Strong skew commutativity preserving maps on von Neumann algebras

TL;DR

This paper characterizes surjective maps on certain von Neumann algebras that preserve a strong skew commutativity property, showing they are essentially multiplication by a central self-adjoint element with square one.

Contribution

It provides a complete characterization of strong skew commutativity preserving maps on von Neumann algebras without type I_1 summands, extending to prime involution rings and algebras.

Findings

Such maps are of the form $ ext{Z}A$ where Z is a central self-adjoint element with Z^2=I.

The maps preserve the strong skew commutativity relation exactly.

Characterizations are extended to prime involution rings and algebras.

Abstract

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