Communications of the Korean Mathematical Society (대한수학회논문집)

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JORDAN *-HOMOMORPHISMS BETWEEN UNITAL C*-ALGEBRAS

  • Received : 2010.09.02
  • Published : 2012.01.31
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Abstract

In this paper, we prove the superstability and the generalized Hyers-Ulam stability of Jordan *-homomorphisms between unital $C^*$-algebras associated with the following functional equation$$f(\frac{-x+y}{3})+f(\frac{x-3z}{c})+f(\frac{3x-y+3z}{3})=f(x)$$. Morever, we investigate Jordan *-homomorphisms between unital $C^*$-algebras associated with the following functional inequality $${\parallel}f(\frac{-x+y}{3})+f(\frac{x-3z}{3})+f(\frac{3x-y+3z}{3}){\parallel}\leq{\parallel}f(x)\parallel$$.

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References

  1. B. Baak, D. Boo, and Th. M. Rassias, Generalized additive mapping in Banach modules and isomorphisms between $C^*$-algebras, J. Math. Anal. Appl. 314 (2006), no. 1, 150-161. https://doi.org/10.1016/j.jmaa.2005.03.099
  2. P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), no. 1-2, 76-86. https://doi.org/10.1007/BF02192660
  3. J. Y. Chung, Distributional methods for a class of functional equations and their stabilities, Acta Math. Sin. (Engl. Ser.) 23 (2007), no. 11, 2017-2026. https://doi.org/10.1007/s10114-007-0977-x
  4. M. Eshaghi Gordji, Stability of an additive-quadratic functional equation of two variables in F-spaces, J. Nonlinear Sci. Appl. 2 (2009), no. 4, 251-259. https://doi.org/10.22436/jnsa.002.01.09
  5. Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991), 431-434. https://doi.org/10.1155/S016117129100056X
  6. P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), no. 3, 431-436. https://doi.org/10.1006/jmaa.1994.1211
  7. D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA 27 (1941), 222-224. https://doi.org/10.1073/pnas.27.4.222
  8. D. H. Hyers, G. Isac, and Th. M. Rassias, Stability of Functional Equations in Several Variables, Birkhauser, Basel, 1998.
  9. D. H. Hyers and Th. M. Rassias, Approximate homomorphisms, Aequationes Math. 44 (1992), no. 2-3, 125-153. https://doi.org/10.1007/BF01830975
  10. G. Isac and Th. M. Rassias, On the Hyers-Ulam stability of $\psi$-additive mappings, J. Approx. Theory 72 (1993), no. 2, 131-137. https://doi.org/10.1006/jath.1993.1010
  11. G. Isac and Th. M. Rassias, Stability of $\psi$-additive mappings: Applications to nonlinear analysis, Internat. J. Math. Math. Sci. 19 (1996), no. 2, 219-228. https://doi.org/10.1155/S0161171296000324
  12. K.-W. Jun and H.-M. Kim, Stability problem for Jensen type functional equations of cubic mappings, Acta Math. Sin. (Engl. Ser.) 22 (2006), no. 6, 1781-1788. https://doi.org/10.1007/s10114-005-0736-9
  13. K. Jun and Y. Lee, A generalization of the Hyers-Ulam-Rassias stability of Jensen's equation, J. Math. Anal. Appl. 238 (1999), no. 1, 305-315. https://doi.org/10.1006/jmaa.1999.6546
  14. S. Jung, Hyers-Ulam-Rassias stability of Jensen's equation and its application, Proc. Amer. Math. Soc. 126 (1998), no. 11, 3137-3143. https://doi.org/10.1090/S0002-9939-98-04680-2
  15. B. E. Johnson, Approximately multiplicative maps between Banach algebras, J. London Math. Soc. (2) 37 (1988), no. 2, 294-316. https://doi.org/10.1112/jlms/s2-37.2.294
  16. R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras. Vol. I, Elementary Theory, Academic Press, New York, 1983.
  17. B. D. Kim, On the derivations of semiprime rings and noncommutative Banach algebras, Acta Math. Sin. (Engl. Ser.) 16 (2000), no. 1, 21-28. https://doi.org/10.1007/s101149900020
  18. B. D. Kim, On Hyers-Ulam-Rassias stability of functional equations, Acta Mathematica Sinica 24 (2008), no. 3, 353-372. https://doi.org/10.1007/s10114-007-7221-6
  19. H.-M. Kim, Stability for generalized Jensen functional equations and isomorphisms between $C^*$-algebras, Bull. Belg. Math. Soc. Simon Stevin 14 (2007), no. 1, 1-14.
  20. M. S. Moslehian, Almost Derivations on $C^*$-Ternary Rings, Bull. Belg. Math. Soc. Simon Stevin 14 (2007), no. 1, 135-142.
  21. A. Najati and C. Park, Stability of a generalized Euler-Lagrange type additive mapping and homomorphisms in $C^*$-algebras, J. Nonlinear Sci. Appl. 3 (2010), no. 2, 123-143. https://doi.org/10.22436/jnsa.003.02.05
  22. C. Park, Homomorphisms between Poisson $JC^*$-algebras, Bull. Braz. Math. Soc. (N.S.) 36 (2005), no. 1, 79-97. https://doi.org/10.1007/s00574-005-0029-z
  23. C. Park, Hyers-Ulam-Rassias stability of a generalized Euler-Lagrange type additive mapping and isomorphisms between $C^*$-algebras, Bull. Belg. Math. Soc. Simon Stevin 13 (2006), no. 4, 619-631.
  24. C. Park, J. An, and J. Cui, Jordan *-derivations on C*-algebras and C*algebras, Abstact and Applied Analasis (in press).
  25. C. Park and J. L. Cui, Approximately linear mappings in Banach modules over a C*-algebra, Acta Math. Sin. (Engl. Ser.) 23 (2007), no. 11, 1919-1936. https://doi.org/10.1007/s10114-007-0964-2
  26. C. Park and W. Park, On the Jensen's equation in Banach modules, Taiwanese J. Math. 6 (2002), no. 4, 523-531. https://doi.org/10.11650/twjm/1500407476
  27. Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), no. 2, 297-300. https://doi.org/10.1090/S0002-9939-1978-0507327-1
  28. Th. M. Rassias, Approximate homomorphisms, Aequationes Math. 44 (1992), no. 2-3, 125-153. https://doi.org/10.1007/BF01830975
  29. Th. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Math. Appl. 62 (2000), no. 1, 23-130. https://doi.org/10.1023/A:1006499223572
  30. Th. M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. 251 (2000), no. 1, 264-284. https://doi.org/10.1006/jmaa.2000.7046
  31. P. K. Sahoo, A generalized cubic functional equation, Acta Math. Sin. (Engl. Ser.) 21 (2005), no. 5, 1159-1166. https://doi.org/10.1007/s10114-005-0551-3
  32. S. Shakeri, Intuitionistic fuzzy stability of Jensen type mapping, J. Nonlinear Sci. Appl. 2 (2009), no. 2, 105-112. https://doi.org/10.22436/jnsa.002.02.05
  33. S. M. Ulam, Problems in Modern Mathematics, Chapter VI, Science ed. Wiley, New York, 1940.
  34. D. H. Zhang and H. X. Cao, Stability of functional equations in several variables, Acta Math. Sin. (Engl. Ser.) 23 (2007), no. 2, 321-326. https://doi.org/10.1007/s10114-005-0862-4

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