## Annals of Functional Analysis

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

#### Abstract

In this article we characterize the form of each 2-local Lie derivation on a von Neumann algebra without central summands of type ${I_{1}}$. We deduce that every 2-local Lie derivation $\delta$ on a finite von Neumann algebra $\mathcal{M}$ without central summands of type ${I_{1}}$ can be written in the form $\delta(A)=AE-EA+h(A)$ for all $A$ in $\mathcal{M}$, where $E$ is an element in $\mathcal{M}$ and $h$ is a center-valued homogenous mapping which annihilates each commutator of $\mathcal{M}$. In particular, every linear 2-local Lie derivation is a Lie derivation on a finite von Neumann algebra without central summands of type ${I_{1}}$. 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

#### 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

#### References

• [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.