Strong skew commutativity preserving maps on von Neumann algebras

Let ${\mathcal M}$ be a von Neumann algebra without central summands of type $I_1$. Assume that $Φ:{\mathcal M}\rightarrow {\mathcal M}$ is a surjective map. It is shown that $Φ$ is strong skew commutativity preserving (that is, satisfies $Φ(A)Φ(B)-Φ(B)Φ(A)^*=AB-BA^*$ for all $A,B\in{\mathcal M}$) if and only if there exists some self-adjoint element $Z$ in the center of ${\mathcal M}$ with $Z^2=I$ such that $Φ(A)=ZA$ for all $A\in{\mathcal M}$. The strong skew commutativity preserving maps on prime involution rings and prime involution algebras are also characterized.

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