Characterizations of Jordan *-derivations on Banach *-algebras

Guangyu An; Ying Yao

doi:doi:10.11648/j.pamj.20200905.13

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Characterizations of Jordan -derivations on Banach -algebras

Guangyu An, Ying Yao

Published in Pure and Applied Mathematics Journal (Volume 9, Issue 5)

Received: 11 August 2020 Accepted: 18 September 2020 Published: 28 October 2020

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Abstract

Suppose that is a real or complex unital Banach *-algebra, is a unital Banach -bimodule, and G ∈ is a left separating point of . In this paper, we investigate whether the additive mapping δ: → satisfies the condition A,B ∈ , AB = G ⇒ Aδ(B)+δ(A)B^*= δ(G) characterize Jordan ^*-derivations. Initially, we prove that if is a real unital C^*-algebra and G = I is the unit element in , then δ (non-necessarily continuous) is a Jordan ^*-derivation. In addition, we prove that if is a real unital C^*-algebra and δ is continuous, then δ is a Jordan ^*-derivation. Finally, we show that if is a complex factor von Neumann algebra and δ is linear, then δ (non-necessarily continuous) is equal to zero.

Published in	Pure and Applied Mathematics Journal (Volume 9, Issue 5)
DOI	10.11648/j.pamj.20200905.13
Page(s)	96-100
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), 2020. Published by Science Publishing Group

Keywords

Jordan *-derivation, Left Separating Point, C^*-algebra, Factor

References

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[3]	Q. Chen, C. Li, X. Fang. Jordan (α,β)-derivations on CSL subalgebras of von Neumann algebras, Acta. Math. Sin., 60 (2017), 537-546.
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[5]	J. He, J. Li, D. Zhao, Derivations, Local and 2-local derivations on some algebras of operators on Hilbert Csup>*-modules, Mediterr. J. Math., 14, 230 (2017).
[6]	S. Kurepa. Quadratic and sesquilinear functionals, Glasnik Mat. Fiz.-Astronom, 20 (1965), 79-92. 106 Guangyu An and Ying Yao: Characterizations of Jordan ∗-derivations on Banach ∗-algebras
[7]	L. Liu. 2-Local Lie derivations of nest subalgebras of factors, Linear and Multilinear algebra, 67 (2019), 448-455.
[8]	D. Liu, J. Zhang. Local Lie derivations on certain operator algebras, Ann. Funct. Anal., 8 (2017), 270-280.
[9]	X. Qi and X. Zhang. Characterizations of Jordan ∗- derivations by local action on rings with involution, Journal of Hyperstructures, 6 (2017), 120-127.
[10]	P. ˇSemrl. On Jordan ∗-derivations and an application,Colloq. Math., 59 (1990), 241-251.
[11]	P.ˇSemrl. Jordan ∗-derivations of standard operator algebras, Proc. Amer. Math. Soc., 120 (1994), 515-518.
[12]	P.ˇSemrl. On quadratic functionals, Bull. Austral. Math. Soc., 37 (1988), 27-28.
[13]	P. ˇSemrl. Quadratic functionals and Jordan ∗-derivations, Studia Math., 97 (1991), 157-165.
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[15]	J. Vukman, Some functional equations in Banach algebras and an application, Proc. Amer. Math. Soc., 100 (1987), 133-136.

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Guangyu An, Ying Yao. (2020). Characterizations of Jordan *-derivations on Banach *-algebras. Pure and Applied Mathematics Journal, 9(5), 96-100. https://doi.org/10.11648/j.pamj.20200905.13

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Guangyu An; Ying Yao. Characterizations of Jordan *-derivations on Banach *-algebras. Pure Appl. Math. J. 2020, 9(5), 96-100. doi: 10.11648/j.pamj.20200905.13

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Guangyu An, Ying Yao. Characterizations of Jordan *-derivations on Banach *-algebras. Pure Appl Math J. 2020;9(5):96-100. doi: 10.11648/j.pamj.20200905.13

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@article{10.11648/j.pamj.20200905.13,
  author = {Guangyu An and Ying Yao},
  title = {Characterizations of Jordan *-derivations on Banach *-algebras},
  journal = {Pure and Applied Mathematics Journal},
  volume = {9},
  number = {5},
  pages = {96-100},
  doi = {10.11648/j.pamj.20200905.13},
  url = {https://doi.org/10.11648/j.pamj.20200905.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20200905.13},
  abstract = {Suppose that  is a real or complex unital Banach *-algebra,  is a unital Banach -bimodule, and G ∈  is a left separating point of . In this paper, we investigate whether the additive mapping δ:  → satisfies the condition A,B ∈ , AB = G ⇒ Aδ(B)+δ(A)B*= δ(G) characterize Jordan *-derivations. Initially, we prove that if  is a real unital C*-algebra and G = I is the unit element in , then δ (non-necessarily continuous) is a Jordan *-derivation. In addition, we prove that if  is a real unital C*-algebra and δ is continuous, then δ is a Jordan *-derivation. Finally, we show that if  is a complex factor von Neumann algebra and δ is linear, then δ (non-necessarily continuous) is equal to zero.},
 year = {2020}
}

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TY  - JOUR
T1  - Characterizations of Jordan *-derivations on Banach *-algebras
AU  - Guangyu An
AU  - Ying Yao
Y1  - 2020/10/28
PY  - 2020
N1  - https://doi.org/10.11648/j.pamj.20200905.13
DO  - 10.11648/j.pamj.20200905.13
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 96
EP  - 100
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20200905.13
AB  - Suppose that  is a real or complex unital Banach *-algebra,  is a unital Banach -bimodule, and G ∈  is a left separating point of . In this paper, we investigate whether the additive mapping δ:  → satisfies the condition A,B ∈ , AB = G ⇒ Aδ(B)+δ(A)B*= δ(G) characterize Jordan *-derivations. Initially, we prove that if  is a real unital C*-algebra and G = I is the unit element in , then δ (non-necessarily continuous) is a Jordan *-derivation. In addition, we prove that if  is a real unital C*-algebra and δ is continuous, then δ is a Jordan *-derivation. Finally, we show that if  is a complex factor von Neumann algebra and δ is linear, then δ (non-necessarily continuous) is equal to zero.
VL  - 9
IS  - 5
ER  -

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Author Information

Guangyu An

Department of Mathematics, Shaanxi University of Science and Technology, Xi’an, China
Ying Yao

Department of Mathematics, Shaanxi University of Science and Technology, Xi’an, China

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APA Style

Guangyu An, Ying Yao. (2020). Characterizations of Jordan *-derivations on Banach *-algebras. Pure and Applied Mathematics Journal, 9(5), 96-100. https://doi.org/10.11648/j.pamj.20200905.13

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ACS Style

Guangyu An; Ying Yao. Characterizations of Jordan *-derivations on Banach *-algebras. Pure Appl. Math. J. 2020, 9(5), 96-100. doi: 10.11648/j.pamj.20200905.13

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AMA Style

Guangyu An, Ying Yao. Characterizations of Jordan *-derivations on Banach *-algebras. Pure Appl Math J. 2020;9(5):96-100. doi: 10.11648/j.pamj.20200905.13

Copy | Download

@article{10.11648/j.pamj.20200905.13,
  author = {Guangyu An and Ying Yao},
  title = {Characterizations of Jordan *-derivations on Banach *-algebras},
  journal = {Pure and Applied Mathematics Journal},
  volume = {9},
  number = {5},
  pages = {96-100},
  doi = {10.11648/j.pamj.20200905.13},
  url = {https://doi.org/10.11648/j.pamj.20200905.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20200905.13},
  abstract = {Suppose that  is a real or complex unital Banach *-algebra,  is a unital Banach -bimodule, and G ∈  is a left separating point of . In this paper, we investigate whether the additive mapping δ:  → satisfies the condition A,B ∈ , AB = G ⇒ Aδ(B)+δ(A)B*= δ(G) characterize Jordan *-derivations. Initially, we prove that if  is a real unital C*-algebra and G = I is the unit element in , then δ (non-necessarily continuous) is a Jordan *-derivation. In addition, we prove that if  is a real unital C*-algebra and δ is continuous, then δ is a Jordan *-derivation. Finally, we show that if  is a complex factor von Neumann algebra and δ is linear, then δ (non-necessarily continuous) is equal to zero.},
 year = {2020}
}

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TY  - JOUR
T1  - Characterizations of Jordan *-derivations on Banach *-algebras
AU  - Guangyu An
AU  - Ying Yao
Y1  - 2020/10/28
PY  - 2020
N1  - https://doi.org/10.11648/j.pamj.20200905.13
DO  - 10.11648/j.pamj.20200905.13
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 96
EP  - 100
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20200905.13
AB  - Suppose that  is a real or complex unital Banach *-algebra,  is a unital Banach -bimodule, and G ∈  is a left separating point of . In this paper, we investigate whether the additive mapping δ:  → satisfies the condition A,B ∈ , AB = G ⇒ Aδ(B)+δ(A)B*= δ(G) characterize Jordan *-derivations. Initially, we prove that if  is a real unital C*-algebra and G = I is the unit element in , then δ (non-necessarily continuous) is a Jordan *-derivation. In addition, we prove that if  is a real unital C*-algebra and δ is continuous, then δ is a Jordan *-derivation. Finally, we show that if  is a complex factor von Neumann algebra and δ is linear, then δ (non-necessarily continuous) is equal to zero.
VL  - 9
IS  - 5
ER  -

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