| Title:
|
Multiplicative Lie triple derivations on standard operator algebras (English) |
| Author:
|
Wani, Bilal Ahmad |
| Language:
|
English |
| Journal:
|
Communications in Mathematics |
| ISSN:
|
1804-1388 (print) |
| ISSN:
|
2336-1298 (online) |
| Volume:
|
29 |
| Issue:
|
3 |
| Year:
|
2021 |
| Pages:
|
357-369 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $\mathcal {X}$ be a Banach space of dimension $n>1$ and $\mathfrak {A} \subset \mathcal {B}(\mathcal {X})$ be a standard operator algebra. In the present paper it is shown that if a mapping $d:\mathfrak {A} \rightarrow \mathfrak {A}$ (not necessarily linear) satisfies $$d([[U,V],W])=[[d(U),V],W]+[[U,d(V)],W]+[[U,V],d(W)]$$ for all $U, V, W \in \mathfrak {A}$, then $d=\psi +\tau $, where $\psi $ is an additive derivation of $\mathfrak {A}$ and $\tau : \mathfrak {A} \rightarrow \mathbb {F}I$ vanishes at second commutator $[[U,V],W]$ for all $U, V, W \in \mathfrak {A}$. Moreover, if $d$ is linear and satisfies the above relation, then there exists an operator $S\in \mathfrak {A}$ and a linear mapping $\tau $ from $\mathfrak {A}$ into $\mathbb {F}I$ satisfying $\tau ([[U,V],W])=0$ for all $U, V, W \in \mathfrak {A}$, such that $d(U)=SU-US+\tau (U)$ for all $U\in \mathfrak {A}$. (English) |
| Keyword:
|
Multiplicative Lie derivation |
| Keyword:
|
multiplicative Lie triple derivation |
| Keyword:
|
standard operator algebra. |
| MSC:
|
16W25 |
| MSC:
|
47B47 |
| MSC:
|
47B48 |
| idZBL:
|
Zbl 07484373 |
| idMR:
|
MR4355418 |
| . |
| Date available:
|
2022-01-10T10:00:17Z |
| Last updated:
|
2022-04-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/149322 |
| . |
| Reference:
|
[1] Cheung, W.: Lie derivations of triangular algebras.Linear Multilinear Algebra, 51, 2003, 299-310, MR 1995661, 10.1080/0308108031000096993 |
| Reference:
|
[2] Chen, L., Zhang, J.H.: Nonlinear Lie derivations on upper triangular matrices.Linear Multilinear Algebra, 56, 6, 2008, 725-730, MR 2457697, 10.1080/03081080701688119 |
| Reference:
|
[3] Daif, M.N.: When is a multiplicative derivation additive?.International Journal of Mathematics and Mathematical Sciences, 14, 3, 1991, 615-618, 10.1155/S0161171291000844 |
| Reference:
|
[4] Halmos, P.: A Hilbert space Problem Book, 2nd ed..1982, Springer-Verlag, New York, |
| Reference:
|
[5] Jing, W., Lu, F.: Lie derivable mappings on prime rings.Linear Multilinear Algebra, 60, 2012, 167-180, MR 2876765, 10.1080/03081087.2011.576343 |
| Reference:
|
[6] Ji, P., Zhao, R. Liu and Y.: Nonlinear Lie triple derivations of triangular algebras.Linear Multilinear Algebra, 60, 2012, 1155-1164, MR 2983757, 10.1080/03081087.2011.652109 |
| Reference:
|
[7] Ji, P.S., Wang, L.: Lie triple derivations of TUHF algebras.Linear Algebra Appl., 403, 2005, 399-408, Zbl 1114.46048, MR 2140293, 10.1016/j.laa.2005.02.004 |
| Reference:
|
[8] Lu, F.: Additivity of Jordan maps on standard operator algebras.Linear Algebra Appl., 357, 2002, 123-131, MR 1935229, 10.1016/S0024-3795(02)00367-1 |
| Reference:
|
[9] Lu, F.: Lie triple derivations on nest algebras.Math. Nachr., 280, 8, 2007, 882-887, Zbl 1124.47054, MR 2326061 |
| Reference:
|
[10] Lu, F., Jing, W.: Characterizations of Lie derivations of $\mathcal{B}(\mathcal{X})$.Linear Algebra Appl., 432, 1, 2010, 89-99, MR 2566460, 10.1016/j.laa.2009.07.026 |
| Reference:
|
[11] Lu, F., Liu, B.: Lie derivable maps on $\mathcal {B}(\mathcal {X})$.Journal of Mathematical Analysis and Applications, 372, 2010, 369-376, MR 2678869, 10.1016/j.jmaa.2010.07.002 |
| Reference:
|
[12] Mathieu, M., Villena, A. R.: The structure of Lie derivations on $C^\ast $-algebras.J. Funct. Anal., 202, 2003, 504-525, MR 1990536, 10.1016/S0022-1236(03)00077-6 |
| Reference:
|
[13] III, W.S. Martindale: When are multiplicative mappings additive?.Proc. Amer. Math. Soc., 21, 1969, 695-698, 10.1090/S0002-9939-1969-0240129-7 |
| Reference:
|
[14] Mires, C.R.: Lie derivations of von Neumann algebras.Duke Math. J., 40, 1973, 403-409, |
| Reference:
|
[15] Mires, C.R.: Lie triple derivations of von Neumann algebras.Proc. Am. Math. Soc., 71, 1978, 57-61, 10.1090/S0002-9939-1978-0487480-9 |
| Reference:
|
[16] Šemrl, P.: Additive derivations of some operator algebras.llinois J. Math., 35, 1991, 234-240, |
| Reference:
|
[17] Villena, A.R.: Lie derivations on Banach algebras.J. Algebra, 226, 2000, 390-409, 10.1006/jabr.1999.8193 |
| Reference:
|
[18] Yu, W., Zhang, J.: Nonlinear Lie derivations of triangular algebras.Linear Algebra Appl., 432, 11, 2010, 2953-2960, MR 2639258, 10.1016/j.laa.2009.12.042 |
| Reference:
|
[19] Zhang, J.H., Wu, B.W., Cao, H.X.: Lie triple derivations of nest algebras.Linear Algebra Appl., 416, 2-3, 2006, 559-567, MR 2242444, 10.1016/j.laa.2005.12.003 |
| Reference:
|
[20] Zhang, F., Zhang, J.: Nonlinear Lie derivations on factor von Neumann algebras.Acta Mathematica Sinica. (Chin. Ser), 54, 5, 2011, 791-802, MR 2918674 |
| . |
