Jordan ${\ast}$-Derivations on ${C}^{\ast}$-Algebras and $J{C}^{\ast}$-Algebras

Jordan ${\ast}$-Derivations on ${C}^{\ast}$-Algebras and $J{C}^{\ast}$-Algebras

Jong Su An, Jianlian Cui, and Choonkil Park

Source: Abstr. Appl. Anal. Volume 2008 (2008), 12 pages.

Abstract

We investigate Jordan ${\ast}$-derivations on ${C}^{\ast}$-algebras and Jordan ${\ast}$-derivations on $J{C}^{\ast}$-algebras associated with the following functional inequality $\Vert f(x)+f(y)+kf(z)\Vert \leq \Vert kf((x+y)/k+z)\Vert $ for some integer $k$ greater than 1. We moreover prove the generalized Hyers-Ulam stability of Jordan ${\ast}$-derivations on $${C}^{\ast}$$-algebras and of Jordan ${\ast}$-derivations on $JC^{\ast}$-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.

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Permanent link to this document: http://projecteuclid.org/euclid.aaa/1234299003
Digital Object Identifier: doi:10.1155/2008/410437
Mathematical Reviews number (MathSciNet): MR2471252

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References

S. M. Ulam, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8, Interscience, New York, NY, USA, 1960.

Mathematical Reviews (MathSciNet): MR0120127

Zentralblatt MATH: 0086.24101

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.

Mathematical Reviews (MathSciNet): MR0004076

Zentralblatt MATH: 0061.26403

Digital Object Identifier: doi:10.1073/pnas.27.4.222

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.

Mathematical Reviews (MathSciNet): MR0040580

Zentralblatt MATH: 0040.35501

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.

Mathematical Reviews (MathSciNet): MR507327

Zentralblatt MATH: 0398.47040

Digital Object Identifier: doi:10.2307/2042795

J. M. Rassias, ``On approximation of approximately linear mappings by linear mappings,'' Journal of Functional Analysis, vol. 46, no. 1, pp. 126--130, 1982.

Mathematical Reviews (MathSciNet): MR654469

Zentralblatt MATH: 0482.47033

Digital Object Identifier: doi:10.1016/0022-1236(82)90048-9

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.

Mathematical Reviews (MathSciNet): MR784679

Zentralblatt MATH: 0599.47106

J. M. Rassias, ``Solution of a problem of Ulam,'' Journal of Approximation Theory, vol. 57, no. 3, pp. 268--273, 1989.

Mathematical Reviews (MathSciNet): MR999861

Zentralblatt MATH: 0672.41027

Digital Object Identifier: doi:10.1016/0021-9045(89)90041-5

J. M. Rassias, ``Solution of a stability problem of Ulam,'' Discussiones Mathematicae, vol. 12, pp. 95--103, 1992.

Mathematical Reviews (MathSciNet): MR1221875

Zentralblatt MATH: 0779.47005

J. M. Rassias, ``Complete solution of the multi-dimensional problem of Ulam,'' Discussiones Mathematicae, vol. 14, pp. 101--107, 1994.

Mathematical Reviews (MathSciNet): MR1323977

Zentralblatt MATH: 0819.39012

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.

Mathematical Reviews (MathSciNet): MR1167074

Zentralblatt MATH: 0753.39003

J. M. Rassias, ``On the stability of the general Euler-Lagrange functional equation,'' Demonstratio Mathematica, vol. 29, no. 4, pp. 755--766, 1996.

Mathematical Reviews (MathSciNet): MR1429514

Zentralblatt MATH: 0884.47040

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.

Mathematical Reviews (MathSciNet): MR2150471

Zentralblatt MATH: 1094.39027

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.

Mathematical Reviews (MathSciNet): MR1110036

Zentralblatt MATH: 0739.39013

Digital Object Identifier: doi:10.1155/S016117129100056X

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.

Mathematical Reviews (MathSciNet): MR1059634

Zentralblatt MATH: 0761.47004

Digital Object Identifier: doi:10.2307/2159617

JSTOR: links.jstor.org

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.

Mathematical Reviews (MathSciNet): MR1281518

Zentralblatt MATH: 0818.46043

Digital Object Identifier: doi:10.1006/jmaa.1994.1211

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.

Mathematical Reviews (MathSciNet): MR1418531

Zentralblatt MATH: 0888.46018

Digital Object Identifier: doi:10.1006/jmaa.1996.0433

S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, River Edge, NJ, USA, 2002.

Mathematical Reviews (MathSciNet): MR1904790

Zentralblatt MATH: 1011.39019

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.

Mathematical Reviews (MathSciNet): MR1639801

Zentralblatt MATH: 0907.39025

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.

Mathematical Reviews (MathSciNet): MR1375983

Zentralblatt MATH: 0843.47036

Digital Object Identifier: doi:10.1155/S0161171296000324

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.

Mathematical Reviews (MathSciNet): MR1415589

Zentralblatt MATH: 0894.39012

Digital Object Identifier: doi:10.1090/S0002-9939-98-04060-X

JSTOR: links.jstor.org

D. G. Bourgin, ``Classes of transformations and bordering transformations,'' Bulletin of the American Mathematical Society, vol. 57, pp. 223--237, 1951.

Mathematical Reviews (MathSciNet): MR0042613

Zentralblatt MATH: 0043.32902

Digital Object Identifier: doi:10.1090/S0002-9904-1951-09511-7

Project Euclid: euclid.bams/1183516108

G. L. Forti, ``Hyers-Ulam stability of functional equations in several variables,'' Aequationes Mathematicae, vol. 50, no. 1-2, pp. 143--190, 1995.

Mathematical Reviews (MathSciNet): MR1336866

Zentralblatt MATH: 0836.39007

Digital Object Identifier: doi:10.1007/BF01831117

A. Gilányi, ``On the stability of monomial functional equations,'' Publicationes Mathematicae Debrecen, vol. 56, no. 1-2, pp. 201--212, 2000.

Mathematical Reviews (MathSciNet): MR1740505

Zentralblatt MATH: 0991.39016

A. Gilányi, ``Eine zur Parallelogrammgleichung äquivalente Ungleichung,'' Aequationes Mathematicae, vol. 62, no. 3, pp. 303--309, 2001.

Mathematical Reviews (MathSciNet): MR1869580

Zentralblatt MATH: 0992.39026

Digital Object Identifier: doi:10.1007/PL00000156

A. Gilányi, ``On a problem by K. Nikodem,'' Mathematical Inequalities & Applications, vol. 5, no. 4, pp. 707--710, 2002.

Mathematical Reviews (MathSciNet): MR1931231

Zentralblatt MATH: 1036.39020

P. M. Gruber, ``Stability of isometries,'' Transactions of the American Mathematical Society, vol. 245, pp. 263--277, 1978.

Mathematical Reviews (MathSciNet): MR511409

Zentralblatt MATH: 0393.41020

Digital Object Identifier: doi:10.2307/1998866

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.

Mathematical Reviews (MathSciNet): MR2324341

Zentralblatt MATH: 1117.39017

Digital Object Identifier: doi:10.1016/j.jmaa.2006.11.024

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.

Mathematical Reviews (MathSciNet): MR2364957

Zentralblatt MATH: 1135.39013

Digital Object Identifier: doi:10.1080/10236190701464590

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.

Mathematical Reviews (MathSciNet): MR1623875

Zentralblatt MATH: 0928.39013

Digital Object Identifier: doi:10.1006/jmaa.1998.5916

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.

Mathematical Reviews (MathSciNet): MR1683116

Zentralblatt MATH: 0926.39013

Digital Object Identifier: doi:10.1006/jmaa.1999.6282

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.

Mathematical Reviews (MathSciNet): MR2377530

Zentralblatt MATH: 1132.39024

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.

Mathematical Reviews (MathSciNet): MR0131187

Zentralblatt MATH: 0096.31501

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.

Mathematical Reviews (MathSciNet): MR2423047

Digital Object Identifier: doi:10.1016/j.aml.2007.07.022

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.

Mathematical Reviews (MathSciNet): MR2320774

Zentralblatt MATH: 1148.39026

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.

Mathematical Reviews (MathSciNet): MR2053888

Zentralblatt MATH: 1051.46052

Digital Object Identifier: doi:10.1016/j.jmaa.2003.10.051

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.

Mathematical Reviews (MathSciNet): MR2147435

Zentralblatt MATH: 1091.39006

C.-G. Park, ``Homomorphisms between Poisson $JC^\ast$-algebras,'' Bulletin of the Brazilian Mathematical Society, vol. 36, no. 1, pp. 79--97, 2005.

Mathematical Reviews (MathSciNet): MR2132832

Zentralblatt MATH: 1091.39007

Digital Object Identifier: doi:10.1007/s00574-005-0029-z

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.

Mathematical Reviews (MathSciNet): MR2300619

Zentralblatt MATH: 1125.39027

Project Euclid: euclid.bbms/1168957339

C. Park, ``Isomorphisms between quasi-Banach algebras,'' Chinese Annals of Mathematics. Series B, vol. 28, no. 3, pp. 353--362, 2007.

Mathematical Reviews (MathSciNet): MR2339439

Zentralblatt MATH: 1127.39056

Digital Object Identifier: doi:10.1007/s11401-005-0427-y

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.

Mathematical Reviews (MathSciNet): MR2274272

Zentralblatt MATH: 1133.39024

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.

Mathematical Reviews (MathSciNet): MR2050156

Zentralblatt MATH: 1058.39025

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.

Mathematical Reviews (MathSciNet): MR2190008

Zentralblatt MATH: 1121.39030

Digital Object Identifier: doi:10.1007/s10114-005-0629-y

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.

Mathematical Reviews (MathSciNet): MR1982070

Zentralblatt MATH: 1053.46029

Digital Object Identifier: doi:10.1142/S0252959903000244

A. Pietrzyk, ``Stability of the Euler-Lagrange-Rassias functional equation,'' Demonstratio Mathematica, vol. 39, no. 3, pp. 523--530, 2006.

Mathematical Reviews (MathSciNet): MR2262001

Zentralblatt MATH: 1113.39034

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.

Mathematical Reviews (MathSciNet): MR1761936

Zentralblatt MATH: 0958.46022

Digital Object Identifier: doi:10.1006/jmaa.2000.6788

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.

Mathematical Reviews (MathSciNet): MR1790409

Zentralblatt MATH: 0964.39026

Digital Object Identifier: doi:10.1006/jmaa.2000.7046

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.

Mathematical Reviews (MathSciNet): MR1778016

Zentralblatt MATH: 0981.39014

Digital Object Identifier: doi:10.1023/A:1006499223572

Th. M. Rassias, Functional Equations, Inequalities and Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2003.

Mathematical Reviews (MathSciNet): MR2042552

Zentralblatt MATH: 1047.39001

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.

Mathematical Reviews (MathSciNet): MR2339517

J. Rätz, ``On inequalities associated with the Jordan-von Neumann functional equation,'' Aequationes Mathematicae, vol. 66, no. 1-2, pp. 191--200, 2003.

Mathematical Reviews (MathSciNet): MR2003465

Zentralblatt MATH: 1078.39026

Digital Object Identifier: doi:10.1007/s00010-003-2684-8

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.

Mathematical Reviews (MathSciNet): MR2214820

Zentralblatt MATH: 1098.39019

Digital Object Identifier: doi:10.1007/s00010-005-2775-9

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.

Mathematical Reviews (MathSciNet): MR2296883

Zentralblatt MATH: 1153.39308

Digital Object Identifier: doi:10.1016/j.jmaa.2006.07.027

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

Mathematical Reviews (MathSciNet): MR2262438

Zentralblatt MATH: 1118.39012

Digital Object Identifier: doi:10.1007/s10114-005-0697-z

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