Lie triple derivations of von Neumann algebras

Abstract:

A Lie triple derivation of an associative algebra M is a linear map $L:M \to M$ such that \[ L[[X,Y],Z] = [ {[L(X),Y],Z} ] + [ {[X,L(Y)],Z} ] + [ {[X,Y],L(Z)} ]\] for all $X,Y,Z \in M$. (Here $[X,Y] = XY - YX$ and [M, M] is the linear subspace of M generated by such terms.) We show that if M is a von Neumann algebra with no central abelian summands then there exists an operator $A \in M$ such that $L(X) = [A,X] + \lambda (X)$ where $\lambda :M \to {Z_M}$ is a linear map which annihilates brackets of operators in M.