Abstract and Applied Analysis

Jordan -Derivations on C -Algebras and J C -Algebras

Jong Su An, Jianlian Cui, and Choonkil Park

Full-text: Open access

Abstract

We investigate Jordan -derivations on C -algebras and Jordan -derivations on J C -algebras associated with the following functional inequality f ( x ) + f ( y ) + k f ( z ) k f ( ( 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 J C -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.

Article information

Source
Abstr. Appl. Anal., Volume 2008 (2008), Article ID 410437, 12 pages.

Dates
First available in Project Euclid: 10 February 2009

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1234299003

Digital Object Identifier
doi:10.1155/2008/410437

Mathematical Reviews number (MathSciNet)
MR2471252

Zentralblatt MATH identifier
1243.46046

Citation

Su An, Jong; Cui, Jianlian; Park, Choonkil. Jordan ${\ast}$ -Derivations on ${C}^{\ast}$ -Algebras and $J{C}^{\ast}$ -Algebras. Abstr. Appl. Anal. 2008 (2008), Article ID 410437, 12 pages. doi:10.1155/2008/410437. https://projecteuclid.org/euclid.aaa/1234299003


Export citation

References

  • S. M. Ulam, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8, Interscience, New York, NY, USA, 1960.
  • 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.
  • 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.
  • 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.
  • J. M. Rassias, ``On approximation of approximately linear mappings by linear mappings,'' Journal of Functional Analysis, vol. 46, no. 1, pp. 126--130, 1982.
  • 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.
  • J. M. Rassias, ``Solution of a problem of Ulam,'' Journal of Approximation Theory, vol. 57, no. 3, pp. 268--273, 1989.
  • J. M. Rassias, ``Solution of a stability problem of Ulam,'' Discussiones Mathematicae, vol. 12, pp. 95--103, 1992.
  • J. M. Rassias, ``Complete solution of the multi-dimensional problem of Ulam,'' Discussiones Mathematicae, vol. 14, pp. 101--107, 1994.
  • P. Găvruţa, ``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. \setlength\itemsep.55pt
  • J. M. Rassias, ``On the stability of the Euler-Lagrange functional equation,'' Chinese Journal of Mathematics, vol. 20, no. 2, pp. 185--190, 1992.
  • J. M. Rassias, ``On the stability of the general Euler-Lagrange functional equation,'' Demonstratio Mathematica, vol. 29, no. 4, pp. 755--766, 1996.
  • 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.
  • Th. M. Rassias, ``Problem 16; 2, Report of the 27th International Symposium on Functional Equations,'' Aequationes Mathematicae, vol. 39, pp. 292--293, 1990.
  • Z. Gajda, ``On stability of additive mappings,'' International Journal of Mathematics and Mathematical Sciences, vol. 14, no. 3, pp. 431--434, 1991.
  • Th. M. Rassias and P. Šemrl, ``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.
  • P. Găvruţa, ``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.
  • 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.
  • S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, River Edge, NJ, USA, 2002.
  • 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.
  • 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.
  • 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.
  • D. G. Bourgin, ``Classes of transformations and bordering transformations,'' Bulletin of the American Mathematical Society, vol. 57, pp. 223--237, 1951.
  • G. L. Forti, ``Hyers-Ulam stability of functional equations in several variables,'' Aequationes Mathematicae, vol. 50, no. 1-2, pp. 143--190, 1995.
  • A. Gilányi, ``On the stability of monomial functional equations,'' Publicationes Mathematicae Debrecen, vol. 56, no. 1-2, pp. 201--212, 2000.
  • A. Gilányi, ``Eine zur Parallelogrammgleichung äquivalente Ungleichung,'' Aequationes Mathematicae, vol. 62, no. 3, pp. 303--309, 2001.
  • A. Gilányi, ``On a problem by K. Nikodem,'' Mathematical Inequalities & Applications, vol. 5, no. 4, pp. 707--710, 2002.
  • P. M. Gruber, ``Stability of isometries,'' Transactions of the American Mathematical Society, vol. 245, pp. 263--277, 1978.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • P. Nakmahachalasint, ``On the generalized Ulam-Găvruţa-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.
  • 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.
  • 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.
  • C.-G. Park, ``Homomorphisms between Poisson $JC^\ast$-algebras,'' Bulletin of the Brazilian Mathematical Society, vol. 36, no. 1, pp. 79--97, 2005.
  • 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.
  • C. Park, ``Isomorphisms between quasi-Banach algebras,'' Chinese Annals of Mathematics. Series B, vol. 28, no. 3, pp. 353--362, 2007.
  • 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.
  • 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.
  • 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.
  • 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.
  • A. Pietrzyk, ``Stability of the Euler-Lagrange-Rassias functional equation,'' Demonstratio Mathematica, vol. 39, no. 3, pp. 523--530, 2006.
  • 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.
  • 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.
  • 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.
  • Th. M. Rassias, Functional Equations, Inequalities and Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2003.
  • K. Ravi and M. Arunkumar, ``On the Ulam-Găvruţa-Rassias stability of the orthogonally Euler-Lagrange type functional equation,'' International Journal of Applied Mathematics & Statistics, vol. 7, no. Fe07, pp. 143--156, 2007.
  • J. Rätz, ``On inequalities associated with the Jordan-von Neumann functional equation,'' Aequationes Mathematicae, vol. 66, no. 1-2, pp. 191--200, 2003.
  • 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.
  • 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.
  • C. Baak, ``Cauchy-Rassias stability of Cauchy-Jensen additive mappings in Banach spaces,'' Acta Mathematica Sinica, vol. 22, no. 6, pp. 1789--1796, 2006. \endthebibliography