Generalized Derivations on Unital and Non-unital <strong>C</strong><sup>★</sup>-Algebras

Dharmendra Kumar; Praveen Sharma; Pawan Kumar

doi:doi:10.11648/j.ajam.20261404.15

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Generalized Derivations on Unital and Non-unital C^★-Algebras

Dharmendra Kumar, Praveen Sharma, Pawan Kumar^*

Published in American Journal of Applied Mathematics (Volume 14, Issue 4)

Received: 26 May 2026 Accepted: 11 June 2026 Published: 23 July 2026

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Abstract

Derivations on C^★-algebras have been widely studied due to their fundamental role in operator algebras and their close connection with classical differentiation. Motivated by the notion of generalized derivations introduced for rings and subsequently investigated in various algebraic settings, this paper studies generalized derivations on C^★-algebras. For a unital C^★-algebra 𝒜, we introduce the notions of generalized derivations and generalized inner derivations and examine their structural properties. We obtain a characterization of generalized derivations on 𝒜 and show that every generalized derivation is necessarily a generalized inner derivation. As a consequence, generalized derivations on C^★-algebras are automatically bounded and therefore continuous. These results extend classical properties of derivations on C^★-algebras to the generalized setting and provide a deeper understanding of their behavior within the framework of operator algebras. We further investigate the algebraic structure of the class 𝒢𝒟 of all generalized derivations on 𝒜. In particular, we derive necessary and sufficient conditions under which the composition of two generalized derivations is again a generalized derivation. Using these conditions, we determine when 𝒢𝒟 is closed under composition and establish that it forms a unital subalgebra of L(A), the algebra of bounded linear operators on 𝒜. Finally, the obtained results are extended to non-unital C^★-algebras, thereby providing a unified treatment of generalized derivations in both unital and non-unital settings.

Published in	American Journal of Applied Mathematics (Volume 14, Issue 4)
DOI	10.11648/j.ajam.20261404.15
Page(s)	220-226
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2026. Published by Science Publishing Group

Keywords

Generalized Derivation, Lie Product, Multiplication Operator, Generalized Inner Derivation

References

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[2]	Bresar, M., On the distance of the composition of two derivations to the generalized derivations, Glasgow Mathematical Journal, 33 (1991), 89-93. https://doi.org/10.1017/S0017089500008077
[3]	Kaushik, K. L., Generalized Inner Derivations, International Journal of Scientific Development and Research, 5(8) (2020), 234-236.
[4]	Kaushik, K. L., Generalized Derivations in Non-Associative Algebra, in Algebra and Its Applications (Recent Developments), Narosa Publishers Pvt. Ltd., 2011.
[5]	Aydin, N., A note on generalized derivations of prime rings, International Journal of Algebra, 5(1-4) (2011), 17-23.
[6]	Posner, E. C., Derivations in prime rings, Proceedings of the American Mathematical Society, 8 (1957), 1093-1100. https://doi.org/10.1090/S0002-9939-1957-0095863-0
[7]	Kadison, R. V., Derivations of operator algebras, Annals of Mathematics, 83(2) (1966), 280-293. https://doi.org/10.2307/1970481
[8]	Sakai, S., On a conjecture of Kaplansky, Tohoku Mathematical Journal, Second Series, 12(1) (1960), 31-33. https://doi.org/10.2748/tmj/1178244586
[9]	Sakai, S., Derivations of W^*-algebras, Annals of Mathematics, 83(2) (1966), 273-279. https://doi.org/10.2307/1970480
[10]	Mamouni, A., Oukhtite, L., and Zerra, M., Some results on derivations and generalized derivations in rings, Mathematica, 65(88)(1) (2023), 94-104. https://doi.org/10.24193/mathcluj.2023.1.10
[11]	Hermas, A., Mamouni, A., and Oukhtite, L., Generalized derivations with power values on rings and Banach algebras, Mathematica Bohemica, 149(4) (2024), 491-502. https://doi.org/10.21136/MB.2024.0079-23
[12]	El Hamdaoui, M. and Boua, A., Algebraic identities and generalized derivations in prime rings, Mathematica, 66(89)(1) (2024), 105-113. https://doi.org/10.24193/mathcluj.2024.1.09
[13]	Mohammed, Z., Bouchannafa, K., and Oukhtite, L., On generalized derivations in factor rings, Georgian Mathematical Journal, 32(1) (2025). https://doi.org/10.1515/gmj-2024-2017
[14]	Hummdi, A. Y., Sogutcu, E. K., Golbasi, O., and Rehman, N. U., Some identities related to semiprime ideals of rings with multiplicative generalized derivations, Axioms, 13(10) (2024), Article 669. https://doi.org/10.3390/axioms13100669
[15]	Hummdi, A. Y., Al-Amery, Z. Z., and Al-Omary, R. M., Factor rings with algebraic identities via generalized derivations, Axioms, 14(1) (2025), Article 15. https://doi.org/10.3390/axioms14010015

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Kumar, D., Sharma, P., Kumar, P. (2026). Generalized Derivations on Unital and Non-unital C★-Algebras. American Journal of Applied Mathematics, 14(4), 220-226. https://doi.org/10.11648/j.ajam.20261404.15

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Kumar, D.; Sharma, P.; Kumar, P. Generalized Derivations on Unital and Non-unital C★-Algebras. Am. J. Appl. Math. 2026, 14(4), 220-226. doi: 10.11648/j.ajam.20261404.15

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AMA Style

Kumar D, Sharma P, Kumar P. Generalized Derivations on Unital and Non-unital C★-Algebras. Am J Appl Math. 2026;14(4):220-226. doi: 10.11648/j.ajam.20261404.15

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@article{10.11648/j.ajam.20261404.15,
  author = {Dharmendra Kumar and Praveen Sharma and Pawan Kumar},
  title = {Generalized Derivations on Unital and Non-unital C★-Algebras},
  journal = {American Journal of Applied Mathematics},
  volume = {14},
  number = {4},
  pages = {220-226},
  doi = {10.11648/j.ajam.20261404.15},
  url = {https://doi.org/10.11648/j.ajam.20261404.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20261404.15},
  abstract = {Derivations on C★-algebras have been widely studied due to their fundamental role in operator algebras and their close connection with classical differentiation. Motivated by the notion of generalized derivations introduced for rings and subsequently investigated in various algebraic settings, this paper studies generalized derivations on C★-algebras. For a unital C★-algebra 𝒜, we introduce the notions of generalized derivations and generalized inner derivations and examine their structural properties. We obtain a characterization of generalized derivations on 𝒜 and show that every generalized derivation is necessarily a generalized inner derivation. As a consequence, generalized derivations on C★-algebras are automatically bounded and therefore continuous. These results extend classical properties of derivations on C★-algebras to the generalized setting and provide a deeper understanding of their behavior within the framework of operator algebras. We further investigate the algebraic structure of the class 𝒢𝒟 of all generalized derivations on 𝒜. In particular, we derive necessary and sufficient conditions under which the composition of two generalized derivations is again a generalized derivation. Using these conditions, we determine when 𝒢𝒟 is closed under composition and establish that it forms a unital subalgebra of L(A), the algebra of bounded linear operators on 𝒜. Finally, the obtained results are extended to non-unital C★-algebras, thereby providing a unified treatment of generalized derivations in both unital and non-unital settings.},
 year = {2026}
}

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TY  - JOUR
T1  - Generalized Derivations on Unital and Non-unital C★-Algebras
AU  - Dharmendra Kumar
AU  - Praveen Sharma
AU  - Pawan Kumar
Y1  - 2026/07/23
PY  - 2026
N1  - https://doi.org/10.11648/j.ajam.20261404.15
DO  - 10.11648/j.ajam.20261404.15
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 220
EP  - 226
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20261404.15
AB  - Derivations on C★-algebras have been widely studied due to their fundamental role in operator algebras and their close connection with classical differentiation. Motivated by the notion of generalized derivations introduced for rings and subsequently investigated in various algebraic settings, this paper studies generalized derivations on C★-algebras. For a unital C★-algebra 𝒜, we introduce the notions of generalized derivations and generalized inner derivations and examine their structural properties. We obtain a characterization of generalized derivations on 𝒜 and show that every generalized derivation is necessarily a generalized inner derivation. As a consequence, generalized derivations on C★-algebras are automatically bounded and therefore continuous. These results extend classical properties of derivations on C★-algebras to the generalized setting and provide a deeper understanding of their behavior within the framework of operator algebras. We further investigate the algebraic structure of the class 𝒢𝒟 of all generalized derivations on 𝒜. In particular, we derive necessary and sufficient conditions under which the composition of two generalized derivations is again a generalized derivation. Using these conditions, we determine when 𝒢𝒟 is closed under composition and establish that it forms a unital subalgebra of L(A), the algebra of bounded linear operators on 𝒜. Finally, the obtained results are extended to non-unital C★-algebras, thereby providing a unified treatment of generalized derivations in both unital and non-unital settings.
VL  - 14
IS  - 4
ER  -

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Author Information

Dharmendra Kumar

Department Of Mathematics, Satyawati College (E) (University of Delhi), Delhi, India

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Praveen Sharma

Department of Mathematics, Shri Vishwakarma Skill University, Haryana, India

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Pawan Kumar

Department of Mathematics, Ram Lal Anand College (University of Delhi), New Delhi, India

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http://orcid.org/0000-0002-6642-1436

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APA Style

Kumar, D., Sharma, P., Kumar, P. (2026). Generalized Derivations on Unital and Non-unital C★-Algebras. American Journal of Applied Mathematics, 14(4), 220-226. https://doi.org/10.11648/j.ajam.20261404.15

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ACS Style

Kumar, D.; Sharma, P.; Kumar, P. Generalized Derivations on Unital and Non-unital C★-Algebras. Am. J. Appl. Math. 2026, 14(4), 220-226. doi: 10.11648/j.ajam.20261404.15

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AMA Style

Kumar D, Sharma P, Kumar P. Generalized Derivations on Unital and Non-unital C★-Algebras. Am J Appl Math. 2026;14(4):220-226. doi: 10.11648/j.ajam.20261404.15

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@article{10.11648/j.ajam.20261404.15,
  author = {Dharmendra Kumar and Praveen Sharma and Pawan Kumar},
  title = {Generalized Derivations on Unital and Non-unital C★-Algebras},
  journal = {American Journal of Applied Mathematics},
  volume = {14},
  number = {4},
  pages = {220-226},
  doi = {10.11648/j.ajam.20261404.15},
  url = {https://doi.org/10.11648/j.ajam.20261404.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20261404.15},
  abstract = {Derivations on C★-algebras have been widely studied due to their fundamental role in operator algebras and their close connection with classical differentiation. Motivated by the notion of generalized derivations introduced for rings and subsequently investigated in various algebraic settings, this paper studies generalized derivations on C★-algebras. For a unital C★-algebra 𝒜, we introduce the notions of generalized derivations and generalized inner derivations and examine their structural properties. We obtain a characterization of generalized derivations on 𝒜 and show that every generalized derivation is necessarily a generalized inner derivation. As a consequence, generalized derivations on C★-algebras are automatically bounded and therefore continuous. These results extend classical properties of derivations on C★-algebras to the generalized setting and provide a deeper understanding of their behavior within the framework of operator algebras. We further investigate the algebraic structure of the class 𝒢𝒟 of all generalized derivations on 𝒜. In particular, we derive necessary and sufficient conditions under which the composition of two generalized derivations is again a generalized derivation. Using these conditions, we determine when 𝒢𝒟 is closed under composition and establish that it forms a unital subalgebra of L(A), the algebra of bounded linear operators on 𝒜. Finally, the obtained results are extended to non-unital C★-algebras, thereby providing a unified treatment of generalized derivations in both unital and non-unital settings.},
 year = {2026}
}

Copy | Download

TY  - JOUR
T1  - Generalized Derivations on Unital and Non-unital C★-Algebras
AU  - Dharmendra Kumar
AU  - Praveen Sharma
AU  - Pawan Kumar
Y1  - 2026/07/23
PY  - 2026
N1  - https://doi.org/10.11648/j.ajam.20261404.15
DO  - 10.11648/j.ajam.20261404.15
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 220
EP  - 226
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20261404.15
AB  - Derivations on C★-algebras have been widely studied due to their fundamental role in operator algebras and their close connection with classical differentiation. Motivated by the notion of generalized derivations introduced for rings and subsequently investigated in various algebraic settings, this paper studies generalized derivations on C★-algebras. For a unital C★-algebra 𝒜, we introduce the notions of generalized derivations and generalized inner derivations and examine their structural properties. We obtain a characterization of generalized derivations on 𝒜 and show that every generalized derivation is necessarily a generalized inner derivation. As a consequence, generalized derivations on C★-algebras are automatically bounded and therefore continuous. These results extend classical properties of derivations on C★-algebras to the generalized setting and provide a deeper understanding of their behavior within the framework of operator algebras. We further investigate the algebraic structure of the class 𝒢𝒟 of all generalized derivations on 𝒜. In particular, we derive necessary and sufficient conditions under which the composition of two generalized derivations is again a generalized derivation. Using these conditions, we determine when 𝒢𝒟 is closed under composition and establish that it forms a unital subalgebra of L(A), the algebra of bounded linear operators on 𝒜. Finally, the obtained results are extended to non-unital C★-algebras, thereby providing a unified treatment of generalized derivations in both unital and non-unital settings.
VL  - 14
IS  - 4
ER  -

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