## Abstract and Applied Analysis

### Characterizations of Nonlinear Lie Derivations of $B(X)$

#### Abstract

Let $X$ be an infinite dimensional Banach space, and $\mathrm{\Phi }:B(X)\to B(X)$ is a nonlinear Lie derivation. Then Φ is the form $\delta +\tau$ where $\delta$ is an additive derivation of $B(X)$ and $\tau$ is a map from $B(X)$ into its center ${Z}_{B(X)}$, which maps commutators into the zero.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 245452, 7 pages.

Dates
First available in Project Euclid: 26 February 2014

https://projecteuclid.org/euclid.aaa/1393449628

Digital Object Identifier
doi:10.1155/2013/245452

Mathematical Reviews number (MathSciNet)
MR3035394

Zentralblatt MATH identifier
1271.47029

#### Citation

Huo, Donghua; Zheng, Baodong; Liu, Hongyu. Characterizations of Nonlinear Lie Derivations of $B(X)$. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 245452, 7 pages. doi:10.1155/2013/245452. https://projecteuclid.org/euclid.aaa/1393449628

#### References

• J. Alaminos, M. Mathieu, and A. R. Villena, “Symmetric amenability and Lie derivations,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 137, no. 2, pp. 433–439, 2004.
• B. E. Johnson, “Symmetric amenability and the nonexistence of Lie and Jordan derivations,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 120, no. 3, pp. 455–473, 1996.
• F. Y. Lu, “Lie derivations of $I$-subspace lattice algebras,” Proceedings of the American Mathematical Society, vol. 135, no. 8, pp. 2581–2590, 2007.
• F. Y. Lu, “Lie derivations of certain CSL algebras,” Israel Journal of Mathematics, vol. 155, pp. 149–156, 2006.
• M. Mathieu and A. R. Villena, “The structure of Lie derivations on ${C}^{\ast\,\!}$-algebras,” Journal of Functional Analysis, vol. 202, no. 2, pp. 504–525, 2003.
• P. Šemrl, “Additive derivations of some operator algebras,” Illinois Journal of Mathematics, vol. 35, no. 2, pp. 234–240, 1991.
• M. I. Berenguer and A. R. Villena, “Continuity of Lie derivations on Banach algebras,” Proceedings of the Edinburgh Mathematical Society. Series II, vol. 41, no. 3, pp. 625–630, 1998.
• D. Benkovič, “Lie derivations on triangular matrices,” Linear and Multilinear Algebra, vol. 55, no. 6, pp. 619–626, 2007.
• W.-S. Cheung, “Lie derivations of triangular algebras,” Linear and Multilinear Algebra, vol. 51, no. 3, pp. 299–310, 2003.
• A. R. Villena, “Lie derivations on Banach algebras,” Journal of Algebra, vol. 226, no. 1, pp. 390–409, 2000.
• F. Y. Lu and W. Jing, “Characterizations of Lie derivations of $B(X)$,” Linear Algebra and its Applications, vol. 432, no. 1, pp. 89–99, 2010.
• W. Y. Yu and J.H. Zhang, “Nonlinear Lie derivations of triangular algebras,” Linear Algebra and its Applications, vol. 432, no. 11, pp. 2953–2960, 2010.
• P. S. Ji and W. Q. Qi, “Characterizations of Lie derivations of triangular algebras,” Linear Algebra and its Applications, vol. 435, no. 5, pp. 1137–1146, 2011.
• Z. F. Bai and S. P. Du, “The structure of nonlinear Lie derivation on von Neumann algebras,” Linear Algebra and its Applications, vol. 436, no. 7, pp. 2701–2708, 2012.
• D. G. Han, “Continuity and linearity of additive derivations of nest algebras on Banach spaces,” Chinese Annals of Mathematics B, vol. 17, no. 2, pp. 227–236, 1996.