MAPS PRESERVING STRONG SKEW LIE PRODUCT ON FACTOR VON NEUMANN ALGEBRAS

Abstract

Let A be a factor von Neumann algebra and Phi be a nonlinear surjective map from A onto itself. We prove that, if Phi satisfies that Phi(A)Phi(B) - Phi(B)Phi(A)* = AB - BA* for all A, B is an element of A, then there exist a linear bijective map Psi: A -> A satisfying Psi(A)Psi(B) - Psi(B)Psi(A)* = AB - BA* for A, B is an element of A and a real functional h on A with h(0) = 0 such that Phi(A) = Psi(A) + h(A)I for every A is an element of A. In particular, if A is a type I factor, then, Phi(A) = cA + h(A)I for every A is an element of A, where c = +/- 1.