Characterization of Lie Derivations on von Neumann Algebras

TL;DR

This paper characterizes additive and linear maps satisfying a specific Lie derivation-like identity on von Neumann algebras, revealing their structure depending on the scalar parameter  and algebra properties.

Contribution

It provides a complete description of such maps on von Neumann algebras, including conditions for derivations, Jordan derivations, and maps involving the center, based on the scalar .

Findings

Additive maps are derivations, Jordan derivations, or sums involving the center, depending on .

Linear maps satisfying the identity are characterized by elements in the algebra and maps into the center.

The structure varies with whether  is rational, irrational, or specific values like 0, 1, or -1.

Abstract

Let ${\mathcal M}$ be a von Neumann algebra without central summands of type $I_1$ and $\xi\in{\mathbb C}$ a scalar. It is shown that an additive map $L$ on $\mathcal M$ satisfies $L(AB-\xi BA)=L(A)B-\xi BL(A)+L(B)A-\xi AL(B)$ whenever $A,B\in{\mathcal M}$ with $AB=0$ if and only if one of the following statements holds: (1) $\xi=1$, $L=\varphi+f$, where $\varphi$ is an additive derivation on $\mathcal M$ and $f$ is an additive map from $\mathcal M$ into its center vanishing on $[A,B]$ with $AB=0$; (2) $\xi=0$, $L(I)\in{\mathcal Z}({\mathcal M})$ and there exists an additive derivation $\varphi$ such that $L(A)=\varphi(A)+L(I)A$ for all $A$; (3) $\xi=-1$, $L$ is a Jordan derivation; (4) $\xi$ is rational and $\xi\not=0, \pm1$, $L$ is an additive derivation; (5) $\xi$ is not rational, there exists an additive derivation $\varphi$ satisfying $\varphi(\xi I)=\xi L(I)$ such that $L(A)=\varphi(A) + L(I)A$ for all $A \in{\mathcal M}$. A linear map $L$ on $\mathcal M$ satisfies $L(AB-\xi BA)=L(A)B-\xi BL(A)+L(B)A-\xi AL(B)$ whenever $A,B\in{\mathcal M}$ with $AB=0$ if and only if there exists a $T\in\mathcal M$ and a linear map $f:{\mathcal M}\rightarrow{\mathcal Z}({\mathcal M})$ vanishing on $[A,B]$ with $AB=0$ such that (i) $\xi=1$, $L(A)=AT-TA+f(A)$ for all $A\in\mathcal M$; (ii) $\xi=0$, $L(I)\in{\mathcal Z}({\mathcal M})$ and $L(A)=AT-(T-L(I))A$ for all $A\in{\mathcal M}$; (iii) $\xi\not=0,1$, $L(A)=AT-TA$ for all $A \in{\mathcal M}$.