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Chinese Annals of Mathematics, Series B

, Volume 39, Issue 5, pp 817–828 | Cite as

Lie Triple Derivations on von Neumann Algebras

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Abstract

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\(\mathcal{A}\).

Keywords

Derivations Lie triple derivations von Neumann algebras 

2000 MR Subject Classification

16W25 47B47 

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Notes

Acknowledgement

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

References

  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.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Benkovic, D., Lie triple derivations of unital algebras with idempotents, Linear Multilinear Algebra, 63(1), (2015), 141–165.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Bresar, M., Characterizing homomorphisms, derivations and multipliers in rings with idempotents, Proc. Roy. Soc. Edinburgh Sect. A, 137, (2007), 9–21.MathSciNetCrossRefzbMATHGoogle Scholar
  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.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Jing, W., Lu, S. and Li, P., Characterizations of derivations on some operator algebras, Bull. Austral. Math. Soc., 66, (2002), 227–232.MathSciNetCrossRefzbMATHGoogle Scholar
  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.zbMATHGoogle Scholar
  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.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Miers, C. R., Lie isomorphisms of operator algebras, Pacific J. Math., 38, (1971), 717–735.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Miers, C. R., Lie triple derivations of von Neumann algebras, Proc. Amer. Math. Soc., 71, (1978), 57–61.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Sakai, S., Derivations of W*-algebras, Ann. Math., 83, (1966), 273–279.MathSciNetCrossRefzbMATHGoogle Scholar
  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.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Zhu, J. and Zhao, S., Characterizations of all-derivable points in nest algebras, Proc. Amer. Math. Soc., 141, (2013), 2343–2350.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Fudan University and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsXidian UniversityXi’anChina
  2. 2.School of Mathematical SciencesFudan UniversityShanghaiChina

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