Nonlinear bi-skew Lie-type derivations on factor von Neumann algebras

Abstract

Let $\mathcal{A}$ be a factor von Neumann algebra with $\mathit{dim}\mathcal{A}\ge 2.$ For any $X_1,X_2,\cdots ,X_n\in \mathcal{A},$ define $p_1(X_1)=X_1,$ $p_2(X_1,X_2)={[X_1,X_2]}_{•}=X_1{X}_2^{\ast }-X_2{X}_1^{\ast },$ and $p_n(X_1,X_2,\cdots ,X_n)={[p_{n-1}(X_1,X_2,\cdots ,X_{n-1}),X_n]}_{•}$ for all integers $n\ge 2.$ In this article, we prove that a map $L:\mathcal{A}\to \mathcal{A}$ satisfies $L(p_n(X_1,X_2,\cdots ,X_n))={\sum }_{i=1}^n{p}_n(X_1,X_2,\cdots ,X_{i-1},L(X_i),X_{i+1},\cdots ,X_n)$ for all $X_1,X_2,\cdots ,X_n\in \mathcal{A}$ if and only if L is an additive *-derivation.

Keywords:

• Additive *-derivation
• bi-skew Lie derivation
• bi-skew Lie-type derivation
• von Neumann algebra

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 16W25
• 46L10

Acknowledgments

The authors would like to express their sincere thanks to the anonymous referee for careful reading of the article and useful suggestions.

Additional information

Funding

This research is partially supported by a research grant from NBHM (No. 02011/5/2020 NBHM(R.P.) R&D II/6243).