Relations Between n-Jordan Homomorphisms and n-Homomorphisms

Abstract

For \(n\ge 2\), an additive map f between two rings A and B is called an n-Jordan homomorphism, or an n-homomorphism if \(f(a^n)=f(a)^n\), for all \(a\in A\), or \(f(a_1a_2\cdots a_n)=f(a_1)f(a_2)\cdots f(a_n)\), for all \(a_1,a_2,\ldots ,a_n\in A\), respectively. In particular, if \(n=2\) then f is simply called a Jordan homomorphism or a homomorphism, respectively. The notion of n-Jordan homomorphism between rings was introduced in 1956 by Herstein and the concept of n-homomorphism between algebras was introduced in 2005 by Hejazian et al. Properties of n-Jordan homomorphisms as well as n-homomorphisms have been studied by many authors since then. One of the main questions is that, “under what conditions n- Jordan homomorphisms are n-homomorphism?”. Another natural question is that “under what conditions certain properties of homomorphisms may be extended to n-homomorphisms”. We provide conditions under which these questions have affirmative answers. We also study the continuity problem for n-Jordan homomorphisms on Banach algebras, while extending some known results in this field.

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References

  1. 1.

    Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions with formulas. Graphs and Mathematical Tables, US Department of Commerce, Washington, DC (1964)

  2. 2.

    An, G.: Characterization of \(n\)-Jordan homomorphisms. Linear Multilinear Algebra 66(4), 1–10 (2017)

    MathSciNet  Google Scholar 

  3. 3.

    Bodaghi, A., Shojaee, B.: \(n\)-Jordan homomorphisms on \(C^{*}\)-algebras. JLTA 01(01), 1–7 (2012)

    MATH  Google Scholar 

  4. 4.

    Bodaghi, A., Inceboz H.: \(n\)-Jordan homomorphisms on commutative algebras. Acta Math. Uni. Comenianae, Vol. LXZZVII, 1, 141–146 (2018)

  5. 5.

    Bračič, J., Moslehian, M.S.: On automatic continuity of 3-homomorphisms on Banach algebras. Bull. Malays. Math. Sci. Soc. (2) 30(2), 195–200 (2007)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Eshaghi, Gordji M.: \(n\)-Jordan homomorphisms. Bull. Aust. Math. Soc. 80, 159–164 (2009)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Eshaghi Gordji, M., Jabbari, A., Karapinar, E.: Automatic continuity of surjective \(n\)-homomorphisms on Banach algebras. Bull. Iran Math. Soc. 41(5), 1207–1211 (2015)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Gselmann, E.: On approximate \(n\)-Jordan homomorphisms. Ann. Math. Sil. 28, 47–58 (2014)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Hejazian, S., Mirzavaziri, M., Moslehian, M.S.: \(n\)-homomorphisms. Bull. Iran. Math. Soc. 31(1), 13–23 (2009)

    MATH  Google Scholar 

  10. 10.

    Herstein, I.N.: Jordan homomorphisms. Trans. Am. Math. Soc. 81, 331–341 (1956)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Honary, T.G., Shayanpour, H.: Automatic continuity of n-homomorphisms between Banach algebras. Quaest. Math. 33, 189–196 (2010)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Jacobson, N., Rickart, C.E.: Jordan homomorphisms of rings. Trans. Am. Math. Soc. 69, 479–502 (1950)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Kaplansky, I.: Semi-automorphisms of rings. Duke Math. J. 14, 521–525 (1947)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Lee Y. H.: Stability of \(n\)-Jordan homomorphisms from a normed algebra to a Banach algebra. Abstr. Appl. Anal. ID 691025, p. 5 (2013)

  15. 15.

    Palmer, T.: Banach algebras and the general theory of *-algebras, vol. I. Cambridge University Press, Cambridge (1994)

    Google Scholar 

  16. 16.

    Żelazko, W.: A characterization of multiplicative linear functionals in complex Banach algebras. Stud. Math. 30, 83–85 (1968)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Zivari-Kazempour, A.: A characterization of \(3\)-Jordan homomorphisms on Banach algebras. Bull. Aust. Math. Soc. 93(2), 301–306 (2016)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Zivari-Kazempour, A.: A characterization of Jordan and \(5\)-Jordan homomorphisms between Banach algebras. Asian Eur. J. Math. 11(2), 1–10 (2018)

    MathSciNet  Article  Google Scholar 

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Acknowledgements

The authors would like to thank the referee whose comments helped us to improve this article.

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Correspondence to Taher Ghasemi Honary.

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Communicated by Mohammad B. Asadi.

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Honary, T.G., Hosseinzadeh, H. & Mohammadi, S. Relations Between n-Jordan Homomorphisms and n-Homomorphisms. Bull. Iran. Math. Soc. (2020). https://doi.org/10.1007/s41980-020-00407-4

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Keywords

  • Homomorphism
  • Jordan homomorphism
  • n-Jordan homomorphism
  • n-Homomorphism
  • Banach algebra
  • Automatic continuity

Mathematics Subject Classification

  • 13B10
  • 47C05
  • 47B48
  • 46H40