A note on nonlinear mixed type product $[E\diamond K,D]_\ast$ on $\ast$-algebras

Year 2025, Volume: 54 Issue: 1, 16 - 25, 28.02.2025

Mohammad Aslam Siddeeque , Raof Ahmad Bhat , Mohammad Shane Alam

https://doi.org/10.15672/hujms.1294965

Cited By: 1

https://izlik.org/JA88TT77LM

Abstract

Let $ \mathfrak{B}$ be a $\ast$-algebra with the unity and a nontrivial projection. In the present paper, we show under certain restrictions that if a map $\Psi$ : $ \mathfrak{B}$ $\to$ $ \mathfrak{B}$ satisfies $ \Psi ( [E \diamond K, D]_\ast ) = [\Psi( E) \diamond K, D]_\ast + [E \diamond \Psi (K), D]_\ast + [E \diamond K, \Psi (D)]_\ast$ for all $ E, K, D \in \mathfrak{B} $, then $\Psi$ is an additive $\ast$-derivation.

Keywords

*-algebra , derivation , mixed *-derivation , additivity

References

• [1] L. Dai and F. Lu, Nonlinear maps preserving Jordan $\ast$-products, J. Math. Anal. Appl. 409 (1), 180-188.
• [2] V. Darvish, H. M. Nazari, H. Rohi and A. Taghavi, Maps preserving $\eta$-product $E^\ast K+ \eta KE^\ast$ on $\mathbb{C}^\ast$-algebras, J. Korean Math. Soc. 54 (3), 867-876, 2017.
• [3] V. Darvish, M. Nouri and M. Razeghi, Nonlinear triple product $ E^{*} K+ K^{*} E $ for derivations on $\ast$-algebras, Math. Notes 108 (1), 179-87, 2020.
• [4] C. J. Li, F. F. Zhao and Q. Y. Chen, Nonlinear skew Lie triple derivations between factors, Acta Math. Sinica (Engl. Ser.) 32 (7), 821830, 2016.
• [5] C. J. Li, Y. Zhao and F. F. Zhao, Nonlinear $\ast$-Jordan-type derivations on -algebras, Rocky Mountain J. Math. 51 (2), 601-612, 2021.
• [6] C. J. Li and D. Zhang, Nonlinear mixed Jordan triple $\ast$-derivations on $\ast$-algebras, Sib. Math. J. 63 (4), 735-742, 2022.
• [7] C. Li, F. Lu and X. Fang, Nonlinear $\xi$-Jordan $\ast$-derivation on von Neumann algebras, Linear Multilinear Algebra 64 (4), 466-472, 2014.
• [8] N. U. Rehman, J. Nisar and M. Nazim, A note on nonlinear mixed Jordan triple derivation on $\ast$-algebras, Comm. Algebra 1-10, 2022.
• [9] A. Taghavi, M. Nouri, M. Razeghi and V. Darvish, Nonlinear $\lambda$-Jordan triple $\ast$- derivation on prime $\ast$-algebras, Rocky Mountain J. Math. 48 (8), 2705-2716, 2018.
• [10] A. Taghavi, H. Rohi and V. Darvish, Nonlinear $\ast$-Jordan derivation on von Neumann algebras, Linear Multilinear Algebra 64, 426439, 2016.
• [11] Z. J. Yang and J. H. Zhang, Nonlinear maps preserving mixed Lie triple products on factor von Neumann algebras, Ann. Funct. Anal. 10 (3), 325-336, 2019.
• [12] L. Yaoxian and Z. Jianhua, Nonlinear mixed Lie triple derivation on factor von Neumann algebras, Acta Math. Sinica (Chin. Ser.) 62 (1), 13-24, 2019.
• [13] F. J. Zhang, Nonlinear skew Jordan derivable maps on factor von Neumann algebras, Linear Multilinear Algebra, 64 (10), 2090-2103, 2016.
• [14] F. F. Zhao and C. J. Li, Nonlinear $\ast$-Jordan triple derivations on von Neumann algebras, Math. Slovaca 68 (1), 163-170, 2018.
• [15] Y. Zhou, Z. J. Yang and J. H. Zhang, Nonlinear mixed Lie triple derivations on prime $\ast$-algebras, Comm. Algebra 47 (1), 4791–4796, 2019.