Nonlinear ξ-skew-Jordan triple higher derivations on prime *-algebras

i

This article has been updated. A Correction to this article was published on .

i

The Original Article was published on

i

The Original Article was published on

i

This article has been updated

i

This article was retracted on

i

This article has been updated

Abstract

Let B denote a unital prime *-algebra over the complex field $\mathbb{C}$, and let $\xi \in \mathbb{C} \backslash\{0, \pm 1\}$. This paper characterized a family $\Psi=\left\{\varphi_{m}\right\}_{m \in \mathbb{N}}$ of maps (not necessarily additive) from B into itself that satisfy the functional identity

$\varphi_{m}\left(A \diamond_{\xi} B \diamond_{\xi} C\right)=\sum_{i+j+k=m} \varphi_{i}(A) \diamond_{\xi} \varphi_{j}(B) \diamond_{\xi} \varphi_{k}(C)$

for all $A, B, C \in \mathfrak{B}$, where $A \diamond_{\xi} B=A B+\xi B A^{*}$. It was proved that Ψ satisfies the above identity if and only if Ψ is an additive higher *-derivation and φ_m(ξA) = ξφ_m(A) holds for every A ∈ $\mathcal{B}$. As an application, the result not only generalizes the structure of nonlinear ξ-skew-Jordan triple derivations on prime *-algebras, but also yields a description of nonlinear ξ-skew-Jordan triple higher derivations on several important operator algebras, including standard operator algebras and factor von Neumann algebras.

Keywords

nonlinear ξ-skew-Jordan triple higher derivation / additive higher *-derivation / unital *-algebras / standard operator algebras / von Neumann algebras

CHANGE HISTORY

A Correction to this paper has been published:
https://doi.org

The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
https://doi.org

References

[1] D. E. N. Kawaia, H. Guzzo Jr., B. L. M. Ferreira, Mappings preserving sum of products {a,b}_*+b^*a or preserving {a,b}_*+a^*b on generalized matrix *-rings , Filomat, 39 (2025), 469-482. https://doi.org/10.2298/FIL2502469K

[2] V. De Filippis, O. M. Di Vincenzo, Generalized skew derivations on semiprime rings, Linear Multilinear Algebra, 63 (2015), 927-939. https://doi.org/10.1080/03081087.2014.909813

[3] A. S. Alali, M. A. Madni, M. R. Mozumder, Nonlinear mixed Jordan-Type derivations on *-algebra, Axioms, 14 (2025), 874. https://doi.org/10.3390/axioms14120874

[4] M. Ashraf, M. S. Akhter, M. A. Ansari, Nonlinear generalized bi-skew Jordan-type derivations on *-algebras, Bull. Malays. Math. Sci. Soc., 47 (2024), 18. https://doi.org/10.1007/s40840-023-01611-1

[5] T. C. Pierin, R. N. Ferreira, F. Borges, B. L. M. Ferreira, Nonlinear mixed *-Jordan type derivations on alternative *-algebras, J. Algebra Appl., 24 (2025), 2550028. https://doi.org/10.1142/S0219498825500288

[6] B. L. M. Ferreira, B. T. Costa, *-Jordan-type maps on C^*-algebras , Commun. Algebra., 49 (2021), 5073-5082. https://doi.org/10.1080/00927872.2021.1937636

[7] I. N. Herstein, Lie and Jordan structures in simple, associative rings, Bull. Am. Math. Soc., 67 (1961), 517-531. https://doi.org/10.1090/S0002-9904-1961-10666-6

[8] A. Taghavi, M. Nouri, M. Razeghi, V. Darvish, A note on non-linear *-Jordan derivations on *-algebras, Math. Slovaca., 69 (2019), 639-646. https://doi.org/10.1515/ms-2017-0253

[9] F. Zhang, Nonlinear ξ-Jordan triple *-derivation on prime *-algebras, Rocky Mountain J. Math., 52 (2022), 323-333. https://doi.org/10.1216/rmj.2022.52.323

[10] D. Huo, B. Zheng, H. Liu, Nonlinear maps preserving Jordan triple η-*-products, J. Math. Anal. Appl., 430 (2015), 830-844. https://doi.org/10.1016/j.jmaa.2015.05.021

[11] A. Taghavi, M. Nouri, M. Razeghi, V. Darvish, Non-linear λ-Jordan triple *-derivations on prime *-algebras, Rocky Mountain J. Math, 48 (2018), 2705-2716. https://doi.org/10.1216/RMJ-2018-48-8-2705

[12] X. Liang, H. Guo, L. Zhao, Nonlinear bi-skew Jordan-type higher derivations on *-algebras, Filomat, 38 (2024), 6087-6098. https://doi.org/10.2298/FIL2417087L

[13] B. A. Wani, M. Ashraf, W. Lin, Multiplicative *-Jordan type higher derivations on von Neumann algebras, Quaest. Math., 43 (2020), 1689-1711. https://doi.org/10.2989/16073606.2019.1649734