Annals of Functional Analysis

The structure of 2-local Lie derivations on von Neumann algebras

Bing Yang and Xiaochun Fang

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Abstract

In this article we characterize the form of each 2-local Lie derivation on a von Neumann algebra without central summands of type I1. We deduce that every 2-local Lie derivation δ on a finite von Neumann algebra M without central summands of type I1 can be written in the form δ(A)=AEEA+h(A) for all A in M, where E is an element in M and h is a center-valued homogenous mapping which annihilates each commutator of M. In particular, every linear 2-local Lie derivation is a Lie derivation on a finite von Neumann algebra without central summands of type I1. We also show that every 2-local Lie derivation on a properly infinite von Neumann algebra is a Lie derivation.

Article information

Source
Ann. Funct. Anal., Volume 10, Number 2 (2019), 242-251.

Dates
Received: 15 May 2018
Accepted: 29 August 2018
First available in Project Euclid: 19 March 2019

Permanent link to this document
https://projecteuclid.org/euclid.afa/1552960864

Digital Object Identifier
doi:10.1215/20088752-2018-0024

Mathematical Reviews number (MathSciNet)
MR3941385

Subjects
Primary: 47B47: Commutators, derivations, elementary operators, etc.
Secondary: 47C15: Operators in $C^*$- or von Neumann algebras

Keywords
2-local Lie derivations Lie derivations von Neumann algebras

Citation

Yang, Bing; Fang, Xiaochun. The structure of 2-local Lie derivations on von Neumann algebras. Ann. Funct. Anal. 10 (2019), no. 2, 242--251. doi:10.1215/20088752-2018-0024. https://projecteuclid.org/euclid.afa/1552960864


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