ON LIE TRIPLE CENTRALIZERS OF VON NEUMANN ALGEBRAS

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