×

Fixed points and stability of the Cauchy functional equation in \(C^{\ast }\)-algebras. (English) Zbl 1167.39019

Summary: Using the fixed point method, we prove the generalized Hyers-Ulam stability of homomorphisms in \(C^{\ast }\)-algebras and Lie \(C^{\ast }\)-algebras and of derivations on \(C^{\ast }\)-algebras and Lie \(C^{\ast }\)-algebras for the Cauchy functional equation.

MSC:

39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
46L57 Derivations, dissipations and positive semigroups in \(C^*\)-algebras
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Ulam SM: A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8. Interscience, New York, NY, USA; 1960:xiii+150.
[2] Hyers DH: On the stability of the linear functional equation.Proceedings of the National Academy of Sciences of the United States of America 1941,27(4):222-224. 10.1073/pnas.27.4.222 · Zbl 0061.26403 · doi:10.1073/pnas.27.4.222
[3] Aoki T: On the stability of the linear transformation in Banach spaces.Journal of the Mathematical Society of Japan 1950, 2: 64-66. 10.2969/jmsj/00210064 · Zbl 0040.35501 · doi:10.2969/jmsj/00210064
[4] Rassias ThM: On the stability of the linear mapping in Banach spaces.Proceedings of the American Mathematical Society 1978,72(2):297-300. 10.1090/S0002-9939-1978-0507327-1 · Zbl 0398.47040 · doi:10.1090/S0002-9939-1978-0507327-1
[5] Găvruţa P: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings.Journal of Mathematical Analysis and Applications 1994,184(3):431-436. 10.1006/jmaa.1994.1211 · Zbl 0818.46043 · doi:10.1006/jmaa.1994.1211
[6] Park C-G: On the stability of the linear mapping in Banach modules.Journal of Mathematical Analysis and Applications 2002,275(2):711-720. 10.1016/S0022-247X(02)00386-4 · Zbl 1021.46037 · doi:10.1016/S0022-247X(02)00386-4
[7] Park C-G: Modified Trif’s functional equations in Banach modules over a [InlineEquation not available: see fulltext.]-algebra and approximate algebra homomorphisms Journal of Mathematical Analysis and Applications 2003,278(1):93-108. 10.1016/S0022-247X(02)00573-5 · Zbl 1046.39022 · doi:10.1016/S0022-247X(02)00573-5
[8] Park C-G: On an approximate automorphism on a [InlineEquation not available: see fulltext.]-algebra Proceedings of the American Mathematical Society 2004,132(6):1739-1745. 10.1090/S0002-9939-03-07252-6 · Zbl 1055.47032 · doi:10.1090/S0002-9939-03-07252-6
[9] Park C-G: Lie [InlineEquation not available: see fulltext.]-homomorphisms between Lie [InlineEquation not available: see fulltext.]-algebras and Lie [InlineEquation not available: see fulltext.]-derivations on Lie [InlineEquation not available: see fulltext.]-algebras Journal of Mathematical Analysis and Applications 2004,293(2):419-434. 10.1016/j.jmaa.2003.10.051 · Zbl 1051.46052 · doi:10.1016/j.jmaa.2003.10.051
[10] Park C-G: Homomorphisms between Lie [InlineEquation not available: see fulltext.]-algebras and Cauchy-Rassias stability of Lie [InlineEquation not available: see fulltext.]-algebra derivations Journal of Lie Theory 2005,15(2):393-414. · Zbl 1091.39006
[11] Park C-G: Homomorphisms between Poisson [InlineEquation not available: see fulltext.]-algebras Bulletin of the Brazilian Mathematical Society 2005,36(1):79-97. 10.1007/s00574-005-0029-z · Zbl 1091.39007 · doi:10.1007/s00574-005-0029-z
[12] Park C-G: Hyers-Ulam-Rassias stability of a generalized Euler-Lagrange type additive mapping and isomorphisms between [InlineEquation not available: see fulltext.]-algebras Bulletin of the Belgian Mathematical Society. Simon Stevin 2006,13(4):619-632. · Zbl 1125.39027
[13] Park, C., Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras, 15 (2007) · Zbl 1167.39018
[14] Park, C.; Cho, YS; Han, M-H, Functional inequalities associated with Jordan-von Neumann-type additive functional equations, 13 (2007) · Zbl 1133.39024
[15] Park, C.; Cui, J., Generalized stability of [InlineEquation not available: see fulltext.]-ternary quadratic mappings, 6 (2007)
[16] Park C-G, Hou J: Homomorphisms between -algebras associated with the Trif functional equation and linear derivations on -algebras.Journal of the Korean Mathematical Society 2004,41(3):461-477. · Zbl 1058.39025 · doi:10.4134/JKMS.2004.41.3.461
[17] Rassias ThM: Problem 16; 2, Report of the 27th International Symposium on Functional Equations.Aequationes Mathematicae 1990,39(2-3):292-293, 309.
[18] Rassias ThM: The problem of S. M. Ulam for approximately multiplicative mappings.Journal of Mathematical Analysis and Applications 2000,246(2):352-378. 10.1006/jmaa.2000.6788 · Zbl 0958.46022 · doi:10.1006/jmaa.2000.6788
[19] Rassias ThM: On the stability of functional equations in Banach spaces.Journal of Mathematical Analysis and Applications 2000,251(1):264-284. 10.1006/jmaa.2000.7046 · Zbl 0964.39026 · doi:10.1006/jmaa.2000.7046
[20] Rassias JM: On approximation of approximately linear mappings by linear mappings.Journal of Functional Analysis 1982,46(1):126-130. 10.1016/0022-1236(82)90048-9 · Zbl 0482.47033 · doi:10.1016/0022-1236(82)90048-9
[21] Rassias JM: On approximation of approximately linear mappings by linear mappings.Bulletin des Sciences Mathématiques 1984,108(4):445-446. · Zbl 0599.47106
[22] Rassias JM: Solution of a problem of Ulam.Journal of Approximation Theory 1989,57(3):268-273. 10.1016/0021-9045(89)90041-5 · Zbl 0672.41027 · doi:10.1016/0021-9045(89)90041-5
[23] Cădariu L, Radu V: Fixed points and the stability of Jensen’s functional equation.Journal of Inequalities in Pure and Applied Mathematics 2003,4(1, article 4):1-7. · Zbl 1043.39010
[24] Diaz JB, Margolis B: A fixed point theorem of the alternative, for contractions on a generalized complete metric space.Bulletin of the American Mathematical Society 1968,74(2):305-309. 10.1090/S0002-9904-1968-11933-0 · Zbl 0157.29904 · doi:10.1090/S0002-9904-1968-11933-0
[25] Fleming RJ, Jamison JE: Isometries on Banach Spaces: Function Spaces, Chapman & Hall/CRC Monographs and Surveys in Pure & Applied Mathematics. Volume 129. Chapman & Hall/CRC, Boca Raton, Fla, USA; 2003:x+197.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.