DOI QR코드

DOI QR Code

ISOMORPHISMS IN QUASI-BANACH ALGEBRAS

  • Published : 2008.02.29

Abstract

Using the Hyers-Ulam-Rassias stability method, we investigate isomorphisms in quasi-Banach algebras and derivations on quasi-Banach algebras associated with the Cauchy-Jensen functional equation $$2f(\frac{x+y}{2}+z)$$=f(x)+f(y)+2f(z), which was introduced and investigated in [2, 17]. The concept of Hyers-Ulam-Rassias stability originated from the Th. M. Rassias' stability theorem that appeared in the paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300. Furthermore, isometries and isometric isomorphisms in quasi-Banach algebras are studied.

Keywords

Cauchy-Jensen functional equation;isomorphism;isometry;derivation;quasi-Banach algebra

References

  1. J. M. Almira and U. Luther, Inverse closedness of approximation algebras, J. Math. Anal. Appl. 314 (2006), no. 1, 30-44 https://doi.org/10.1016/j.jmaa.2005.03.067
  2. C. Baak, Cauchy-Rassias stability of Cauchy-Jensen additive mappings in Banach spaces, Acta Math. Sin. (Engl. Ser.) 22 (2006), no. 6, 1789-1796 https://doi.org/10.1007/s10114-005-0697-z
  3. C. Baak and M. S. Moslehian, On the stability of ${\theta}$-derivations on JB*-triples, Bull. Braz. Math. Soc. (N.S.) 38 (2007), no. 1, 115-127 https://doi.org/10.1007/s00574-007-0039-0
  4. J. Baker, Isometries in normed spaces, Amer. Math. Monthly 78 (1971), 655-658 https://doi.org/10.2307/2316577
  5. Y. Benyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis. Vol. 1, American Mathematical Society Colloquium Publications, 48. American Mathematical Society, Providence, RI, 2000
  6. J. Bourgain, Real isomorphic complex Banach spaces need not be complex isomorphic, Proc. Amer. Math. Soc. 96 (1986), no. 2, 221-226 https://doi.org/10.1090/S0002-9939-1986-0818448-2
  7. A. Gilanyi, Eine zur Parallelogrammgleichung aquivalente Ungleichung, Aequationes Math. 62 (2001), no. 3, 303-309 https://doi.org/10.1007/PL00000156
  8. A. Gilanyi, On a problem by K. Nikodem, Math. Inequal. Appl. 5 (2002), no. 4, 707-710
  9. D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U. S. A. 27 (1941), 222-224 https://doi.org/10.1073/pnas.27.4.222
  10. N. Kalton, An elementary example of a Banach space not isomorphic to its complex conjugate, Canad. Math. Bull. 38 (1995), no. 2, 218-222 https://doi.org/10.4153/CMB-1995-031-4
  11. S. Mazur and S. Ulam, Sur les transformation d'espaces vectoriels norme, C. R. Acad. Sci. Paris 194 (1932), 946-948
  12. C. Park, On an approximate automorphism on a C*-algebra, Proc. Amer. Math. Soc. 132 (2004), no. 6, 1739-1745 https://doi.org/10.1090/S0002-9939-03-07252-6
  13. 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
  14. C. Park, A generalized Jensen's mapping and linear mappings between Banach modules, Bull. Braz. Math. Soc. (N.S.) 36 (2005), no. 3, 333-362 https://doi.org/10.1007/s00574-005-0043-1
  15. C. Park, Isomorphisms between C*-ternary algebras, J. Math. Phys. 47, no. 10, Article ID 103512 (2006), 12 pages
  16. 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
  17. Th. M. Rassias, Problem 16; 2, Report of the 27th Internat. Symp. on Functional Equations, Aequationes Math. 39 (1990), 292-293; 309
  18. Th. M. Rassias, Properties of isometic mappings, J. Math. Anal. Appl. 235 (1997), 108-121 https://doi.org/10.1006/jmaa.1999.6363
  19. Th. M. Rassias, The problem of S. M. Ulam for approximately multiplicative mappings, J. Math. Anal. Appl. 246 (2000), no. 2, 352-378 https://doi.org/10.1006/jmaa.2000.6788
  20. Th. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math. 62 (2000), no. 1, 23-130 https://doi.org/10.1023/A:1006499223572
  21. Th. M. Rassias, Functional Equations, Inequalities and Applications, Kluwer Academic Publishers, Dordrecht, 2003
  22. J. Ratz, On inequalities associated with the Jordan-von Neumann functional equation, Aequationes Math. 66 (2003), no. 1-2, 191-200 https://doi.org/10.1007/s00010-003-2684-8
  23. S. Rolewicz, Metric Linear Spaces, Second edition. PWN-Polish Scientific Publishers, Warsaw; D. Reidel Publishing Co., Dordrecht, 1984
  24. S. M. Ulam, Problems in Modern Mathematics, Science Editions John Wiley & Sons, Inc., New York, 1964
  25. W. Fechner, Stability of a functional inequality associated with the Jordan-von Neumann functional equation, Aequationes Math. 71 (2006), no. 1-2, 149-161 https://doi.org/10.1007/s00010-005-2775-9
  26. C. Park, Y. Cho, and M. Han, Functional inequalities associated with Jordan-von Neumann type additive functional equations, J. Inequal. Appl. 2007, Article ID 41820 (2007), 13 pages https://doi.org/10.1155/2007/41820
  27. 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
  28. Th. M. Rassias and P. Semrl, On the Mazur-Ulam theorem and the Aleksandrov problem for unit distance preserving mappings, Proc. Amer. Math. Soc. 118 (1993), no. 3, 919-925 https://doi.org/10.1090/S0002-9939-1993-1111437-6
  29. M. Mirzavaziri and M. S. Moslehian, A fixed point approach to stability of a quadratic equation, Bull. Braz. Math. Soc. (N.S.) 37 (2006), no. 3, 361-376 https://doi.org/10.1007/s00574-006-0016-z

Cited by

  1. CHARACTERIZATIONS OF REAL HYPERSURFACES OF TYPE A IN A COMPLEX SPACE FORM vol.47, pp.1, 2010, https://doi.org/10.4134/BKMS.2010.47.1.001
  2. APPROXIMATE BI-HOMOMORPHISMS AND BI-DERIVATIONS IN C*-TERNARY ALGEBRAS vol.47, pp.1, 2010, https://doi.org/10.4134/BKMS.2010.47.1.195
  3. Hybrid fixed point result for lipschitz homomorphisms on quasi-Banach algebras vol.27, pp.2, 2011, https://doi.org/10.1007/s10496-011-0109-4
  4. Stability of a Bi-Additive Functional Equation in Banach Modules Over aC⋆-Algebra vol.2012, 2012, https://doi.org/10.1155/2012/835893