Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

CHARACTERIZATIONS OF JORDAN DERIVABLE MAPPINGS AT THE UNIT ELEMENT

  • Li, Jiankui (Department of Mathematics East China University of Science and Technology) ;
  • Li, Shan (Department of Mathematics East China University of Science and Technology) ;
  • Luo, Kaijia (Department of Mathematics East China University of Science and Technology)
  • Received : 2020.09.09
  • Accepted : 2021.11.05
  • Published : 2022.03.31
Download PDF
⟨ Previous Next ⟩

Abstract

Let 𝒜 be a unital Banach algebra, 𝓜 a unital 𝒜-bimodule, and 𝛿 a linear mapping from 𝒜 into 𝓜. We prove that if 𝛿 satisfies 𝛿(A)A-1+A-1𝛿(A)+A𝛿(A-1)+𝛿(A-1)A = 0 for every invertible element A in 𝒜, then 𝛿 is a Jordan derivation. Moreover, we show that 𝛿 is a Jordan derivable mapping at the unit element if and only if 𝛿 is a Jordan derivation. As an application, we answer the question posed in [4, Problem 2.6].

Keywords

Acknowledgement

We are grateful to the referee for his/her insightful comments and valuable suggestions. This research was partly supported by the National Natural Science Foundation of China (Grant No.11871021).

References

  1. T. J. Barton and Y. Friedman, Bounded derivations of JB*-triples, Quart. J. Math. Oxford Ser. (2) 41 (1990), no. 163, 255-268. https://doi.org/10.1093/qmath/41.3.255
  2. M. Burgos, F. J. Fernandez-Polo, J. J. Garces, and A. M. Peralta, Local triple derivations on C*-algebras, Comm. Algebra 42 (2014), no. 3, 1276-1286. https://doi.org/10.1080/00927872.2012.737191
  3. M. Burgos, F. J. Fernandez-Polo, and A. M. Peralta, Local triple derivations on C*-algebras and JB*-triples, Bull. Lond. Math. Soc. 46 (2014), no. 4, 709-724. https://doi.org/10.1112/blms/bdu024
  4. A. B. A. Essaleh and A. M. Peralta, Linear maps on C*-algebras which are derivations or triple derivations at a point, Linear Algebra Appl. 538 (2018), 1-21. https://doi.org/10.1016/j.laa.2017.10.009
  5. T. Ho, J. Martinez-Moreno, A. M. Peralta, and B. Russo, Derivations on real and complex JB *-triples, J. London Math. Soc. (2) 65 (2002), no. 1, 85-102. https://doi.org/10.1112/S002461070100271X
  6. B. E. Johnson, Symmetric amenability and the nonexistence of Lie and Jordan derivations, Math. Proc. Cambridge Philos. Soc. 120 (1996), no. 3, 455-473. https://doi.org/10.1017/S0305004100075010
  7. W. Kaup, A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces, Math. Z. 183 (1983), no. 4, 503-529. https://doi.org/10.1007/BF01173928
  8. J. Li and J. Zhou, Characterizations of Jordan derivations and Jordan homomorphisms, Linear Multilinear Algebra 59 (2011), no. 2, 193-204. https://doi.org/10.1080/03081080903304093
  9. C.-K. Liu, The structure of triple derivations on semisimple Banach *-algebras, Q. J. Math. 68 (2017), no. 3, 759-779. https://doi.org/10.1093/qmath/haw061
  10. A. M. Peralta and B. Russo, Automatic continuity of derivations on C*-algebras and JB*-triples, J. Algebra 399 (2014), 960-977. https://doi.org/10.1016/j.jalgebra.2013.10.017