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

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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 $U\in \mathfrak {A}$.

How to cite

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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},
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
UR - http://eudml.org/doc/298199
ER -

References

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  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

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