# American Institute of Mathematical Sciences

December  2011, 31(4): 1469-1477. doi: 10.3934/dcds.2011.31.1469

## Hyers--Ulam--Rassias stability of derivations in proper Jordan $CQ^{*}$-algebras

 1 Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran, Iran

Received  October 2009 Revised  February 2010 Published  September 2011

In this paper, we investigate derivation in proper Jordan $CQ^{*}$-algebras associated with the following Pexiderized Jensen type functional equation $kf(\frac{x+y}{k}) = f_{0}(x)+ f_{1} (y).$ This is applied to investigate derivations and their Hyers--Ulam--Rassias stability in proper Jordan $CQ^{*}$-algebras.
Citation: Golamreza Zamani Eskandani, Hamid Vaezi. Hyers--Ulam--Rassias stability of derivations in proper Jordan $CQ^{*}$-algebras. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1469-1477. doi: 10.3934/dcds.2011.31.1469
##### References:
 [1] J.-P. Antoine, A. Inoue and C. Trapani, "Partial *-Algebras and Their Operator Realizations," Mathematics and its Applications, 553, Kluwer Academic Publishers, Dordrecht, 2002. [2] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2 (1950), 64-66. [3] F. Bagarello, A. Inoue and C. Trapani, Some classes of topological quasi *-algebras, Proc. Amer. Math. Soc., 129 (2001), 2973-2980. doi: 10.1090/S0002-9939-01-06019-1. [4] F. Bagarello and G. Morchio, Dynamics of mean-field spin models from basic results in abstract differential equations, J. Stat. Phys., 66 (1992), 849-866. doi: 10.1007/BF01055705. [5] F. Bagarello and C. Trapani, States and representations of $CQ$*-algebras, Ann. Inst. H. Poincaré Phys. Théor., 61 (1994), 103-133. [6] F. Bagarello and C. Trapani, $CQ$*-algebras: Structure properties, Publ. Res. Inst. Math. Sci., 32 (1996), 85-116. doi: 10.2977/prims/1195163181. [7] F. Bagarello and C. Trapani, Morphisms of certain Banach $C$*-modules, Publ. Res. Inst. Math. Sci., 36 (2000), 681-705. doi: 10.2977/prims/1195139642. [8] S. Czerwik, "Stability of Functional Equations of Ulam-Hyers-Rassias Type," Hadronic Press, Inc., Palm Harbor, USA, pp. 200. [9] S. Czerwik, "Functional Equations and Inequalities in Several Variables," World Scientific Publishing Co., Inc., River Edge, NJ, 2002. [10] G. O. S. Ekhaguere, Partial $W$*-dynamical systems, in "Current Topics in Operator Algebras" (Nara, 1990), World Scientific Publ., River Edge, NJ, (1991), 202-217. [11] G. Z. Eskandani, On the Hyers-–Ulam-–Rassias stability of an additive functional equation in quasi-Banach spaces, J. Math. Anal. Appl., 345 (2008), 405-409. doi: 10.1016/j.jmaa.2008.03.039. [12] G. Z. Eskandani, H. Vaezi and Y. N. Dehghan, Stability of a mixed additive and quadratic functional equation in non-Archimedean Banach modules, Taiwanese J. Math., 14 (2010), 1309-1324. [13] P. Găvruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184 (1994), 431-436. doi: 10.1006/jmaa.1994.1211. [14] R. Haag and D. Kastler, An algebraic approach to quantum field theory, J. Math. Phys., 5 (1964), 848-861. doi: 10.1063/1.1704187. [15] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA, 27 (1941), 222-224. doi: 10.1073/pnas.27.4.222. [16] D. H. Hyers, G. Isac Th. M. Rassias, "Stability of Functional Equations in Several Variables," Progress in Nonlinear Differential Equations and their Applications, 34, Birkhäuser Boston, Inc., Boston, MA, 1998. [17] D. H. Hyers and Th. M. Rassias, Approximate homomorphisms, Aequationes Math., 44 (1992), 125-153. doi: 10.1007/BF01830975. [18] S.-M. Jung, "Hyers-Ulam-Rassias Stability of Functional Equations in Mathimatical Analysis," Hadronic Press, Palm Harbor, FL, 2001. [19] Z. Kominek, On a local stability of the Jensen functional equation, Demonstratio Math., 22 (1989), 499-507. [20] G. Lassner, Topological algebras and their applications in quantum statistics, Wiss. Z. KMU, Leipzig, Math.-Nat. R., 30 (1981), 572-595. [21] G. Lassner and G. A. Lassner, Quasi* -algebras and twisted product, Publ. RIMS, 25 (1989), 279-299. doi: 10.2977/prims/1195173612. [22] F. Moradlou, H. Vaezi and C. Park, Fixed points and stability of an additive functional equation of $n$-Apollonius type in $C$*-algebras, Abstract and Applied Analysis, 2008, Article ID 672618, 13 pp. doi: 10.1155/2008/672618. [23] F. Moradlou, H. Vaezi and G. Z. Eskandani, Hyers-–Ulam-–Rassias stability of a quadratic and additive functional equation in quasi-Banach spaces, Mediterr. J. of Math., 6 (2009), 233-248. [24] A. Najati and G. Z. Eskandani, Stability of a mixed additive and cubic functional equation in quasi-Banach spaces, J. Math. Anal. Appl., 342 (2008), 1318-1331. doi: 10.1016/j.jmaa.2007.12.039. [25] C. Park, Homomorphisms between Poisson $JC$*-algebras, Bull. Braz. Math. Soc., 36 (2005), 79-97. doi: 10.1007/s00574-005-0029-z. [26] C. Park and Th. M. Rassias, Homomorphisms and derivations in proper $JCQ$*-triples, J. Math. Anal. Appl., 337 (2008), 1404-1414. doi: 10.1016/j.jmaa.2007.04.063. [27] J. C. Parnami and H. L. Vasudeva, On Jensen’s functional equation, Aequationes Math., 43 (1992), 211-218. doi: 10.1007/BF01835703. [28] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297-300. doi: 10.1090/S0002-9939-1978-0507327-1. [29] Th. M. Rassias, On a modified Hyers-Ulam sequence, J. Math. Anal. Appl., 158 (1991), 106-113. doi: 10.1016/0022-247X(91)90270-A. [30] Th. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Applicandae Mathematicae, 62 (2000), 23-130. doi: 10.1023/A:1006499223572. [31] Th. M. Rassias and P. Šemrl, On the Hyers-Ulam stability of linear mappings, J. Math. Anal. Appl., 173 (1993), 325-338. doi: 10.1006/jmaa.1993.1070. [32] G. L. Sewell, "Quantum Mechanics and its Emergent Macrophysics," Princeton Univ. Press, Princeton, NJ, 2002. [33] C. Trapani, Quasi-*-algebras of operators and their applications, Rev. Math. Phys., 7 (1995), 1303-1332. doi: 10.1142/S0129055X95000475. [34] S. M. Ulam, "A Collection of the Mathematical Problems," Interscience Tracts in Pure and Applied Mathematics, 8, Interscience Publ., New York-London, 1960.

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##### References:
 [1] J.-P. Antoine, A. Inoue and C. Trapani, "Partial *-Algebras and Their Operator Realizations," Mathematics and its Applications, 553, Kluwer Academic Publishers, Dordrecht, 2002. [2] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2 (1950), 64-66. [3] F. Bagarello, A. Inoue and C. Trapani, Some classes of topological quasi *-algebras, Proc. Amer. Math. Soc., 129 (2001), 2973-2980. doi: 10.1090/S0002-9939-01-06019-1. [4] F. Bagarello and G. Morchio, Dynamics of mean-field spin models from basic results in abstract differential equations, J. Stat. Phys., 66 (1992), 849-866. doi: 10.1007/BF01055705. [5] F. Bagarello and C. Trapani, States and representations of $CQ$*-algebras, Ann. Inst. H. Poincaré Phys. Théor., 61 (1994), 103-133. [6] F. Bagarello and C. Trapani, $CQ$*-algebras: Structure properties, Publ. Res. Inst. Math. Sci., 32 (1996), 85-116. doi: 10.2977/prims/1195163181. [7] F. Bagarello and C. Trapani, Morphisms of certain Banach $C$*-modules, Publ. Res. Inst. Math. Sci., 36 (2000), 681-705. doi: 10.2977/prims/1195139642. [8] S. Czerwik, "Stability of Functional Equations of Ulam-Hyers-Rassias Type," Hadronic Press, Inc., Palm Harbor, USA, pp. 200. [9] S. Czerwik, "Functional Equations and Inequalities in Several Variables," World Scientific Publishing Co., Inc., River Edge, NJ, 2002. [10] G. O. S. Ekhaguere, Partial $W$*-dynamical systems, in "Current Topics in Operator Algebras" (Nara, 1990), World Scientific Publ., River Edge, NJ, (1991), 202-217. [11] G. Z. Eskandani, On the Hyers-–Ulam-–Rassias stability of an additive functional equation in quasi-Banach spaces, J. Math. Anal. Appl., 345 (2008), 405-409. doi: 10.1016/j.jmaa.2008.03.039. [12] G. Z. Eskandani, H. Vaezi and Y. N. Dehghan, Stability of a mixed additive and quadratic functional equation in non-Archimedean Banach modules, Taiwanese J. Math., 14 (2010), 1309-1324. [13] P. Găvruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184 (1994), 431-436. doi: 10.1006/jmaa.1994.1211. [14] R. Haag and D. Kastler, An algebraic approach to quantum field theory, J. Math. Phys., 5 (1964), 848-861. doi: 10.1063/1.1704187. [15] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA, 27 (1941), 222-224. doi: 10.1073/pnas.27.4.222. [16] D. H. Hyers, G. Isac Th. M. Rassias, "Stability of Functional Equations in Several Variables," Progress in Nonlinear Differential Equations and their Applications, 34, Birkhäuser Boston, Inc., Boston, MA, 1998. [17] D. H. Hyers and Th. M. Rassias, Approximate homomorphisms, Aequationes Math., 44 (1992), 125-153. doi: 10.1007/BF01830975. [18] S.-M. Jung, "Hyers-Ulam-Rassias Stability of Functional Equations in Mathimatical Analysis," Hadronic Press, Palm Harbor, FL, 2001. [19] Z. Kominek, On a local stability of the Jensen functional equation, Demonstratio Math., 22 (1989), 499-507. [20] G. Lassner, Topological algebras and their applications in quantum statistics, Wiss. Z. KMU, Leipzig, Math.-Nat. R., 30 (1981), 572-595. [21] G. Lassner and G. A. Lassner, Quasi* -algebras and twisted product, Publ. RIMS, 25 (1989), 279-299. doi: 10.2977/prims/1195173612. [22] F. Moradlou, H. Vaezi and C. Park, Fixed points and stability of an additive functional equation of $n$-Apollonius type in $C$*-algebras, Abstract and Applied Analysis, 2008, Article ID 672618, 13 pp. doi: 10.1155/2008/672618. [23] F. Moradlou, H. Vaezi and G. Z. Eskandani, Hyers-–Ulam-–Rassias stability of a quadratic and additive functional equation in quasi-Banach spaces, Mediterr. J. of Math., 6 (2009), 233-248. [24] A. Najati and G. Z. Eskandani, Stability of a mixed additive and cubic functional equation in quasi-Banach spaces, J. Math. Anal. Appl., 342 (2008), 1318-1331. doi: 10.1016/j.jmaa.2007.12.039. [25] C. Park, Homomorphisms between Poisson $JC$*-algebras, Bull. Braz. Math. Soc., 36 (2005), 79-97. doi: 10.1007/s00574-005-0029-z. [26] C. Park and Th. M. Rassias, Homomorphisms and derivations in proper $JCQ$*-triples, J. Math. Anal. Appl., 337 (2008), 1404-1414. doi: 10.1016/j.jmaa.2007.04.063. [27] J. C. Parnami and H. L. Vasudeva, On Jensen’s functional equation, Aequationes Math., 43 (1992), 211-218. doi: 10.1007/BF01835703. [28] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297-300. doi: 10.1090/S0002-9939-1978-0507327-1. [29] Th. M. Rassias, On a modified Hyers-Ulam sequence, J. Math. Anal. Appl., 158 (1991), 106-113. doi: 10.1016/0022-247X(91)90270-A. [30] Th. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Applicandae Mathematicae, 62 (2000), 23-130. doi: 10.1023/A:1006499223572. [31] Th. M. Rassias and P. Šemrl, On the Hyers-Ulam stability of linear mappings, J. Math. Anal. Appl., 173 (1993), 325-338. doi: 10.1006/jmaa.1993.1070. [32] G. L. Sewell, "Quantum Mechanics and its Emergent Macrophysics," Princeton Univ. Press, Princeton, NJ, 2002. [33] C. Trapani, Quasi-*-algebras of operators and their applications, Rev. Math. Phys., 7 (1995), 1303-1332. doi: 10.1142/S0129055X95000475. [34] S. M. Ulam, "A Collection of the Mathematical Problems," Interscience Tracts in Pure and Applied Mathematics, 8, Interscience Publ., New York-London, 1960.
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