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

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.