Strong Bi-skew Commutativity Preserving Maps on von Neumann Algebras

Let M be a von Neumann algebra with no central summands of type I1. Assume that : M → M is a surjective map and (I ) is an unitary operator. It is shown that is strong bi-skew commutativity preserving (that is, satisfies (A) (B)^∗− (B) (A) ∗ = AB^∗−BA∗ for all A, B ∈M) if and only if there exists a self-adjoint central operator Z ∈Mwith Z2 = I such that (A) = Z A (I ) for all A ∈M. The strong bi-skew commutativity preserving maps on prime algebras with involution are also characterized.