Maps preserving products on von Neumann algebras

Let M , N be von Neumann algebras with no central summands of type I 1 . For T , S ∈ M , define [ T , S ] ⁎ = T S − S T ⁎ a new product of T and S. In this note, it is proved that a not necessarily linear bijective map Φ : M → N satisfies Φ ( [ T , S ] ⁎ ) = [ Φ ( T ) , Φ ( S ) ] ⁎ for all T , S ∈ M if and only if Φ is the direct sum of a linear ⁎-isomorphism and a conjugate linear ⁎-isomorphism.