2-local Lie derivations of nest subalgebras of factors

Abstract

Let $\mathit{\beta }$ be a non-trivial nest in an arbitrary factor $\mathcal{M}$. We prove that every additive 2-local Lie derivation $\mathit{\delta }$ from nest subalgebra $\mathrm{a}l{g}_{\mathcal{M}}\mathit{\beta }$ into $\mathcal{M}$ can be written in the form $\mathit{\delta }(A)=AT-TA+h(A)I$ for all $A\in \mathrm{a}l{g}_{\mathcal{M}}\mathit{\beta }$, where T is an element in $\mathcal{M}$ and h is a linear functional vanishing on each commutator. We also provide an example to show the additivity of $\mathit{\delta }$ is necessary.

Keywords:

• Derivations
• 2-local Lie derivations
• nest subalgebras

AMS Subject Classifications:

• Primary: 47B47
• Secondary: 46L10

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

This work was supported by National Natural Science Foundation of China [grant number 11671078]; the Fundamental Research Funds for the Central Universities.