On Lie triple centralizers of von Neumann algebras

‎Let U be a von Neumann algebra endowed with the Lie product [A,B]=AB-BA (A,B in U)‎. ‎In this article‎, ‎we consider the subsequent condition on an additive mapping \phi on the von Neumann algebra U with a suitable projection P in U‎: ‎\phi([ [A‎ , ‎B]‎ , ‎C ]) = [ [\phi(A)‎, ‎B]‎ , ‎C ] = [ [ A‎, ‎\phi (B)‎ ] , ‎C ] ‎ ‎for all A,B‎, ‎C in U with AB=P and we show that \phi(A)=WA+\xi(A) for all A in U$, ‎where W in Z(U)‎, ‎and \xi: U \to Z(U) (Z(U)$ is the center of U) is an additive map in which \xi([[A‎, ‎B ]‎, ‎C] )=0 for any A,B,C in U with AB=P‎. ‎We also give some results of the conclusion‎.