*}\).
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References
An, R., Hou, J.: A characterization of \(\ast \)-automorphism on \({{\mathscr {B}}}(H)\). Acta Math. Sin. 26(2), 287–294 (2010)
Bai, Z., Du, S.: Maps preserving products \(XY-YX^{\ast }\) on von Neumann algebras. J. Math. Anal. Appl. 386(1), 103–109 (2012)
Chen, L., Zhang, J.: Nonlinear Lie derivations on upper triangular matrices. Linear Multilinear Algebra 56(6), 725–730 (2008)
Cui, J., Li, C.K.: Maps preserving product \(XY-YX^{\ast }\) on factor von Neumann algebras. Linear Algebra Appl. 431(5-7), 833–842 (2009)
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)
Taghavi, A., Darvish, V., Rohi, H.: Additivity of maps preserving products \(AP\pm PA^{\ast }\) on \(C^{\ast }\)-algebras. Math. Slovaca 67(7), 213–220 (2017)
Taghavi, A., Rohi, H.: A note on skew product preserving maps on factor von Neumann algebras. Rocky Mountain J. Math. 47(6), 2083–2094 (2017)
Yu, W., Zhang, J.: Nonlinear \(\ast \)-Lie derivations on factor von Neumann algebras. Linear Algebra Appl. 437(8), 1979–1991 (2012)
Yu, W., Zhang, J.: Nonlinear Lie derivations of triangular algebras. Linear Algebra Appl. 432(11), 2953–2960 (2010)
Zhang, F., Qi, X., Zhang, J.: Nonlinear \(\ast \)-Lie higher derivations on factor von Neumann algebras. Bull. Iran. Math. Soc. 42(3), 659–678 (2016)
Acknowledgements
The authors wish to thank the anonymous referees for their valuable comments and suggestions that improved the presentation of the paper. This research is supported by Scientific Research Project of Shangluo University (21SKY104).
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Communicated by Tony Joseph Puthenpuraka.
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Kong, L., Zhang, J. Nonlinear skew Lie derivations on prime \(*\)-rings. Indian J Pure Appl Math 54, 475–484 (2023). https://doi.org/10.1007/s13226-022-00269-y
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DOI: https://doi.org/10.1007/s13226-022-00269-y