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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\(\mathcal{A}\).

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Acknowledgement

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).

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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

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Keywords

  • Derivations
  • Lie triple derivations
  • von Neumann algebras

2000 MR Subject Classification

  • 16W25
  • 47B47