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

Mohammad Aslam Siddeeque; Raof Ahmad Bhat; Mohammad Shane Alam

doi:10.15672/hujms.1294965

Araştırma Makalesi

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

Yıl 2025, Cilt: 54 Sayı: 1, 16 - 25, 28.02.2025

Mohammad Aslam Siddeeque , Raof Ahmad Bhat , Mohammad Shane Alam

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

Öz

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.

Anahtar Kelimeler

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

Kaynakça

[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.

Yıl 2025, Cilt: 54 Sayı: 1, 16 - 25, 28.02.2025

Mohammad Aslam Siddeeque , Raof Ahmad Bhat , Mohammad Shane Alam

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

Öz

Kaynakça

[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.

Toplam 15 adet kaynakça vardır.

Ayrıntılar

Birincil Dil	İngilizce
Konular	Matematik
Bölüm	Matematik
Yazarlar	Mohammad Aslam Siddeeque 0000-0002-2275-8328 Raof Ahmad Bhat 0009-0009-4601-0937 Mohammad Shane Alam 0000-0002-8605-2674
Erken Görünüm Tarihi	14 Nisan 2024
Yayımlanma Tarihi	28 Şubat 2025
Yayımlandığı Sayı	Yıl 2025 Cilt: 54 Sayı: 1

Kaynak Göster

APA	Siddeeque, M. A., Bhat, R. A., & Alam, M. S. (2025). A note on nonlinear mixed type product $[E\diamond K,D]_\ast$ on $\ast$-algebras. Hacettepe Journal of Mathematics and Statistics, 54(1), 16-25. https://doi.org/10.15672/hujms.1294965
AMA	Siddeeque MA, Bhat RA, Alam MS. A note on nonlinear mixed type product $[E\diamond K,D]_\ast$ on $\ast$-algebras. Hacettepe Journal of Mathematics and Statistics. Şubat 2025;54(1):16-25. doi:10.15672/hujms.1294965
Chicago	Siddeeque, Mohammad Aslam, Raof Ahmad Bhat, ve Mohammad Shane Alam. “A Note on Nonlinear Mixed Type Product $[E\diamond K,D]_\ast$ on $\ast$-Algebras”. Hacettepe Journal of Mathematics and Statistics 54, sy. 1 (Şubat 2025): 16-25. https://doi.org/10.15672/hujms.1294965.
EndNote	Siddeeque MA, Bhat RA, Alam MS (01 Şubat 2025) A note on nonlinear mixed type product $[E\diamond K,D]_\ast$ on $\ast$-algebras. Hacettepe Journal of Mathematics and Statistics 54 1 16–25.
IEEE	M. A. Siddeeque, R. A. Bhat, ve M. S. Alam, “A note on nonlinear mixed type product $[E\diamond K,D]_\ast$ on $\ast$-algebras”, Hacettepe Journal of Mathematics and Statistics, c. 54, sy. 1, ss. 16–25, 2025, doi: 10.15672/hujms.1294965.
ISNAD	Siddeeque, Mohammad Aslam vd. “A Note on Nonlinear Mixed Type Product $[E\diamond K,D]_\ast$ on $\ast$-Algebras”. Hacettepe Journal of Mathematics and Statistics 54/1 (Şubat 2025), 16-25. https://doi.org/10.15672/hujms.1294965.
JAMA	Siddeeque MA, Bhat RA, Alam MS. A note on nonlinear mixed type product $[E\diamond K,D]_\ast$ on $\ast$-algebras. Hacettepe Journal of Mathematics and Statistics. 2025;54:16–25.
MLA	Siddeeque, Mohammad Aslam vd. “A Note on Nonlinear Mixed Type Product $[E\diamond K,D]_\ast$ on $\ast$-Algebras”. Hacettepe Journal of Mathematics and Statistics, c. 54, sy. 1, 2025, ss. 16-25, doi:10.15672/hujms.1294965.
Vancouver	Siddeeque MA, Bhat RA, Alam MS. A note on nonlinear mixed type product $[E\diamond K,D]_\ast$ on $\ast$-algebras. Hacettepe Journal of Mathematics and Statistics. 2025;54(1):16-25.