Article
Full entry |
PDF
(0.4 MB)
Feedback
PDF
(0.4 MB)
Feedback
Keywords:
Nonlinear $\ast $-Lie derivation; nonlinear $\ast $-Lie higher derivation; additive $\ast $-higher derivation; standard operator algebra.
Nonlinear $\ast $-Lie derivation; nonlinear $\ast $-Lie higher derivation; additive $\ast $-higher derivation; standard operator algebra.
Summary:
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.
References:
[1] Brešar, M.: Commuting traces of biadditive mappings, commutativity preserving mappings and Lie mappings. Trans. Amer. Math. Soc., 335, 2, 1993, 525-546, DOI 10.1090/S0002-9947-1993-1069746-X | MR 1069746
[2] Chen, L., Zhang, J. H.: Nonlinear Lie derivations on upper triangular matrices. Linear Multilinear Algebra, 56, 6, 2008, 725-730, DOI 10.1080/03081080701688119 | MR 2457697
[3] Ferrero, M., Haetinger, C.: Higher derivations of semiprime rings. Comm. Algebra, 30, 2002, 2321-2333, DOI 10.1081/AGB-120003471 | MR 1904640 | Zbl 1010.16028
[4] Jing, Wu: Nonlinear $\ast $-Lie derivations of standard operator algebras. Quaestiones Mathematicae, 39, 8, 2016, 1037-1046, MR 3589132
[5] Jing, W., Lu, F.: Lie derivable mappings on prime rings. Linear Multilinear Algebra, 60, 2012, 167-180, DOI 10.1080/03081087.2011.576343 | MR 2876765
[6] Lu, F. Y., Jing, W.: Characterizations of Lie derivations of $\mathcal {B}(\mathcal {X})$. Linear Algebra Appl., 432, 1, 2010, 89-99, DOI 10.1016/j.laa.2009.07.026 | MR 2566460
[7] III, W. S. Martindale: Lie derivations of primitive rings. Michigan Math. J., 11, 1964, 183-187, DOI 10.1307/mmj/1028999091 | MR 0166234
[8] Mires, C. R.: Lie derivations of von Neumann algebras. Duke Math. J., 40, 1973, 403-409, DOI 10.1215/S0012-7094-73-04032-5 | MR 0315466
[9] Nowicki, A.: Inner derivations of higher orders. Tsukuba J. Math., 8, 2, 1984, 219-225, MR 0767958
[10] Qi, X. F., Hou, J. C.: Lie higher derivations on nest algebras. Commun. Math. Res., 26, 2, 2010, 131-143, MR 2662638
[11] Qi, X. F., Hou, J. C.: Characterization of Lie derivations on prime rings. Comm. Algebra, 39, 10, 2011, 3824-3835, DOI 10.1080/00927872.2010.512588 | MR 2845604
[12] Šemrl, P.: Additive derivations of some operator algebras. Illinois J. Math., 35, 1991, 234-240, DOI 10.1215/ijm/1255987893 | MR 1091440
[13] Wei, F., Xiao, Z. K.: Higher derivations of triangular algebras and its generalizations. Linear Algebra Appl., 435, 2011, 1034-1054, DOI 10.1016/j.laa.2011.02.027 | MR 2807218
[14] Xiao, Z. K., Wei, F.: Nonlinear Lie higher derivations on triangular algebras. Linear Multilinear Algebra, 60, 8, 2012, 979-994, MR 2955278
[15] Yu, W., Zhang, J.: Nonlinear Lie derivations of triangular algebras. Linear Algebra Appl., 432, 11, 2010, 2953-2960, DOI 10.1016/j.laa.2009.12.042 | MR 2639258
[16] Yu, W., Zhang, J.: Nonlinear $\ast $-Lie derivations on factor von Neumann algebras. Linear Algebra Appl., 437, 2012, 1979-1991, DOI 10.1016/j.laa.2012.05.032 | MR 2950465
[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 3518210
[18] Zhang, F., Zhang, J.: Nonlinear Lie derivations on factor von Neumann algebras. Acta Mathematica Sinica. (Chin. Ser), 54, 5, 2011, 791-802, MR 2918674
Search
Partner of
