Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
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APPROXIMATE BI-HOMOMORPHISMS AND BI-DERIVATIONS IN C*-TERNARY ALGEBRAS: REVISITED

  • Cho, Young (Faculty of Electrical and Electronics Engineering Ulsan College West Campus) ;
  • Jang, Sun Young (Department of Mathematics University of Ulsan) ;
  • Kwon, Su Min (Department of Mathematics Hanyang University) ;
  • Park, Choonkil (Department of Mathematics Research Institute for Natural Sciences Hanyang University) ;
  • Park, Won-Gil (Department of Mathematics Education Mokwon University)
  • Received : 2013.03.14
  • Accepted : 2013.04.25
  • Published : 2013.06.30
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Abstract

Bae and W. Park [3] proved the Hyers-Ulam stability of bi-homomorphisms and bi-derivations in $C^*$-ternary algebras. It is easy to show that the definitions of bi-homomorphisms and bi-derivations, given in [3], are meaningless. So we correct the definitions of bi-homomorphisms and bi-derivations. Under the conditions in the main theorems, we can show that the related mappings must be zero. In this paper, we correct the statements and the proofs of the results, and prove the corrected theorems.

Keywords

Acknowledgement

Supported by : National Research Foundation of Korea

References

  1. M. Amyari and M.S. Moslehian, Approximately ternary semigroup homomor-phisms, Lett. Math. Phys. 77 (2006), 1-9. https://doi.org/10.1007/s11005-005-0042-6
  2. T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64-66. https://doi.org/10.2969/jmsj/00210064
  3. J. Bae and W. Park, Approximate bi-homomorphisms and bi-derivations in C*-ternary algebras, Bull. Korean Math. Soc. 47 (2010), 195-209. https://doi.org/10.4134/BKMS.2010.47.1.195
  4. A. Ebadian, N. Ghobadipour and H. Baghban, Stability of bi-${\theta}$-derivations on JB*-triples, Int. J. Geom. Methods Mod. Phys. 9 (2012), No. 7, Art. ID 1250051, 12 pages.
  5. A. Ebadian, N. Ghobadipour and M. Eshaghi Gordji, A xed point method for perturbation of bimultipliers and Jordan bimultipliers in C*-ternary algebras, J. Math. Phys. 51 (2010), No. 10, Art. ID 103508, 10 pages.
  6. A. Ebadian, I. Nikoufar and M. Eshaghi Gordji, Nearly (${\theta}_1,{\theta}_2,{\theta}_3,{\phi}$)-derivations on C*-modules, Int. J. Geom. Methods Mod. Phys. 9 (2012), No. 3, Art. ID 1250019, 12 pages.
  7. M. Eshaghi Gordji, A. Fazeli and C. Park, 3-Lie multipliers on Banach 3-Lie algebras, Int. J. Geom. Methods Mod. Phys. 9 (2012), No. 7, Art. ID 1250052, 15 pages.
  8. M. Eshaghi Gordji, M.B. Ghaemi and B. Alizadeh, A fixed point method for perturbation of higher ring derivationsin non-Archimedean Banach algebras, Int. J. Geom. Methods Mod. Phys. 8 (2011), 1611-1625. https://doi.org/10.1142/S021988781100583X
  9. M. Eshaghi Gordji, M.B. Ghaemi, S. Kaboli Gharetapeh, S. Shams and A. Ebadian, On the stability of J*-derivations, J. Geom. Phys. 60 (2010), 454-459. https://doi.org/10.1016/j.geomphys.2009.11.004
  10. M. Eshaghi Gordji and N. Ghobadipour, Stability of ($-\alpha}$, $-\beta}$, $-\gamma}$)-derivations on Lie C*-algebras, Int. J. Geom. Methods Mod. Phys. 7 (2010), 1097-1102.
  11. D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA. 27 (1941), 222-224. https://doi.org/10.1073/pnas.27.4.222
  12. J.M. Rassias, On approximation of approximately linear mappings by linear mappings, J. Funct. Anal. 46 (1982), 126-130. https://doi.org/10.1016/0022-1236(82)90048-9
  13. Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300. https://doi.org/10.1090/S0002-9939-1978-0507327-1
  14. J. Shokri, A. Ebadian and R. Aghalari, Approximate bihomomorphisms and biderivations in 3-Lie algebras, Int. J. Geom. Methods Mod. Phys. 10 (2013), No. 1, Art. ID 1220020, 13 pages.
  15. S.M. Ulam, Problems in Modern Mathematics, Chapter VI, Science ed., Wiley, New York, 1940.
  16. H. Zettl, A characterization of ternary rings of operators, Adv. Math. 48 (1983), 117-143. https://doi.org/10.1016/0001-8708(83)90083-X

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