Continuity of Generalized Jordan Derivations on Semisimple Banach Algebras

  • Mohammad Gholampour
  • Shirin Hejazian


A generalized Jordan derivation D on a Banach algebra \({\mathcal {A}}\) is a linear mapping \(D:{\mathcal {A}}\rightarrow {\mathcal {A}}\) for which there exists a positive \(\varepsilon \) satisfying
$$\begin{aligned}\Vert D(a\circ b)-a\circ D(b)-D(a)\circ b\Vert \le \varepsilon \Vert a\Vert \Vert b\Vert \;\;(a, b \in {\mathcal {A}}),\end{aligned}$$
where \(\circ \) denotes the Jordan product on \({\mathcal {A}}\). We study generalized Jordan derivations on Banach algebras and prove that every generalized Jordan derivation on a semisimple Banach algebra is automatically continuous.


Generalized homomorphism Jordan derivation Generalized Jordan derivation Automatic continuity 

Mathematics Subject Classification

46H40 47B47 



The authors thank the referees for their careful reading of the article and suggesting valuable comments that improved the quality of this work.


  1. 1.
    Alaminos, J., Extremera, J., Villena, A.R.: Approximately spectrum-preserving maps. J. Funct. Anal. 261, 233–266 (2011)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Aupetit, B.: The uniqueness of the complete norm topology in Banach algebras and Banach Jordan algebras. J. Funct. Anal. 47, 1–6 (1982)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Brešar, M.: Jordan derivations revisited. Math. Proc. Camb. Philos. Soc. 139, 411–425 (2005)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Cabrera, M., Rodríguez, Á.: Non-associative Normed Algebras, Volume 1, the Vidav–Palmer and Gelfand–Naimark Theorems, Encyclopedia of Mathematics and Its Applications, vol. 154. Cambridge University Press, Cambridge (2014)MATHGoogle Scholar
  5. 5.
    Cusack, J.M.: Jordan derivations on rings. Proc. Am. Math. Soc. 53, 321–324 (1975)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Dales, H.G.: Banach Algebras and Automatic Continuity, London Mathematical Society Monographs, vol. 24. Oxford University Press, New York (2000)Google Scholar
  7. 7.
    Garcés, J.J., Peralta, A.M.: Generalized triple homomorphisms and derivations. Can. J. Math. 65, 783–807 (2013)CrossRefMATHGoogle Scholar
  8. 8.
    Gholampour, M., Hejazian, S.: Continuity of generalized derivations on \(\textit{JB}^*\)-algebras. Quaest. Math. 41, 227–238 (2018)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Hejazian, S., Niknam, A.: Modules, annihilators and module derivations of \(\textit{JB}^*\)-algebras. Indian J. Pure Appl. Math. 27, 129–140 (1996)MathSciNetMATHGoogle Scholar
  10. 10.
    Herstein, I.N.: Topics in Ring Theory. University of Chicago Press, Chicago (1969)MATHGoogle Scholar
  11. 11.
    Jacobson, N.: Structure and Representation of Jordan Algebras, vol. 39. American Mathematical Society Colloquium Publications, Providence (1968)MATHGoogle Scholar
  12. 12.
    Jarosz, K.: Perturbations of Banach Algebras, Lecture Notes in Mathematics, vol. 1120. Springer, Berlin (1985)CrossRefMATHGoogle Scholar
  13. 13.
    Johnson, B.E.: The uniqueness of the (complete) norm topology. Bull. Am. Math. Soc. 73, 537–539 (1967)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Johnson, B.E.: Continuity of generalized homomorphisms. Bull. Lond. Math. Soc. 19, 67–71 (1987)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Johnson, B.E.: Approximately multiplicative maps between Banach algebras. J. Lond. Math. Soc. 37, 294–316 (1988)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Johnson, B.E., Sinclair, A.M.: Continuity of derivations and a problem of Kaplansky. Am. J. Math. 90, 1067–1073 (1968)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Peralta, A.M., Russo, B.: Automatic continuity of derivations on \(C^*\)-algebras and \(\textit{JB}^*\)-triples. J. Algebra 399, 960–977 (2014)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Rodríguez-Palacios, Á.: The uniqueness of the complete norm topology in complete normed nonassociative algebras. J. Funct. Anal. 60, 1–15 (1985)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Sinclair, A.M.: Jordan homomorphisms and derivations on semisimple Banach algebras. Proc. Am. Math. Soc. 24, 209–214 (1970)MathSciNetMATHGoogle Scholar
  20. 20.
    Villena, A.R.: Derivations on Jordan–Banach algebras. Studia Math. 118, 205–229 (1996)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2018

Authors and Affiliations

  1. 1.Department of Pure MathematicsFerdowsi University of MashhadMashhadIran

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