Local Lie $n$-derivations on certain algebras

TL;DR

This paper proves that local Lie $n$-derivations are actually Lie $n$-derivations in certain unital algebras and applies this result to various algebra classes, providing a comprehensive understanding of their structure.

Contribution

It establishes that local Lie $n$-derivations coincide with Lie $n$-derivations under mild conditions and characterizes them in multiple algebra contexts.

Findings

Local Lie $n$-derivations are Lie $n$-derivations in specified algebras

Descriptions of local Lie $n$-derivations on generalized matrix and triangular algebras

Applications to von Neumann and related operator algebras

Abstract

We prove that each local Lie $n$-derivation is a Lie $n$-derivation under mild assumptions on the unital algebras with a nontrivial idempotent. As applications, we obtain descriptions of local Lie $n$-derivations on generalized matrix algebras, triangular algebras, nest algebras, von Neumann algebras, and the algebras of locally measurable operators affiliated with a von Neumann algebra.