Let $ \mathcal{A} $ be a unital $ \ast $-algebra containing a non-trivial projection. In this paper, we prove that if a map $ \Omega $ : $ \mathcal{A} $ $ \to $ $ \mathcal{A} $ such that
$ \begin{equation} \nonumber \Omega( [ \mathscr{K}, \mathscr{F}]_\ast \odot \mathscr{D}) = [\Omega(\mathscr{K}), \mathscr{F}]_\ast \odot \mathscr{D} + [ \mathscr{K}, \Omega (\mathscr{F})]_\ast \odot \mathscr{D} + [ \mathscr{K}, \mathscr{F}]_\ast \odot \Omega (\mathscr{D}), \end{equation} $
where $ [\mathscr{K}, \mathscr{F}]_{\ast} = \mathscr{K}\mathscr{F}- \mathscr{F}\mathscr{K}^\ast $ and $ \mathscr{K} \odot \mathscr{F} = \mathscr{K}^\ast \mathscr{F}+ \mathscr{F}\mathscr{K}^\ast $ for all $ \mathscr{K}, \mathscr{F}, \mathscr{D} \in \mathcal{A}, $ then $ \Omega $ is an additive $ \ast $-derivation. Furthermore, we extend its results on factor von Neumann algebras, standard operator algebras and prime $ \ast $-algebras. Additionally, we provide an example illustrating the existence of such maps.
Citation: Junaid Nisar, Turki Alsuraiheed, Nadeem ur Rehman. Nonlinear mixed type product $ [\mathscr{K}, \mathscr{F}]_\ast \odot \mathscr{D} $ on $ \ast $-algebras[J]. AIMS Mathematics, 2024, 9(8): 21596-21608. doi: 10.3934/math.20241049
Let $ \mathcal{A} $ be a unital $ \ast $-algebra containing a non-trivial projection. In this paper, we prove that if a map $ \Omega $ : $ \mathcal{A} $ $ \to $ $ \mathcal{A} $ such that
$ \begin{equation} \nonumber \Omega( [ \mathscr{K}, \mathscr{F}]_\ast \odot \mathscr{D}) = [\Omega(\mathscr{K}), \mathscr{F}]_\ast \odot \mathscr{D} + [ \mathscr{K}, \Omega (\mathscr{F})]_\ast \odot \mathscr{D} + [ \mathscr{K}, \mathscr{F}]_\ast \odot \Omega (\mathscr{D}), \end{equation} $
where $ [\mathscr{K}, \mathscr{F}]_{\ast} = \mathscr{K}\mathscr{F}- \mathscr{F}\mathscr{K}^\ast $ and $ \mathscr{K} \odot \mathscr{F} = \mathscr{K}^\ast \mathscr{F}+ \mathscr{F}\mathscr{K}^\ast $ for all $ \mathscr{K}, \mathscr{F}, \mathscr{D} \in \mathcal{A}, $ then $ \Omega $ is an additive $ \ast $-derivation. Furthermore, we extend its results on factor von Neumann algebras, standard operator algebras and prime $ \ast $-algebras. Additionally, we provide an example illustrating the existence of such maps.
| [1] | M. Ashraf, M. S. Akhter, M. A. Ansari, Nonlinear bi-skew Lie-type derivations on factor von Neumann algebras, Commun. Algebra, 50 (2022), 4766–4780. https://doi.org/10.1080/00927872.2022.2074027 |
| [2] |
D. Huo, B. Zheng, J. Xu, H. Liu, Nonlinear mappings preserving Jordan multiple $\ast$-product on factor von-neumann algebras, Linear Multilinear A., 63 (2015), 1026–1036. https://doi.org/10.1080/03081087.2014.915321 doi: 10.1080/03081087.2014.915321
|
| [3] |
L. Kong, J. Zhang, Nonlinear skew Lie derivations on prime $\ast$-rings, Indian J. Pure Appl. Math., 54 (2023), 475–484. https://doi.org/10.1007/s13226-022-00269-y doi: 10.1007/s13226-022-00269-y
|
| [4] |
A. N. Khan, Multiplicative biskew Lie triple derivations on factor von Neumann algebras, Rocky Mountain J. Math., 51 (2021), 2103–2114. https://doi.org/10.1216/rmj.2021.51.2103 doi: 10.1216/rmj.2021.51.2103
|
| [5] |
C. J. Li, F. Y. Lu, Nonlinear maps preserving the Jordan triple 1 $\ast$-product on von Neumann algebras, Complex Anal. Oper. Theory, 11 (2017), 109–117. https://doi.org/10.1007/s11785-016-0575-y doi: 10.1007/s11785-016-0575-y
|
| [6] |
C. J. Li, D. Zhang, Nonlinear mixed Jordan triple $\ast $-derivations on $\ast $-algebras, Sib. Math. J., 63 (2022), 735–742. https://doi.org/10.1134/S0037446622040140 doi: 10.1134/S0037446622040140
|
| [7] |
C. J. Li, F. F. Zhao, Q. Y. Chen, Nonlinear skew Lie triple derivations between factors, Acta Math. Sin. English Ser., 32 (2016), 821–830. https://doi.org/10.1007/s10114-016-5690-1 doi: 10.1007/s10114-016-5690-1
|
| [8] |
C. J. Li, Y. Zhao, F. Zhao, Nonlinear maps preserving the mixed product $[A \odot B, C]_{\ast}$ on von Neumann algebras, Filomat, 35 (2021), 2775–2781. https://doi.org/10.2298/FIL2108775L doi: 10.2298/FIL2108775L
|
| [9] |
C. J. Li, Q. Y. Chen, T. Wang, Nonlinear maps preserving the Jordan triple $\ast$-product on factor von Neumann algebras, Chin. Ann. Math. Ser. B, 39 (2018), 633–642. https://doi.org/10.1007/s11401-018-0086-4 doi: 10.1007/s11401-018-0086-4
|
| [10] |
C. J. Li, Y. Zhao, F. F. Zhao, Nonlinear $\ast$-Jordan-type derivations on $\ast$-algebras, Rocky Mountain J. Math., 51 (2021), 601–612. https://doi.org/10.1216/rmj.2021.51.601 doi: 10.1216/rmj.2021.51.601
|
| [11] |
Y. Pang, D. Zhang, D. Ma, The second nonlinear mixed Jordan triple derivable mapping on factor von Neumann algebras, Bull. Iran. Math. Soc., 48 (2022), 951–962. https://doi.org/10.1007/s41980-021-00555-1 doi: 10.1007/s41980-021-00555-1
|
| [12] |
N. Rehman, J. Nisar, M. Nazim, A note on nonlinear mixed Jordan triple derivation on $\ast$-algebras, Commun. Algebra, 51 (2023), 1334–1343. https://doi.org/10.1080/00927872.2022.2134410 doi: 10.1080/00927872.2022.2134410
|
| [13] | A. Taghavi, M. Nouri, M. Razeghi, V. Darvish, Non-linear $\lambda $-Jordan triple $\ast $-derivation on prime $\ast $-algebras, Rocky Mountain J. Math., 48 (2018), 2705–2716. https://doi.org/10.1216/RMJ-2018-48-8-2705 |
| [14] |
L. Y. Xian, Z. J. Hua, Nonlinear mixed Lie triple derivation on factor von neumann algebras, Acta Math. Sin. Chinese Ser., 62 (2019), 13–24. https://doi.org/10.12386/A2019sxxb0002 doi: 10.12386/A2019sxxb0002
|
| [15] |
F. Zhang, Nonlinear $\eta$-Jordan triple $\ast$-derivation on prime $\ast$-algebras, Rocky Mountain J. Math., 52 (2022), 323–333. https://doi.org/10.1216/rmj.2022.52.323 doi: 10.1216/rmj.2022.52.323
|
| [16] |
F. F. Zhao, C. J. Li, Nonlinear $\ast$-Jordan triple derivations on von Neumann algebras, Math. Slovaca, 68 (2018), 163–170. https://doi.org/10.1515/ms-2017-0089 doi: 10.1515/ms-2017-0089
|
Junaid Nisar, Turki Alsuraiheed, Nadeem ur Rehman. Nonlinear mixed type product $ [\mathscr{K}, \mathscr{F}]_\ast \odot \mathscr{D} $ on $ \ast $-algebras[J]. AIMS Mathematics, 2024, 9(8): 21596-21608. doi: 10.3934/math.20241049
DownLoad: