Geodesic
Multiplicative Lie triple derivations on standard operator algebras
Communications in Mathematics, Tome 29 (2021) no. 3, pp. 357-369 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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 $U\in \mathfrak {A}$.
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 $U\in \mathfrak {A}$.
Classification : 16W25, 47B47, 47B48
Keywords: Multiplicative Lie derivation; multiplicative Lie triple derivation; standard operator algebra. 
@article{COMIM_2021_29_3_a2,
     author = {Wani, Bilal Ahmad},
     title = {Multiplicative {Lie} triple derivations on standard operator algebras},
     journal = {Communications in Mathematics},
     pages = {357--369},
     year = {2021},
     volume = {29},
     number = {3},
     mrnumber = {4355418},
     zbl = {07484373},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/COMIM_2021_29_3_a2/}
}
TY  - JOUR
AU  - Wani, Bilal Ahmad
TI  - Multiplicative Lie triple derivations on standard operator algebras
JO  - Communications in Mathematics
PY  - 2021
SP  - 357
EP  - 369
VL  - 29
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/COMIM_2021_29_3_a2/
LA  - en
ID  - COMIM_2021_29_3_a2
ER  - 
%0 Journal Article
%A Wani, Bilal Ahmad
%T Multiplicative Lie triple derivations on standard operator algebras
%J Communications in Mathematics
%D 2021
%P 357-369
%V 29
%N 3
%U http://geodesic.mathdoc.fr/item/COMIM_2021_29_3_a2/
%G en
%F COMIM_2021_29_3_a2
Wani, Bilal Ahmad. Multiplicative Lie triple derivations on standard operator algebras. Communications in Mathematics, Tome 29 (2021) no. 3, pp. 357-369. http://geodesic.mathdoc.fr/item/COMIM_2021_29_3_a2/

[1] Cheung, W.: Lie derivations of triangular algebras. Linear Multilinear Algebra, 51, 2003, 299-310, | DOI | MR

[2] Chen, L., Zhang, J.H.: Nonlinear Lie derivations on upper triangular matrices. Linear Multilinear Algebra, 56, 6, 2008, 725-730, | DOI | MR

[3] Daif, M.N.: When is a multiplicative derivation additive?. International Journal of Mathematics and Mathematical Sciences, 14, 3, 1991, 615-618, | DOI

[4] Halmos, P.: A Hilbert space Problem Book, 2nd ed. 1982, Springer-Verlag, New York,

[5] Jing, W., Lu, F.: Lie derivable mappings on prime rings. Linear Multilinear Algebra, 60, 2012, 167-180, | DOI | MR

[6] Ji, P., Zhao, R. Liu and Y.: Nonlinear Lie triple derivations of triangular algebras. Linear Multilinear Algebra, 60, 2012, 1155-1164, | DOI | MR

[7] Ji, P.S., Wang, L.: Lie triple derivations of TUHF algebras. Linear Algebra Appl., 403, 2005, 399-408, | DOI | MR | Zbl

[8] Lu, F.: Additivity of Jordan maps on standard operator algebras. Linear Algebra Appl., 357, 2002, 123-131, | DOI | MR

[9] Lu, F.: Lie triple derivations on nest algebras. Math. Nachr., 280, 8, 2007, 882-887, | MR | Zbl

[10] Lu, F., Jing, W.: Characterizations of Lie derivations of $\mathcal{B}(\mathcal{X})$. Linear Algebra Appl., 432, 1, 2010, 89-99, | DOI | MR

[11] Lu, F., Liu, B.: Lie derivable maps on $\mathcal {B}(\mathcal {X})$. Journal of Mathematical Analysis and Applications, 372, 2010, 369-376, | DOI | MR

[12] Mathieu, M., Villena, A. R.: The structure of Lie derivations on $C^\ast $-algebras. J. Funct. Anal., 202, 2003, 504-525, | DOI | MR

[13] III, W.S. Martindale: When are multiplicative mappings additive?. Proc. Amer. Math. Soc., 21, 1969, 695-698, | DOI

[14] Mires, C.R.: Lie derivations of von Neumann algebras. Duke Math. J., 40, 1973, 403-409,

[15] Mires, C.R.: Lie triple derivations of von Neumann algebras. Proc. Am. Math. Soc., 71, 1978, 57-61, | DOI

[16] Šemrl, P.: Additive derivations of some operator algebras. llinois J. Math., 35, 1991, 234-240,

[17] Villena, A.R.: Lie derivations on Banach algebras. J. Algebra, 226, 2000, 390-409, | DOI

[18] Yu, W., Zhang, J.: Nonlinear Lie derivations of triangular algebras. Linear Algebra Appl., 432, 11, 2010, 2953-2960, | DOI | MR

[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, | DOI | MR

[20] Zhang, F., Zhang, J.: Nonlinear Lie derivations on factor von Neumann algebras. Acta Mathematica Sinica. (Chin. Ser), 54, 5, 2011, 791-802, | MR