References
- A. D. Aleksandrov, Mappings of families of sets, Soviet Math. Dokl. 11 (1970), 376-380
- J. Baker, Isometries in normed spaces, Amer. Math. Monthly 78 (1971), 655-658 https://doi.org/10.2307/2316577
- F. F. Bonsall and J. Duncan, Complete normed algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 80. Springer-Verlag, New York-Heidelberg, 1973
- D. Boo and C. Park, The fundamental group of the automorphism group of a noncommutative torus, Chinese Ann. Math. Ser. B 21 (2000), no. 4, 441-452 https://doi.org/10.1142/S0252959900000443
- Y. J. Cho, P. C. S. Lin, S. S. Kim, and A. Misiak, Theory of 2-inner product spaces, Nova Science Publishers, Inc., Huntington, NY, 2001
- H. Chu, K. Lee, and C. Park, On the Aleksandrov problem in linear n-normed spaces, Nonlinear Anal.-TMA 59 (2004), no. 7, 1001-1011
- H. Chu, C. Park, and W. Park, The Aleksandrov problem in linear 2-normed spaces, J. Math. Anal. Appl. 289 (2004), no. 2, 666-672 https://doi.org/10.1016/j.jmaa.2003.09.009
- S. Czerwik, Functional equations and inequalities in several variables, World Scientific Publishing Co., Inc., River Edge, NJ, 2002
- S. Czerwik, Stability of functional equations of Ulam-Hyers-Rassias type, Hadronic Press Inc., Palm Harbor, Florida, 2003
- J. Dixmier, C*-algebras, North-Holland Mathematical Library, Vol. 15. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977
- G. Dolinar, Generalized stability of isometries, J. Math. Anal. Appl. 242 (2000), no. 1, 39-56 https://doi.org/10.1006/jmaa.1999.6649
-
V. A. Faizev, Th. M. Rassias, and P. K. Sahoo, The space of (
${\psi},{\gamma}$ )-additive mappings on semigroups, Trans. Amer. Math. Soc. 354 (2002), no. 11, 4455-4472 https://doi.org/10.1090/S0002-9947-02-03036-2 - R. J. Fleming and J. E. Jamison, Isometries on Banach spaces: function spaces, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 129. Chapman & Hall/CRC, Boca Raton, FL, 2003
- Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991), no. 3, 431-434 https://doi.org/10.1155/S016117129100056X
- 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
- J. Gevirtz, Stability of isometries on Banach spaces, Proc. Amer. Math. Soc. 89 (1983), no. 4, 633-636 https://doi.org/10.2307/2044596
- K. R. Goodearl and E. S. Letzter, Quantum n-space as a quotient of classical n-space, Trans. Amer. Math. Soc. 352 (2000), no. 12, 5855-5876 https://doi.org/10.1090/S0002-9947-00-02639-8
- P. Gruber, Stability of isometries, Trans. Amer. Math. Soc. 245 (1978), 263-277 https://doi.org/10.2307/1998866
- H. Haruki and Th. M. Rassias, A new functional equation of Pexider type related to the complex exponential function, Trans. Amer. Math. Soc. 347 (1995), no. 8, 3111-3119 https://doi.org/10.2307/2154775
- D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U. S. A. 27 (1941), 222-224
- D. H. Hyers, G. Isac, and Th. M. Rassias, Stability of functional equations in several variables, Progress in Nonlinear Differential Equations and their Applications, 34. Birkhauser Boston, Inc., Boston, MA, 1998
- D. H. Hyers and Th. M. Rassias, Approximate homomorphisms, Aequationes Math. 44 (1992), no. 2-3, 125-153 https://doi.org/10.1007/BF01830975
- S. Jung, On the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 204 (1996), no. 1, 221-226 https://doi.org/10.1006/jmaa.1996.0433
- S. Jung, Hyers-Ulam-Rassias stability of functional equations in mathematical analysis, Hadronic Press, Inc., Palm Harbor, FL, 2001
- R. V. Kadison and J. R. Ringrose, Fundamentals of the theory of operator algebras. Vol. I, Elementary theory. Pure and Applied Mathematics, 100. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983
- 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
- Y. Ma, The Aleksandrov problem for unit distance preserving mapping, Acta Math. Sci. Ser. B Engl. Ed. 20 (2000), no. 3, 359-364
- S. Mazur and S. Ulam, Sur les transformations analytiques des domaines cercles et semi-cercles bornes, Math. Ann. 106 (1932), no. 1, 540-573 https://doi.org/10.1007/BF01455901
- B. Mielnik and Th. M. Rassias, On the Aleksandrov problem of conservative distances, Proc. Amer. Math. Soc. 116 (1992), no. 4, 1115-1118 https://doi.org/10.2307/2159497
- S. Oh, C. Park, and Y. Shin, Quantum n-space and Poisson n-space, Comm. Algebra 30 (2002), no. 9, 4197-4209 https://doi.org/10.1081/AGB-120013313
- S. Oh, C. Park, and Y. Shin, A Poincare-Birkhoff-Witt theorem for Poisson enveloping algebras, Comm. Algebra 30 (2002), no. 10, 4867-4887 https://doi.org/10.1081/AGB-120014673
- C. Park, On the stability of the linear mapping in Banach modules, J. Math. Anal. Appl. 275 (2002), no. 2, 711-720 https://doi.org/10.1016/S0022-247X(02)00386-4
- C. Park, Functional equations in Banach modules, Indian J. Pure Appl. Math. 33 (2002), no. 7, 1077-1086
- C. Park, Generalized simple noncommutative tori, Chinese Ann. Math. Ser. B 23 (2002), no. 4, 539-544 https://doi.org/10.1142/S025295990200050X
- 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
- C. Park, Lie -homomorphisms between Lie C-algebras and Lie -derivations on Lie C-algebras, J. Math. Anal. Appl. 293 (2004), no. 2, 419-434 https://doi.org/10.1016/j.jmaa.2003.10.051
- C. Park, Universal Jensen's equations in Banach modules over a C*-algebra and its unitary group, Acta Math. Sin. (Engl. Ser.) 20 (2004), no. 6, 1047-1056 https://doi.org/10.1007/s10114-004-0409-0
- 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
- C. Park, Approximate homomorphisms on JB*-triples, J. Math. Anal. Appl. 306 (2005), no. 1, 375-381 https://doi.org/10.1016/j.jmaa.2004.12.043
- C. Park, Homomorphisms between Lie JC*-algebras and Cauchy-Rassias stability of Lie JC*-algebra derivations, J. Lie Theory 15 (2005), no. 2, 393-414
- C. Park, Isomorphisms between unital C*-algebras, J. Math. Anal. Appl. 307 (2005), no. 2, 753-762 https://doi.org/10.1016/j.jmaa.2005.01.059
- C. Park, Linear *-derivations on JB*-algebras, Acta Math. Sci. Ser. B Engl. Ed. 25 (2005), no. 3, 449-454
- C. Park and J. Hou, Homomorphisms between C*-algebras associated with the Trif functional equation and linear derivations on C*-algebras, J. Korean Math. Soc. 41 (2004), no. 3, 461-477 https://doi.org/10.4134/JKMS.2004.41.3.461
- C. Park, J. Hou, and S. Oh, Homomorphisms between JC*-algebras and Lie C*-algebras, Acta Math. Sin. (Engl. Ser.) 21 (2005), no. 6, 1391-1398 https://doi.org/10.1007/s10114-005-0629-y
- C. Park and Th. M. Rassias, On a generalized Trif 's mapping in Banach modules over a C*-algebra, J. Korean Math. Soc. 43 (2006), no. 2, 323-356 https://doi.org/10.4134/JKMS.2006.43.2.323
- C. Park and Th. M. Rassias, Isometries on linear n-normed spaces, J. Inequal. Pure Appl. Math. 7 (2006), no. 5, Article 168, 7 pp
- 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/10.2307/2042795
- Th. M. Rassias, Problem 16; 2, Report of the 27th International Symp. on Functional Equations, Aequationes Math. 39 (1990), 292-293
- Th. M. Rassias, Properties of isometric mappings, J. Math. Anal. Appl. 235 (1999), no. 1, 108-121 https://doi.org/10.1006/jmaa.1999.6363
- 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
- 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
- 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
- Th. M. Rassias, On the A. D. Aleksandrov problem of conservative distances and the Mazur-Ulam theorem, Nonlinear Anal.-TMA 47 (2001), no. 4, 2597-2608 https://doi.org/10.1016/S0362-546X(01)00381-9
- Th. M. Rassias and P. Semrl , On the behavior of mappings which do not satisfy Hyers-Ulam stability, Proc. Amer. Math. Soc. 114 (1992), no. 4, 989-993 https://doi.org/10.2307/2159617
- 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/10.2307/2160142
- Th. M. Rassias and P. Semrl, On the Hyers-Ulam stability of linear mappings, J. Math. Anal. Appl. 173 (1993), no. 2, 325-338 https://doi.org/10.1006/jmaa.1993.1070
- Th. M. Rassias and S. Xiang, On mappings with conservative distances and the Mazur-Ulam theorem, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 11 (2000), 1-8 (2001)
- S. M. Ulam, Problems in modern mathematics, Science Editions John Wiley & Sons, Inc., New York, 1964
- H. Upmeier, Jordan algebras in analysis, operator theory, and quantum mechanics, CBMS Regional Conference Series in Mathematics, 67. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1987
- S. Xiang, Mappings of conservative distances and the Mazur-Ulam theorem, J. Math. Anal. Appl. 254 (2001), no. 1, 262-274 https://doi.org/10.1006/jmaa.2000.7276
Cited by
- 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
- 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