2-Local Lie triple isomorphisms of nest algebras

Abstract

Let $\mathcal{N}$ be a nontrivial nest on a complex separable Hilbert space $\mathcal{H}$ with dim $\mathcal{H}>2$, and Alg $\mathcal{N}$ be the associated nest algebra. Suppose that $\delta :\text{Alg}\mathcal{N}\to \text{Alg}\mathcal{N}$ is an additive surjective 2-local Lie triple isomorphism. If $\text{Alg}\mathcal{N}$ is not of infinite multiplicity, we prove that δ is of the form $\delta (x)=\varphi (x)+\tau (x)$ for any $x\in \text{Alg}\mathcal{N}$, where $\varphi$ is an isomorphism or the negative of an anti-isomorphism and $\tau :\text{Alg}\mathcal{N}\to \mathbb{C}I$ is a linear map with $\tau ([[x,y],z])=0$ for all $x,y,z\in \text{Alg}\mathcal{N}$.

Keywords:

• 2-local Lie triple isomorphism
• Lie triple isomorphism
• Nest algebra

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 16W10
• 47L35

Acknowledgments

The authors are highly grateful to the referees for their careful reading and valuable suggestions on this paper.

Additional information

Funding

This work is supported by the Scientific and Technological Innovation Programs of Higher Education Programs in Shanxi (Grant no. 2021L015) and Fundamental Research Program of Shanxi Province (No. 202103021223038).