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**Jordan *-derivations on \(C^*\)-algebras and \(JC^*\)-algebras.**
*(English)*
Zbl 1243.46046

Summary: We investigate Jordan *-derivations on \(C^*\)-algebras and Jordan *-derivations on \(JC^*\)-algebras associated with the following functional inequality \(\| f(x)+f(y)+kf(z)\| \leq \| kf((x+y)/k+z)\| \) for some integer \(k\) greater than 1. We moreover prove the generalized Hyers-Ulam stability of Jordan *-derivations on \(C^*\)-algebras and of Jordan *-derivations on \(JC^*\)-algebras associated with the following functional equation \(f((x+y)/k+z)=(f(x)+f(y))/k+f(z)\) for some integer \(k\) greater than 1.

### MSC:

46L05 | General theory of \(C^*\)-algebras |

46L70 | Nonassociative selfadjoint operator algebras |

39B82 | Stability, separation, extension, and related topics for functional equations |

47B47 | Commutators, derivations, elementary operators, etc. |

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\textit{J. S. An} et al., Abstr. Appl. Anal. 2008, Article ID 410437, 12 p. (2008; Zbl 1243.46046)

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### References:

[1] | S. M. Ulam, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8, Interscience, New York, NY, USA, 1960. · Zbl 0086.24101 |

[2] | D. H. Hyers, “On the stability of the linear functional equation,” Proceedings of the National Academy of Sciences of the United States of America, vol. 27, no. 4, pp. 222-224, 1941. · Zbl 0061.26403 |

[3] | T. Aoki, “On the stability of the linear transformation in Banach spaces,” Journal of the Mathematical Society of Japan, vol. 2, pp. 64-66, 1950. · Zbl 0040.35501 |

[4] | Th. M. Rassias, “On the stability of the linear mapping in Banach spaces,” Proceedings of the American Mathematical Society, vol. 72, no. 2, pp. 297-300, 1978. · Zbl 0398.47040 |

[5] | J. M. Rassias, “On approximation of approximately linear mappings by linear mappings,” Journal of Functional Analysis, vol. 46, no. 1, pp. 126-130, 1982. · Zbl 0482.47033 |

[6] | J. M. Rassias, “On approximation of approximately linear mappings by linear mappings,” Bulletin des Sciences Mathématiques. Deuxième Série, vol. 108, no. 4, pp. 445-446, 1984. · Zbl 0599.47106 |

[7] | J. M. Rassias, “Solution of a problem of Ulam,” Journal of Approximation Theory, vol. 57, no. 3, pp. 268-273, 1989. · Zbl 0672.41027 |

[8] | J. M. Rassias, “Solution of a stability problem of Ulam,” Discussiones Mathematicae, vol. 12, pp. 95-103, 1992. · Zbl 0779.47005 |

[9] | J. M. Rassias, “Complete solution of the multi-dimensional problem of Ulam,” Discussiones Mathematicae, vol. 14, pp. 101-107, 1994. · Zbl 0819.39012 |

[10] | P. G\uavru\cta, “An answer to a question of John M. Rassias concerning the stability of Cauchy equation,” in Advances in Equations and Inequalities, Hadronic Mathematics, pp. 67-71, Hadronic Press, Palm Harbor, Fla, USA, 1999. |

[11] | J. M. Rassias, “On the stability of the Euler-Lagrange functional equation,” Chinese Journal of Mathematics, vol. 20, no. 2, pp. 185-190, 1992. · Zbl 0753.39003 |

[12] | J. M. Rassias, “On the stability of the general Euler-Lagrange functional equation,” Demonstratio Mathematica, vol. 29, no. 4, pp. 755-766, 1996. · Zbl 0884.47040 |

[13] | M. J. Rassias and J. M. Rassias, “On the Ulam stability for Euler-Lagrange type quadratic functional equations,” The Australian Journal of Mathematical Analysis and Applications, vol. 2, no. 1, pp. 1-10, 2005. · Zbl 1094.39027 |

[14] | Th. M. Rassias, “Problem 16; 2, Report of the 27th International Symposium on Functional Equations,” Aequationes Mathematicae, vol. 39, pp. 292-293, 1990. |

[15] | Z. Gajda, “On stability of additive mappings,” International Journal of Mathematics and Mathematical Sciences, vol. 14, no. 3, pp. 431-434, 1991. · Zbl 0739.39013 |

[16] | Th. M. Rassias and P. \vSemrl, “On the behavior of mappings which do not satisfy Hyers-Ulam stability,” Proceedings of the American Mathematical Society, vol. 114, no. 4, pp. 989-993, 1992. · Zbl 0761.47004 |

[17] | P. G\uavru\cta, “A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings,” Journal of Mathematical Analysis and Applications, vol. 184, no. 3, pp. 431-436, 1994. · Zbl 0818.46043 |

[18] | S.-M. Jung, “On the Hyers-Ulam-Rassias stability of approximately additive mappings,” Journal of Mathematical Analysis and Applications, vol. 204, no. 1, pp. 221-226, 1996. · Zbl 0888.46018 |

[19] | S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, River Edge, NJ, USA, 2002. · Zbl 1011.39019 |

[20] | D. H. Hyers, G. Isac, and Th. M. Rassias, Stability of Functional Equations in Several Variables, vol. 34 of Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser, Boston, Mass, USA, 1998. · Zbl 0907.39025 |

[21] | G. Isac and Th. M. Rassias, “Stability of \psi -additive mappings: applications to nonlinear analysis,” International Journal of Mathematics and Mathematical Sciences, vol. 19, no. 2, pp. 219-228, 1996. · Zbl 0843.47036 |

[22] | D. H. Hyers, G. Isac, and Th. M. Rassias, “On the asymptoticity aspect of Hyers-Ulam stability of mappings,” Proceedings of the American Mathematical Society, vol. 126, no. 2, pp. 425-430, 1998. · Zbl 0894.39012 |

[23] | D. G. Bourgin, “Classes of transformations and bordering transformations,” Bulletin of the American Mathematical Society, vol. 57, pp. 223-237, 1951. · Zbl 0043.32902 |

[24] | G. L. Forti, “Hyers-Ulam stability of functional equations in several variables,” Aequationes Mathematicae, vol. 50, no. 1-2, pp. 143-190, 1995. · Zbl 0836.39007 |

[25] | A. Gilányi, “On the stability of monomial functional equations,” Publicationes Mathematicae Debrecen, vol. 56, no. 1-2, pp. 201-212, 2000. · Zbl 0991.39016 |

[26] | A. Gilányi, “Eine zur Parallelogrammgleichung äquivalente Ungleichung,” Aequationes Mathematicae, vol. 62, no. 3, pp. 303-309, 2001. · Zbl 0992.39026 |

[27] | A. Gilányi, “On a problem by K. Nikodem,” Mathematical Inequalities & Applications, vol. 5, no. 4, pp. 707-710, 2002. · Zbl 1036.39020 |

[28] | P. M. Gruber, “Stability of isometries,” Transactions of the American Mathematical Society, vol. 245, pp. 263-277, 1978. · Zbl 0393.41020 |

[29] | K.-W. Jun and H.-M. Kim, “On the stability of Euler-Lagrange type cubic mappings in quasi-Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 332, no. 2, pp. 1335-1350, 2007. · Zbl 1117.39017 |

[30] | K.-W. Jun, H.-M. Kim, and J. M. Rassias, “Extended Hyers-Ulam stability for Cauchy-Jensen mappings,” Journal of Difference Equations and Applications, vol. 13, no. 12, pp. 1139-1153, 2007. · Zbl 1135.39013 |

[31] | S.-M. Jung, “On the Hyers-Ulam stability of the functional equations that have the quadratic property,” Journal of Mathematical Analysis and Applications, vol. 222, no. 1, pp. 126-137, 1998. · Zbl 0928.39013 |

[32] | S.-M. Jung, “On the Hyers-Ulam-Rassias stability of a quadratic functional equation,” Journal of Mathematical Analysis and Applications, vol. 232, no. 2, pp. 384-393, 1999. · Zbl 0926.39013 |

[33] | H.-M. Kim, J. M. Rassias, and Y.-S. Cho, “Stability problem of Ulam for Euler-Lagrange quadratic mappings,” Journal of Inequalities and Applications, vol. 2007, Article ID 10725, 15 pages, 2007. · Zbl 1132.39024 |

[34] | S. Kurepa, “On the quadratic functional,” Publications de l’Institut Mathématique de l’Académie Serbe des Sciences et des Arts, vol. 13, pp. 57-72, 1961. · Zbl 0096.31501 |

[35] | Y.-S. Lee and S.-Y. Chung, “Stability of an Euler-Lagrange-Rassias equation in the spaces of generalized functions,” Applied Mathematics Letters, vol. 21, no. 7, pp. 694-700, 2008. · Zbl 1152.39318 |

[36] | P. Nakmahachalasint, “On the generalized Ulam-G\uavru\cta-Rassias stability of mixed-type linear and Euler-Lagrange-Rassias functional equations,” International Journal of Mathematics and Mathematical Sciences, vol. 2007, Article ID 63239, 10 pages, 2007. · Zbl 1148.39026 |

[37] | C.-G. Park, “Lie \ast -homomorphisms between Lie C\ast -algebras and Lie \ast -derivations on Lie C\ast -algebras,” Journal of Mathematical Analysis and Applications, vol. 293, no. 2, pp. 419-434, 2004. · Zbl 1051.46052 |

[38] | C.-G. Park, “Homomorphisms between Lie JC\ast -algebras and Cauchy-Rassias stability of Lie JC\ast -algebra derivations,” Journal of Lie Theory, vol. 15, no. 2, pp. 393-414, 2005. · Zbl 1091.39006 |

[39] | C.-G. Park, “Homomorphisms between Poisson JC\ast -algebras,” Bulletin of the Brazilian Mathematical Society, vol. 36, no. 1, pp. 79-97, 2005. · Zbl 1091.39007 |

[40] | C.-G. Park, “Hyers-Ulam-Rassias stability of a generalized Euler-Lagrange type additive mapping and isomorphisms between C\ast -algebras,” Bulletin of the Belgian Mathematical Society, vol. 13, no. 4, pp. 619-632, 2006. · Zbl 1125.39027 |

[41] | C. Park, “Isomorphisms between quasi-Banach algebras,” Chinese Annals of Mathematics. Series B, vol. 28, no. 3, pp. 353-362, 2007. · Zbl 1127.39056 |

[42] | C. Park, Y. S. Cho, and M.-H. Han, “Functional inequalities associated with Jordan-von Neumann-type additive functional equations,” Journal of Inequalities and Applications, vol. 2007, Article ID 41820, 13 pages, 2007. · Zbl 1133.39024 |

[43] | C.-G. Park and J. Hou, “Homomorphisms between C\ast -algebras associated with the Trif functional equation and linear derivations on C\ast -algebras,” Journal of the Korean Mathematical Society, vol. 41, no. 3, pp. 461-477, 2004. · Zbl 1058.39025 |

[44] | C. G. Park, J. C. Hou, and S. Q. Oh, “Homomorphisms between JC\ast -algebras and Lie C\ast -algebras,” Acta Mathematica Sinica, vol. 21, no. 6, pp. 1391-1398, 2005. · Zbl 1121.39030 |

[45] | C. Park, J. Park, and J. Shin, “Hyers-Ulam-Rassias stability of quadratic functional equations in Banach modules over a C\ast -algebra,” Chinese Annals of Mathematics. Series B, vol. 24, no. 2, pp. 261-266, 2003. · Zbl 1053.46029 |

[46] | A. Pietrzyk, “Stability of the Euler-Lagrange-Rassias functional equation,” Demonstratio Mathematica, vol. 39, no. 3, pp. 523-530, 2006. · Zbl 1113.39034 |

[47] | Th. M. Rassias, “The problem of S. M. Ulam for approximately multiplicative mappings,” Journal of Mathematical Analysis and Applications, vol. 246, no. 2, pp. 352-378, 2000. · Zbl 0958.46022 |

[48] | Th. M. Rassias, “On the stability of functional equations in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 251, no. 1, pp. 264-284, 2000. · Zbl 0964.39026 |

[49] | Th. M. Rassias, “On the stability of functional equations and a problem of Ulam,” Acta Applicandae Mathematicae, vol. 62, no. 1, pp. 23-130, 2000. · Zbl 0981.39014 |

[50] | Th. M. Rassias, Functional Equations, Inequalities and Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2003. · Zbl 1047.39001 |

[51] | K. Ravi and M. Arunkumar, “On the Ulam-G\uavru\cta-Rassias stability of the orthogonally Euler-Lagrange type functional equation,” International Journal of Applied Mathematics & Statistics, vol. 7, no. Fe07, pp. 143-156, 2007. |

[52] | J. Rätz, “On inequalities associated with the Jordan-von Neumann functional equation,” Aequationes Mathematicae, vol. 66, no. 1-2, pp. 191-200, 2003. · Zbl 1078.39026 |

[53] | W. Fechner, “Stability of a functional inequality associated with the Jordan-von Neumann functional equation,” Aequationes Mathematicae, vol. 71, no. 1-2, pp. 149-161, 2006. · Zbl 1098.39019 |

[54] | M. Amyari, F. Rahbarnia, and Gh. Sadeghi, “Some results on stability of extended derivations,” Journal of Mathematical Analysis and Applications, vol. 329, no. 2, pp. 753-758, 2007. · Zbl 1153.39308 |

[55] | C. Baak, “Cauchy-Rassias stability of Cauchy-Jensen additive mappings in Banach spaces,” Acta Mathematica Sinica, vol. 22, no. 6, pp. 1789-1796, 2006. · Zbl 1118.39012 |

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