Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Lie Triple Derivations on von Neumann Algebras

  • 23 Accesses


Let \(\mathcal{A}\) be a von Neumann algebra with no central abelian projections. It is proved that if an additive map δ : \(\mathcal{A}\)\(\mathcal{A}\) satisfies δ([[a, b], c]) = [[δ(a), b], c]+[[a, δ(b)], c]+ [[a, b], δ(c)] for any a, b, c\(\mathcal{A}\) with ab = 0 (resp. ab = P, where P is a fixed nontrivial projection in \(\mathcal{A}\)), then there exist an additive derivation d from \(\mathcal{A}\) into itself and an additive map f : \(\mathcal{A}\)\(\mathcal{Z}_\mathcal{A}\) vanishing at every second commutator [[a, b], c] with ab = 0 (resp. ab = P) such that δ(a) = d(a) + f(a) for any a\(\mathcal{A}\).

This is a preview of subscription content, log in to check access.


  1. [1]

    Alaminos, J., Extremera, J., Villena, A. R. and Bresar, M., Characterizing homomorphisms and derivations on C*-algebras, Proc. Roy. Soc. Edinburgh Sect. A, 137, (2007), 1–7.

  2. [2]

    Benkovic, D., Lie triple derivations of unital algebras with idempotents, Linear Multilinear Algebra, 63(1), (2015), 141–165.

  3. [3]

    Bresar, M., Characterizing homomorphisms, derivations and multipliers in rings with idempotents, Proc. Roy. Soc. Edinburgh Sect. A, 137, (2007), 9–21.

  4. [4]

    Chebotar, M. A., Ke, W.-F. and Lee, P.-H., Maps characterized by action on zero products, Pacific J. Math., 216, (2004), 217–228.

  5. [5]

    Jing, W., Lu, S. and Li, P., Characterizations of derivations on some operator algebras, Bull. Austral. Math. Soc., 66, (2002), 227–232.

  6. [6]

    Kadison, R. V. and Ringrose, J. R., Fundamentals of the Theory of Operator Algebras, Vol. I, Academic Press, New York, 1983, Vol. II, Academic Press, New York, 1986.

  7. [7]

    Li, J. and Shen, Q., Characterizations of Lie higher and Lie triple derivations on triangular algebras, J. Korean Math. Soc., 49(2), (2012), 419–433.

  8. [8]

    Miers, C. R., Lie isomorphisms of operator algebras, Pacific J. Math., 38, (1971), 717–735.

  9. [9]

    Miers, C. R., Lie triple derivations of von Neumann algebras, Proc. Amer. Math. Soc., 71, (1978), 57–61.

  10. [10]

    Sakai, S., Derivations of W*-algebras, Ann. Math., 83, (1966), 273–279.

  11. [11]

    Zhang, Y., Hou, J. and Qi, X., Characterizing derivations for any nest algebras on Banach spaces by their behaviors at an injective operator, Linear Algebra Appl., 449, (2014), 312–333.

  12. [12]

    Zhu, J. and Zhao, S., Characterizations of all-derivable points in nest algebras, Proc. Amer. Math. Soc., 141, (2013), 2343–2350.

Download references


The author wishes to give his thanks to the referees and the editor for their helpful comments and suggestions.

Author information

Correspondence to Lei Liu.

Additional information

This work was supported by the National Natural Science Foundation of China (No. 11401452) and the China Postdoctoral Science Foundation (No. 2015M581513).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Liu, L. Lie Triple Derivations on von Neumann Algebras. Chin. Ann. Math. Ser. B 39, 817–828 (2018). https://doi.org/10.1007/s11401-018-0098-0

Download citation


  • Derivations
  • Lie triple derivations
  • von Neumann algebras

2000 MR Subject Classification

  • 16W25
  • 47B47