Strong skew commutativity preserving maps on von Neumann algebras

Abstract

Let $\mathcal{M}$ be a von Neumann algebra without central summands of type $I_1$. Assume that $\Phi :\mathcal{M}\to \mathcal{M}$ is a surjective map. It is shown that $\Phi$ is strong skew commutativity preserving (that is, satisfies $\Phi (A)\Phi (B)-\Phi (B)\Phi (A{)}^{\ast }=AB-B{A}^{\ast }$ 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 $\Phi (A)=ZA$ for all $A\in \mathcal{M}$. The strong skew commutativity preserving maps on prime involution rings and prime involution algebras are also characterized.

16 pages