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Additive mappings on von Neumann algebras acting as Lie triple centralizer via local actions and related mappings

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Abstract

Let \( {\mathcal {M}} \) be an arbitrary von Neumann algebra, and \( \phi : {\mathcal {M}} \rightarrow {\mathcal {M}} \) be an additive map. We show that \(\phi \) satisfies

$$\begin{aligned} \phi ([ [A, B], C ]) = [ [\phi (A), B], C ] = [ [ A, \phi (B) ], C ] \end{aligned}$$

for all \(A,B, C \in \mathcal {M}\) with \(AB=0\) if and only if \( \phi (A) = W A + \xi (A) \) for any \( A \in {\mathcal {M}} \), where \( W \in {\textrm{Z}}( {\mathcal {M}} ) \) and \( \xi : {\mathcal {M}} \rightarrow {\textrm{Z}}({\mathcal {M}} ) \) is an additive mapping such that \(\xi ([[A, B ], C] )=0\) for any \(A,B, C \in \mathcal {M}\) with \(AB=0\). Then we present various applications of this result to determine other types of additive mappings on von Neumann algebras such as Lie triple centralizers, Lie centralizers, generalized Lie triple derivations at zero products, generalized Lie derivations, Jordan centralizers and Jordan generalized derivations. Some of our results are generalizations of some previously known results.

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Acknowledgements

The work of Behrooz Fadaee has been supported financially by Vice Chancellorship of Research and Technology at University of Kurdistan under research Project No. 01/9/11/17885.

The authors thank the referee(s) for careful reading of the manuscript and for helpful suggestions.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, University of Kurdistan, P.O. Box 416, Sanandaj, Kurdistan, Iran

    Behrooz Fadaee & Hoger Ghahramani

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  1. Behrooz Fadaee
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Fadaee, B., Ghahramani, H. Additive mappings on von Neumann algebras acting as Lie triple centralizer via local actions and related mappings. Acta Sci. Math. (Szeged) (2024). https://doi.org/10.1007/s44146-024-00123-z

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