# Nonlinear $*$-Lie higher derivations on factor von Neumann algebras

Document Type : Research Paper

Authors

1 School of Science‎, ‎Xi'an University of Posts and Telecommunications‎, ‎Xi'an 710121‎, ‎P‎. ‎R. China.

2 Department of Mathematics‎, ‎Shanxi University‎, ‎Taiyuan 030006‎, ‎P‎. ‎R. China.

3 College of Mathematics and Information Science‎, ‎Shaanxi Normal University‎, ‎Xi'an 710062‎, ‎P‎. ‎R China.

Abstract

Let $\mathcal M$ be a factor von Neumann algebra. It is shown that every nonlinear $*$-Lie higher derivation $D={\phi_{n}}_{n\in\mathbb{N}}$ on $\mathcal M$ is additive. In particular, if $\mathcal M$ is infinite type $I$ factor, a concrete characterization of $D$ is given.

Keywords

Main Subjects

#### References

M. Brešar, Commuting traces of biadditive mappings, commutativity preserving mappings and Lie mappings, Trans. Amer. Math. Soc. 335 (1993), no. 2, 525--546.
Z. F. Bai and S. P. Du, The structure of nonlinear Lie derivation on von Neumann algebras, Linear Algebra Appl. 436 (2012), no. 7, 2701--2708.
L. Chen and J. H. Zhang, Nonlinear Lie derivations on upper triangular matrices, Linear Multilinear Algebra 56 (2008), no. 6, 725--730.
Y. Q. Du and Y. Wang, Lie derivations of generalized matrix algebras, Linear Algebra Appl. 437 (2012), no. 11, 2719--2726.
M. Ferrero and C. Haetinger, Higher derivations and a theorem by Herstein, Quaest. Math. 25 (2002), no. 2, 249--257.
M. Ferrero and C. Haetinger, Higher derivations of semiprime rings, Comm. Algebra 30 (2002), no. 5, 2321--2333.
N. Heerema, Higher derivations and automorphisms of complete local rings, Bull. Amer. Math. Soc. 76 (1970) 1212--1225.
F. Y. Lu and W. Jing, Characterizations of Lie derivations of B(X), Linear Algebra Appl. 432 (2010), no. 1, 89--99.
M. Mathieu and A. R. Villena, The structure of Lie derivations on C-algebras, J. Funct. Anal. 202 (2003), no. 2, 504--525.
A. Nakajima, On generalized higher derivations, Turkish J. Math. 24 (2000), no. 3, 295--311.
A. Nowicki, Inner derivations of higher orders, Tsukuba J. Math. 8 (1984), no. 2, 219--225.
X. F. Qi and J. C. Hou, Characterization of Lie derivations on prime rings, Comm. Algebra 39 (2011), no. 10, 3824--3835.
X. F. Qi and J. C. Hou, Lie higher derivations on nest algebras, Commun. Math. Res. 26 (2010), no. 2, 131--143.
A. Roy and R. Sridharan, Higher derivations and central simple algebras, Nagoya Math. J. 32 (1968) 21--30.
P. Šemrl, Additive derivations of some operator algebras, Illinois J. Math. 35 (1991), no. 2, 234--240.
F. Wei and Z. K. Xiao, Higher derivations of triangular algebras and its generalizations, Linear Algebra Appl. 435 (2011), no. 5, 1034--1054.
Z. K. Xiao and F. Wei, Nonlinear Lie higher derivations on triangular algebras, Linear Multilinear Algebra 60 (2012), no. 8, 979--994.
W. Y. Yu and J. H. Zhang, Nonlinear *-Lie derivations on factor von Neumann algebras, Linear Algebra Appl. 437 (2012), no. 8, 1979--1991.
W. Y. Yu and J. H. Zhang, Nonlinear Lie derivations of triangular algebras, Linear Algebra Appl. 432 (2010), no. 11, 2953--2960.
F. J. Zhang and J. H. Zhang, Nonlinear Lie derivations on factor von Neumann algebras, Acta Mathematica Sinica. (Chin. Ser) 54 (2011), no. 5, 791--802.

### History

• Receive Date: 26 September 2014
• Revise Date: 12 March 2015
• Accept Date: 17 March 2015
• First Publish Date: 01 June 2016