Nonlinear mixed skew bi-skew Jordan $n$-derivations on $\ast$-algebras

Abstract

Let $\mathcal{A}$ be a unital $\ast$-algebra containing nontrivial projections $P_j~~~~(j=1,2)$. In this article, we examine the nature of nonlinear mixed skew bi-skew Jordan $n$-derivations on $\mathcal{A}$. Further, we extend our main results to some special classes of $\ast$-algebras such as prime $\ast$-algebras, factor von Neumann algebras and standard operator algebras and prove that on these $\ast$-algebras every nonlinear mixed skew bi-skew Jordan $n$-derivation is an additive $\ast$-derivation.

Keywords

• Mixed skew bi-skew Jordan $n$-product
• $\ast$-derivation
• $\ast$-algebra

References

1. A. Ali, A. S. Alali and M. Tasleem, Characterization of nonlinear bi-skew Jordan $n$-derivations on prime $\ast$-algebras, Axioms, 12(8) (2023), 753, 1-13.
2. A. Ali, M. Tasleem and A. N. Khan, Non-linear mixed Jordan bi-skew Lie triple derivations on $\ast$-algebras, Filomat, 38(6) (2024), 2079-2090.
3. D. Benkovic and N. Sirovnik, Jordan derivations of unital algebras with idempotents, Linear Algebra Appl., 437(9) (2012), 2271-2284.
4. L. Dai and F. Lu, Nonlinear maps preserving Jordan $\ast$-products, J. Math. Anal. Appl., 409(1) (2014), 180-188.
5. V. Darvish, M. Nouri and M. Razeghi, Nonlinear triple product $A^*B+B^*A$ for derivations on $\ast$-algebras, Math. Notes, 108 (2020), 179-187.
6. V. Darvish, M. Nouri, M. Razeghi and A. Taghavi, Nonlinear $\ast$-Jordan triple derivation on prime $\ast$-algebras, Rocky Mountain J. Math., 50(2) (2020), 543-549.
7. V. Darvish, M. Nouri and M. Razeghi, Non-linear bi-skew Jordan derivations on $\ast$-algebra, Filomat, 36(10) (2022), 3231-3239.
8. B. L. M. Ferreira, H. Guzzo and R. N. Ferreira, An approach between the multiplicative and additive structure of a Jordan ring, Bull. Iranian Math. Soc., 47(4) (2021), 961-975.

9. B. L. M. Ferreira and F. Wei, Mixed $\ast$-Jordan-type derivations on $\ast$-algebras, J. Algebra Appl., 22(5) (2023), 2350100 (14 pp).
10. A. N. Khan and H. Alhazmi, Multiplicative bi-skew Jordan triple derivations on prime $\ast$-algebra, Georgian Math. J., 30(3) (2023), 389-396.
11. J. Li and F. Lu, Additive Jordan derivations of reflexive algebras, J. Math. Anal. Appl., 329(1) (2007), 102-111.
12. C. Li, F. Lu and X. Fang, Nonlinear mappings preserving product $XY+ YX^*$ on factor von Neumann algebras, Linear Algebra Appl., 438(5) (2013), 2339-2345.
13. C. Li and F. Lu, Nonlinear maps preserving the Jordan triple 1-$\ast$-product on von Neumann Algebras, Complex Anal. Oper. Theory, 11 (2017), 109-117.
14. C. Li, F. Zhao and Q. Chen, Nonlinear maps preserving product $X^*Y+Y^*X$ on von Neumann algebras, Bull. Iranian Math. Soc., 44(3) (2018), 729-738.
15. C. Li, Y. Zhao and F. Zhao, Nonlinear $\ast$-Jordan-type derivations on $\ast$-algebras, Rocky Mountain J. Math., 51(2) (2021), 601-612.
16. C. Li and D. Zhang, Nonlinear mixed Jordan triple $\ast$-derivations on factor von Neumann algebras, Filomat, 36(8) (2022), 2637-2644.
17. C. Li and D. Zhang, Nonlinear mixed Jordan triple $\ast$-derivations on $\ast$-algebras, Sib. Math. J., 63(4) (2022), 735-742.
18. F. Lu, Jordan derivable maps of prime rings, Comm. Algebra, 38(12) (2010), 4430-4440.
19. Y. Pang, D. Zhang and D. Ma, The second nonlinear mixed Jordan triple derivable mapping on factor von Neumann algebras, Bull. Iranian Math. Soc., 48(3) (2022), 951-962.
20. X. Qi, Z. Guo and T. Zhang, Characterizing Jordan n-derivations of unital rings containing idempotents, Bull. Iranian Math. Soc., 46(6) (2020), 1639-1658.
21. A. Taghavi and S. Gholampoor, Maps preserving product $A^*B+B^*A$ on $C^*$-algebras, Bull. Iranian Math. Soc., 48 (2022), 757-767.
22. A. Taghavi, H. Rohi and V. Darvish, Nonlinear $\ast$-Jordan derivations on von Neumann algebras, Linear Multilinear Algebra, 64(3) (2016), 426-439.
23. J. Wang, C. Li, Y. Liang and L. Chen, The nonlinear mixed bi-skew lie triple derivations on $\ast$-algebras, Filomat, 37(29) (2023), 9981-9989.
24. D. Zhang, C. Li and Y. Zhao, Nonlinear maps preserving bi-skew Jordan triple product on factor von Neumann algebras, Period. Math. Hungar., 86(2) (2023), 578-586.
25. F. Zhao and C. Li, Nonlinear maps preserving the Jordan triple $\ast$-product between factors, Indag. Math. (N.S.), 29(2) (2018), 619-627.
26. F. Zhao, D. Zhang and C. Li, Nonlinear bi-skew Jordan-type derivations on $\ast$-algebras, Filomat, 37(13) (2023), 4211-4219.