Local Lie derivations on von Neumann algebras and algebras of locally measurable operators

TL;DR

This paper proves that local Lie derivations on von Neumann algebras are actually Lie derivations, extending the understanding of derivation structures in operator algebras and their measurable operator algebras.

Contribution

It establishes that all local Lie derivations on von Neumann algebras are genuine Lie derivations, including on algebras of locally measurable operators for certain types.

Findings

Every local Lie derivation on von Neumann algebras is a Lie derivation.

Local Lie derivations on $LS(rak M)$ are Lie derivations for type I von Neumann algebras with atomic projections.

Results extend the structure theory of derivations in operator algebras.

Abstract

Let $\mathcal{A}$ be a unital associative algebra and $\mathcal{M}$ be an $\mathcal{A}$-bimodule. A linear mapping $\varphi$ from $\mathcal{A}$ into an $\mathcal{A}$-bimodule $\mathcal{M}$ is called a Lie derivation if $\varphi[A,B]=[\varphi(A),B]+[A,\varphi(B)]$ for each $A,B$ in $\mathcal{A}$, and $\varphi$ is called a \emph{local Lie derivation} if for every $A$ in $\mathcal{A}$, there exists a Lie derivation $\varphi_{A}$ (depending on $A$) from $\mathcal{A}$ into $\mathcal{M}$ such that $\varphi(A)=\varphi_{A}(A)$. In this paper, we prove that every local Lie derivation on von Neumann algebras is a Lie derivation; and we show that if $\mathcal M$ is a type I von Neumann algebra with atomic lattice of projections, then every local Lie derivation on $LS(\mathcal M)$ is a Lie derivation.