Strong Bi-skew Commutativity Preserving Maps on von Neumann Algebras

Abstract

Let ${\mathcal {M}}$ be a von Neumann algebra with no central summands of type $I_1$. Assume that $\Phi :{\mathcal {M}} \rightarrow {\mathcal {M}}$ is a surjective map and $\Phi (I)$ is an unitary operator. It is shown that $\Phi$ is strong bi-skew commutativity preserving (that is, $\Phi$ satisfies $\Phi (A)\Phi (B)^*- \Phi (B)\Phi (A)^*= AB^*- BA^*$ for all $A, B \in \mathcal M$) if and only if there exists a self-adjoint central operator $Z \in {\mathcal {M}}$ with $Z^2=I$ such that $\Phi (A) = ZA\Phi (I)$ for all $A \in {\mathcal {M}}$. The strong bi-skew commutativity preserving maps on prime algebras with involution are also characterized.