Multiplicative Lie triple derivations on standard operator algebras

Bilal Ahmad Wani

Communications in Mathematics (2021)

  • Volume: 29, Issue: 3, page 357-369
  • ISSN: 1804-1388

Abstract

top
Let 𝒳 be a Banach space of dimension n > 1 and 𝔄 ( 𝒳 ) be a standard operator algebra. In the present paper it is shown that if a mapping d : 𝔄 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 𝔄 , then d = ψ + τ , where ψ is an additive derivation of 𝔄 and τ : 𝔄 𝔽 I vanishes at second commutator [ [ U , V ] , W ] for all U , V , W 𝔄 . Moreover, if d is linear and satisfies the above relation, then there exists an operator S 𝔄 and a linear mapping τ from 𝔄 into 𝔽 I satisfying τ ( [ [ U , V ] , W ] ) = 0 for all U , V , W 𝔄 , such that d ( U ) = S U - U S + τ ( U ) for all U 𝔄 .

How to cite

top

Wani, Bilal Ahmad. "Multiplicative Lie triple derivations on standard operator algebras." Communications in Mathematics 29.3 (2021): 357-369. <http://eudml.org/doc/298199>.

@article{Wani2021,
abstract = {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 $U\in \mathfrak \{A\}$.},
author = {Wani, Bilal Ahmad},
journal = {Communications in Mathematics},
keywords = {Multiplicative Lie derivation; multiplicative Lie triple derivation; standard operator algebra},
language = {eng},
number = {3},
pages = {357-369},
publisher = {University of Ostrava},
title = {Multiplicative Lie triple derivations on standard operator algebras},
url = {http://eudml.org/doc/298199},
volume = {29},
year = {2021},
}

TY - JOUR
AU - Wani, Bilal Ahmad
TI - Multiplicative Lie triple derivations on standard operator algebras
JO - Communications in Mathematics
PY - 2021
PB - University of Ostrava
VL - 29
IS - 3
SP - 357
EP - 369
AB - 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 $U\in \mathfrak {A}$.
LA - eng
KW - Multiplicative Lie derivation; multiplicative Lie triple derivation; standard operator algebra
UR - http://eudml.org/doc/298199
ER -

References

top
  1. Cheung, W., 10.1080/0308108031000096993, Linear Multilinear Algebra, 51, 2003, 299-310, (2003) MR1995661DOI10.1080/0308108031000096993
  2. Chen, L., Zhang, J.H., 10.1080/03081080701688119, Linear Multilinear Algebra, 56, 6, 2008, 725-730, (2008) MR2457697DOI10.1080/03081080701688119
  3. Daif, M.N., 10.1155/S0161171291000844, International Journal of Mathematics and Mathematical Sciences, 14, 3, 1991, 615-618, (1991) DOI10.1155/S0161171291000844
  4. Halmos, P., A Hilbert space Problem Book, 2nd ed., 1982, Springer-Verlag, New York, (1982) 
  5. Jing, W., Lu, F., 10.1080/03081087.2011.576343, Linear Multilinear Algebra, 60, 2012, 167-180, (2012) MR2876765DOI10.1080/03081087.2011.576343
  6. Ji, P., Zhao, R. Liu and Y., 10.1080/03081087.2011.652109, Linear Multilinear Algebra, 60, 2012, 1155-1164, (2012) MR2983757DOI10.1080/03081087.2011.652109
  7. Ji, P.S., Wang, L., 10.1016/j.laa.2005.02.004, Linear Algebra Appl., 403, 2005, 399-408, (2005) Zbl1114.46048MR2140293DOI10.1016/j.laa.2005.02.004
  8. Lu, F., 10.1016/S0024-3795(02)00367-1, Linear Algebra Appl., 357, 2002, 123-131, (2002) MR1935229DOI10.1016/S0024-3795(02)00367-1
  9. Lu, F., Lie triple derivations on nest algebras, Math. Nachr., 280, 8, 2007, 882-887, (2007) Zbl1124.47054MR2326061
  10. Lu, F., 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
  11. Lu, F., Liu, B., 10.1016/j.jmaa.2010.07.002, Journal of Mathematical Analysis and Applications, 372, 2010, 369-376, (2010) MR2678869DOI10.1016/j.jmaa.2010.07.002
  12. Mathieu, M., Villena, A. R., 10.1016/S0022-1236(03)00077-6, J. Funct. Anal., 202, 2003, 504-525, (2003) MR1990536DOI10.1016/S0022-1236(03)00077-6
  13. III, W.S. Martindale, 10.1090/S0002-9939-1969-0240129-7, Proc. Amer. Math. Soc., 21, 1969, 695-698, (1969) DOI10.1090/S0002-9939-1969-0240129-7
  14. Mires, C.R., Lie derivations of von Neumann algebras, Duke Math. J., 40, 1973, 403-409, (1973) 
  15. Mires, C.R., 10.1090/S0002-9939-1978-0487480-9, Proc. Am. Math. Soc., 71, 1978, 57-61, (1978) DOI10.1090/S0002-9939-1978-0487480-9
  16. Šemrl, P., Additive derivations of some operator algebras, llinois J. Math., 35, 1991, 234-240, (1991) 
  17. Villena, A.R., 10.1006/jabr.1999.8193, J. Algebra, 226, 2000, 390-409, (2000) DOI10.1006/jabr.1999.8193
  18. 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
  19. Zhang, J.H., Wu, B.W., Cao, H.X., 10.1016/j.laa.2005.12.003, Linear Algebra Appl., 416, 2-3, 2006, 559-567, (2006) MR2242444DOI10.1016/j.laa.2005.12.003
  20. 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.