Geodesic
Nonlinear $\ast $-Lie higher derivations of standard operator algebras
Communications in Mathematics, Tome 26 (2018) no. 1, pp. 15-29 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

Let $\mathcal {H}$ be an infinite-dimensional complex Hilbert space and $\mathfrak {A}$~be a standard operator algebra on $\mathcal {H}$ which is closed under the adjoint operation. It is shown that every nonlinear $\ast $-Lie higher derivation $\mathcal {D}=\{{\delta _n}\}_{n\in \mathbb {N}}$ of $\mathfrak {A}$ is automatically an additive higher derivation on $\mathfrak {A}$. Moreover, $\mathcal {D}=\{{\delta _n}\}_{n\in \mathbb {N}}$ is an inner $\ast $-higher derivation.
Let $\mathcal {H}$ be an infinite-dimensional complex Hilbert space and $\mathfrak {A}$~be a standard operator algebra on $\mathcal {H}$ which is closed under the adjoint operation. It is shown that every nonlinear $\ast $-Lie higher derivation $\mathcal {D}=\{{\delta _n}\}_{n\in \mathbb {N}}$ of $\mathfrak {A}$ is automatically an additive higher derivation on $\mathfrak {A}$. Moreover, $\mathcal {D}=\{{\delta _n}\}_{n\in \mathbb {N}}$ is an inner $\ast $-higher derivation.
Classification : 16W25, 46K15, 47B47
Keywords: Nonlinear $\ast $-Lie derivation; nonlinear $\ast $-Lie higher derivation; additive $\ast $-higher derivation; standard operator algebra. 
@article{COMIM_2018_26_1_a2,
     author = {Ashraf, Mohammad and Ali, Shakir and Wani, Bilal Ahmad},
     title = {Nonlinear $\ast ${-Lie} higher derivations of standard operator algebras},
     journal = {Communications in Mathematics},
     pages = {15--29},
     year = {2018},
     volume = {26},
     number = {1},
     mrnumber = {3827141},
     zbl = {1410.16038},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/COMIM_2018_26_1_a2/}
}
TY  - JOUR
AU  - Ashraf, Mohammad
AU  - Ali, Shakir
AU  - Wani, Bilal Ahmad
TI  - Nonlinear $\ast $-Lie higher derivations of standard operator algebras
JO  - Communications in Mathematics
PY  - 2018
SP  - 15
EP  - 29
VL  - 26
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/COMIM_2018_26_1_a2/
LA  - en
ID  - COMIM_2018_26_1_a2
ER  - 
%0 Journal Article
%A Ashraf, Mohammad
%A Ali, Shakir
%A Wani, Bilal Ahmad
%T Nonlinear $\ast $-Lie higher derivations of standard operator algebras
%J Communications in Mathematics
%D 2018
%P 15-29
%V 26
%N 1
%U http://geodesic.mathdoc.fr/item/COMIM_2018_26_1_a2/
%G en
%F COMIM_2018_26_1_a2
Ashraf, Mohammad; Ali, Shakir; Wani, Bilal Ahmad. Nonlinear $\ast $-Lie higher derivations of standard operator algebras. Communications in Mathematics, Tome 26 (2018) no. 1, pp. 15-29. http://geodesic.mathdoc.fr/item/COMIM_2018_26_1_a2/

[1] Brešar, M.: Commuting traces of biadditive mappings, commutativity preserving mappings and Lie mappings. Trans. Amer. Math. Soc., 335, 2, 1993, 525-546, | 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] Ferrero, M., Haetinger, C.: Higher derivations of semiprime rings. Comm. Algebra, 30, 2002, 2321-2333, | DOI | MR | Zbl

[4] Jing, Wu: Nonlinear $\ast $-Lie derivations of standard operator algebras. Quaestiones Mathematicae, 39, 8, 2016, 1037-1046, | MR

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

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

[7] III, W. S. Martindale: Lie derivations of primitive rings. Michigan Math. J., 11, 1964, 183-187, | DOI | MR

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

[9] Nowicki, A.: Inner derivations of higher orders. Tsukuba J. Math., 8, 2, 1984, 219-225, | MR

[10] Qi, X. F., Hou, J. C.: Lie higher derivations on nest algebras. Commun. Math. Res., 26, 2, 2010, 131-143, | MR

[11] Qi, X. F., Hou, J. C.: Characterization of Lie derivations on prime rings. Comm. Algebra, 39, 10, 2011, 3824-3835, | DOI | MR

[12] Šemrl, P.: Additive derivations of some operator algebras. Illinois J. Math., 35, 1991, 234-240, | DOI | MR

[13] Wei, F., Xiao, Z. K.: Higher derivations of triangular algebras and its generalizations. Linear Algebra Appl., 435, 2011, 1034-1054, | DOI | MR

[14] Xiao, Z. K., Wei, F.: Nonlinear Lie higher derivations on triangular algebras. Linear Multilinear Algebra, 60, 8, 2012, 979-994, | MR

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

[16] Yu, W., Zhang, J.: Nonlinear $\ast $-Lie derivations on factor von Neumann algebras. Linear Algebra Appl., 437, 2012, 1979-1991, | DOI | MR

[17] Zhang, F., Qi, X., Zhang, J.: Nonlinear $\ast $-Lie higher derivations on factor von Neumann algebras. Bull. Iranian Math. Soc., 42, 3, 2016, 659-678, | MR

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