A note on strong skew Jordan product preserving maps on von Neumann algebras

Let A be a von Neumann algebra acting on the complex Hilbert space H and Phi : A -> A be a surjective map that satisfies the condition Phi(T) Phi(P) + Phi(P) Phi(T)* = T P + PT* for all T and all projections P in A. We characterize the concrete form of Phi on selfadjoint elements of A. Also when A is a factor von Neumann algebra, it is shown that Phi is either of the form Phi(T) = T + i tau(T) I or of the form Phi(T) = -T + i tau(T) I, where tau : A -> R is a real map.