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Bulletin of the Iranian Mathematical Society

, Volume 45, Issue 2, pp 569–581 | Cite as

Nonlinear Generalized Lie n-Derivations on von Neumann Algebras

  • Xiaoxue Feng
  • Xiaofei QiEmail author
Original Paper
  • 158 Downloads

Abstract

Let \({\mathcal {M}}\) be a von Neumann algebra without central summands of type \(I_1\). Assume that \(G:{{\mathcal {M}}}\rightarrow {{\mathcal {M}}}\) is a nonlinear map. It is shown that G is a generalized Lie n-derivation (\(n\ge 2\)) if and only if \(G(A)=\varphi (A)+\tau (A)\) holds for all \(A\in {{\mathcal {M}}}\), where \(\varphi :{\mathcal M}\rightarrow {{\mathcal {M}}}\) is an additive generalized derivation and \(\tau :{{\mathcal {M}}}\rightarrow {{\mathcal {Z}}}({{\mathcal {M}}})\) is a central-valued map annihilating all \((n-1)\)th commutators. This generalizes some related known results.

Keywords

Generalized Lie n-derivations Lie n-derivations Lie derivations von Neumann algebras 

Mathematics Subject Classification

47B47 47B49 

Notes

Acknowledgements

The authors wish to give their thanks to the referees for careful reading and many valued comments.

References

  1. 1.
    Abdullaev, I.Z.: \(n\)-Lie derivations on von Neumann algebras. Uzbek. Mat. Zh. 5–6, 3–9 (1992)MathSciNetGoogle Scholar
  2. 2.
    Ashraf, M., Jabeen, A.: Nonlinear generalized Lie triple derivation on triangular algebras. Commun. Algebra 45, 4380–4395 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bai, Z.-F., Du, S.-P.: The structure of nonlinear Lie derivation on von Neumann algebras. Linear Algebra Appl. 436, 2701–2708 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Benkovič, D.: Generalized Lie derivations of unital algebras with idempotents. Oper. Matrices 12, 357–367 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Benkovič, D., Eremita, D.: Multiplicative Lie \(n\)-derivations of triangular rings. Linear Algebra Appl. 436(11), 4223–4240 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Brešar, M.: Commuting traces of biadditive mappings, commutativity-preserving mappings and Lie mappings. Trans. Am. Math. Soc 335, 525–546 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cheung, W.-S.: Lie derivations of triangular algebras. Linear Multilinear Algebra 51, 299–310 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Fei, X.-H., Zhang, J.-H.: Nonlinear generalized Lie derivations on triangular algebras. Linear Multilinear Algebra 65, 1158–1170 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Fošner, A., Wei, F., Xiao, Z.-K.: Nonlinear Lie-type derivations of von Neumann algebras and related topics. Colloq. Math. 132, 53–71 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kadison, R.V., Ringrose, J.R.: Fundamentals of the Theory of Operator Algebras, vol. I. Academic, New York (1983)zbMATHGoogle Scholar
  11. 11.
    Kadison, R.V., Ringrose, J.R.: Fundamentals of the Theory of Operator Algebras, vol. II. Academic, New York (1986)zbMATHGoogle Scholar
  12. 12.
    Lin, W.-H.: Nonlinear generalized Lie \(n\)-derivations on triangular algebras. Commun. Algebra 46, 2368–2383 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lu, F.-Y., Liu, B.-H.: Lie derivations of reflexive algebras. Integr. Equ. Oper. Theory 64(2), 261–271 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Mathieu, M., Villena, A.R.: The structure of Lie derivations on C*-algebras. J. Funct. Anal. 202, 504–525 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Miers, C.R.: Lie isomorphisms of operator algebras. Pac. J. Math. 38, 717–735 (1971)CrossRefzbMATHGoogle Scholar
  16. 16.
    Wang, Y., Wang, Y.: Multiplicative Lie \(n\)-derivations of generalized matrix algebras. Linear Algebra Appl. 438, 2599–2616 (2013)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Iranian Mathematical Society 2018

Authors and Affiliations

  1. 1.Department of MathematicsShanxi UniversityTaiyuanPeople’s Republic of China

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