Nonlinear * -Lie higher derivations of standard operator algebras

Mohammad Ashraf; Shakir Ali; Bilal Ahmad Wani

Communications in Mathematics (2018)

  • Volume: 26, Issue: 1, page 15-29
  • ISSN: 1804-1388

Abstract

top
Let ℋ be an infinite-dimensional complex Hilbert space and 𝔄  be a standard operator algebra on ℋ which is closed under the adjoint operation. It is shown that every nonlinear * -Lie higher derivation 𝒟 = { δ n } n ∈ ℕ of 𝔄 is automatically an additive higher derivation on 𝔄 . Moreover, 𝒟 = { δ n } n ∈ ℕ is an inner * -higher derivation.

How to cite

top

Ashraf, Mohammad, Ali, Shakir, and Wani, Bilal Ahmad. "Nonlinear $\ast $-Lie higher derivations of standard operator algebras." Communications in Mathematics 26.1 (2018): 15-29. <http://eudml.org/doc/294361>.

@article{Ashraf2018,
abstract = {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 $\mathcal \{D\}=\lbrace \{\delta _n\}\rbrace _\{n\in \mathbb \{N\}\}$ of $\mathfrak \{A\}$ is automatically an additive higher derivation on $\mathfrak \{A\}$. Moreover, $\mathcal \{D\}=\lbrace \{\delta _n\}\rbrace _\{n\in \mathbb \{N\}\}$ is an inner $\ast $-higher derivation.},
author = {Ashraf, Mohammad, Ali, Shakir, Wani, Bilal Ahmad},
journal = {Communications in Mathematics},
keywords = {Nonlinear $\ast $-Lie derivation; nonlinear $\ast $-Lie higher derivation; additive $\ast $-higher derivation; standard operator algebra},
language = {eng},
number = {1},
pages = {15-29},
publisher = {University of Ostrava},
title = {Nonlinear $\ast $-Lie higher derivations of standard operator algebras},
url = {http://eudml.org/doc/294361},
volume = {26},
year = {2018},
}

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
PB - University of Ostrava
VL - 26
IS - 1
SP - 15
EP - 29
AB - 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 $\mathcal {D}=\lbrace {\delta _n}\rbrace _{n\in \mathbb {N}}$ of $\mathfrak {A}$ is automatically an additive higher derivation on $\mathfrak {A}$. Moreover, $\mathcal {D}=\lbrace {\delta _n}\rbrace _{n\in \mathbb {N}}$ is an inner $\ast $-higher derivation.
LA - eng
KW - Nonlinear $\ast $-Lie derivation; nonlinear $\ast $-Lie higher derivation; additive $\ast $-higher derivation; standard operator algebra
UR - http://eudml.org/doc/294361
ER -

References

top
  1. Brešar, M., 10.1090/S0002-9947-1993-1069746-X, Trans. Amer. Math. Soc., 335, 2, 1993, 525-546, (1993) MR1069746DOI10.1090/S0002-9947-1993-1069746-X
  2. Chen, L., Zhang, J. H., 10.1080/03081080701688119, Linear Multilinear Algebra, 56, 6, 2008, 725-730, (2008) MR2457697DOI10.1080/03081080701688119
  3. Ferrero, M., Haetinger, C., 10.1081/AGB-120003471, Comm. Algebra, 30, 2002, 2321-2333, (2002) Zbl1010.16028MR1904640DOI10.1081/AGB-120003471
  4. Jing, Wu, Nonlinear * -Lie derivations of standard operator algebras, Quaestiones Mathematicae, 39, 8, 2016, 1037-1046, (2016) MR3589132
  5. Jing, W., Lu, F., 10.1080/03081087.2011.576343, Linear Multilinear Algebra, 60, 2012, 167-180, (2012) MR2876765DOI10.1080/03081087.2011.576343
  6. Lu, F. Y., Jing, W., 10.1016/j.laa.2009.07.026, Linear Algebra Appl., 432, 1, 2010, 89-99, (2010) MR2566460DOI10.1016/j.laa.2009.07.026
  7. III, W. S. Martindale, 10.1307/mmj/1028999091, Michigan Math. J., 11, 1964, 183-187, (1964) MR0166234DOI10.1307/mmj/1028999091
  8. Mires, C. R., 10.1215/S0012-7094-73-04032-5, Duke Math. J., 40, 1973, 403-409, (1973) MR0315466DOI10.1215/S0012-7094-73-04032-5
  9. Nowicki, A., Inner derivations of higher orders, Tsukuba J. Math., 8, 2, 1984, 219-225, (1984) MR0767958
  10. Qi, X. F., Hou, J. C., Lie higher derivations on nest algebras, Commun. Math. Res., 26, 2, 2010, 131-143, (2010) MR2662638
  11. Qi, X. F., Hou, J. C., 10.1080/00927872.2010.512588, Comm. Algebra, 39, 10, 2011, 3824-3835, (2011) MR2845604DOI10.1080/00927872.2010.512588
  12. Šemrl, P., 10.1215/ijm/1255987893, Illinois J. Math., 35, 1991, 234-240, (1991) MR1091440DOI10.1215/ijm/1255987893
  13. Wei, F., Xiao, Z. K., 10.1016/j.laa.2011.02.027, Linear Algebra Appl., 435, 2011, 1034-1054, (2011) MR2807218DOI10.1016/j.laa.2011.02.027
  14. Xiao, Z. K., Wei, F., Nonlinear Lie higher derivations on triangular algebras, Linear Multilinear Algebra, 60, 8, 2012, 979-994, (2012) MR2955278
  15. Yu, W., Zhang, J., 10.1016/j.laa.2009.12.042, Linear Algebra Appl., 432, 11, 2010, 2953-2960, (2010) MR2639258DOI10.1016/j.laa.2009.12.042
  16. Yu, W., Zhang, J., 10.1016/j.laa.2012.05.032, Linear Algebra Appl., 437, 2012, 1979-1991, (2012) MR2950465DOI10.1016/j.laa.2012.05.032
  17. Zhang, F., Qi, X., Zhang, J., Nonlinear * -Lie higher derivations on factor von Neumann algebras, Bull. Iranian Math. Soc., 42, 3, 2016, 659-678, (2016) MR3518210
  18. Zhang, F., Zhang, J., Nonlinear Lie derivations on factor von Neumann algebras, Acta Mathematica Sinica. (Chin. Ser), 54, 5, 2011, 791-802, (2011) MR2918674

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.