Annals of Functional Analysis

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

Bing Yang and Xiaochun Fang

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

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

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

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

2-local Lie derivations Lie derivations von Neumann algebras


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.

Export citation


  • [1] S. A. Ayupov and K. Kudaybergenov, $2$-local derivations on von Neumann algebras, Positivity 19 (2015), no. 3, 445–455.
  • [2] J. C. Cabello and A. M. Peralta, On a generalized Šemrl’s theorem for weak $2$-local derivations on $B(H)$, Banach J. Math. Anal. 11 (2017), no. 2, 382–397.
  • [3] A. B. A. Essaleh, A. M. Peralta, and M. I. Ramírez, Weak-local derivations and homomorphisms on $C^{*}$-algebras, Linear Multilinear Algebra 64 (2016), no. 2, 169–186.
  • [4] J. He, J. Li, G. An, and W. Huang, Characterization of $2$-local derivations and local Lie derivations of certain algebras (in Russian), Sibirsk. Mat. Zh. 59 (2018), no. 4, 912–926; English translation in Sib. Math. J. 59 (2018), no. 4, 721–730.
  • [5] B. E. Johnson, Local derivations on $C^{*}$-algebras are derivations, Trans. Amer. Math. Soc. 353 (2001), no. 1, 313–325.
  • [6] R. V. Kadison, Local derivations, J. Algebra 130 (1990), no. 2, 494–509.
  • [7] D. Liu and J. Zhang, Local Lie derivations on certain operator algebras, Ann. Funct. Anal. 8 (2017), no. 2, 270–280.
  • [8] L. Liu, $2$-local Lie derivations on semi-finite factor von Neumann algebras, Linear Multilinear Algebra 64 (2016), no. 9, 1679–1686.
  • [9] C. R. Miers, Lie homomorphisms of operator algebras, Pacific J. Math. 38 (1971), 717–735.
  • [10] C. R. Miers, Lie derivations of von Neumann algebras, Duke Math. J. 40 (1973), 403–409.
  • [11] M. Niazi and A. M. Peralta, Weak-$2$-local derivations on $\mathbb{M}_{n}$, Filomat 31 (2017), no. 6, 1687–1708.
  • [12] G. K. Pedersen, $C^{*}$-algebras and Their Automorphism Groups, London Math. Soc. Monogr. 14, Academic Press, London, 1979.
  • [13] X. Qi, J. Ji, and J. Hou, Characterization of additive maps $\xi$-Lie derivable at zero on von Neumann algebras, Publ. Math. Debrecen 86 (2015), nos. 1–2, 99–117.
  • [14] P. Šemrl, Local automorphisms and derivations on $B(H)$, Proc. Amer. Math. Soc. 125 (1997), no. 9, 2677–2680.
  • [15] H. Sunouchi, Infinite Lie rings, Tohoku Math. J. (2) 8 (1956), no. 3, 291–307.
  • [16] B. Yang and X. Fang, Weak $2$-local derivations on finite von Neumann algebras, Linear Multilinear Algebra 66 (2018), no. 8, 1520–1529.