# 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 & Continuous Dynamical Systems - A, 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 (2002). Google Scholar [2] T. Aoki, On the stability of the linear transformation in Banach spaces,, J. Math. Soc. Japan, 2 (1950), 64. Google Scholar [3] F. Bagarello, A. Inoue and C. Trapani, Some classes of topological quasi *-algebras,, Proc. Amer. Math. Soc., 129 (2001), 2973. doi: 10.1090/S0002-9939-01-06019-1. Google Scholar [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. doi: 10.1007/BF01055705. Google Scholar [5] F. Bagarello and C. Trapani, States and representations of $CQ$*-algebras,, Ann. Inst. H. Poincaré Phys. Théor., 61 (1994), 103. Google Scholar [6] F. Bagarello and C. Trapani, $CQ$*-algebras: Structure properties,, Publ. Res. Inst. Math. Sci., 32 (1996), 85. doi: 10.2977/prims/1195163181. Google Scholar [7] F. Bagarello and C. Trapani, Morphisms of certain Banach $C$*-modules,, Publ. Res. Inst. Math. Sci., 36 (2000), 681. doi: 10.2977/prims/1195139642. Google Scholar [8] S. Czerwik, "Stability of Functional Equations of Ulam-Hyers-Rassias Type,", Hadronic Press, (). Google Scholar [9] S. Czerwik, "Functional Equations and Inequalities in Several Variables,", World Scientific Publishing Co., (2002). Google Scholar [10] G. O. S. Ekhaguere, Partial $W$*-dynamical systems,, in, (1991), 202. Google Scholar [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. doi: 10.1016/j.jmaa.2008.03.039. Google Scholar [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. Google Scholar [13] P. Găvruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings,, J. Math. Anal. Appl., 184 (1994), 431. doi: 10.1006/jmaa.1994.1211. Google Scholar [14] R. Haag and D. Kastler, An algebraic approach to quantum field theory,, J. Math. Phys., 5 (1964), 848. doi: 10.1063/1.1704187. Google Scholar [15] D. H. Hyers, On the stability of the linear functional equation,, Proc. Nat. Acad. Sci. USA, 27 (1941), 222. doi: 10.1073/pnas.27.4.222. Google Scholar [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 (1998). Google Scholar [17] D. H. Hyers and Th. M. Rassias, Approximate homomorphisms,, Aequationes Math., 44 (1992), 125. doi: 10.1007/BF01830975. Google Scholar [18] S.-M. Jung, "Hyers-Ulam-Rassias Stability of Functional Equations in Mathimatical Analysis,", Hadronic Press, (2001). Google Scholar [19] Z. Kominek, On a local stability of the Jensen functional equation,, Demonstratio Math., 22 (1989), 499. Google Scholar [20] G. Lassner, Topological algebras and their applications in quantum statistics,, Wiss. Z. KMU, 30 (1981), 572. Google Scholar [21] G. Lassner and G. A. Lassner, Quasi* -algebras and twisted product,, Publ. RIMS, 25 (1989), 279. doi: 10.2977/prims/1195173612. Google Scholar [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 (6726). doi: 10.1155/2008/672618. Google Scholar [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. Google Scholar [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. doi: 10.1016/j.jmaa.2007.12.039. Google Scholar [25] C. Park, Homomorphisms between Poisson $JC$*-algebras,, Bull. Braz. Math. Soc., 36 (2005), 79. doi: 10.1007/s00574-005-0029-z. Google Scholar [26] C. Park and Th. M. Rassias, Homomorphisms and derivations in proper $JCQ$*-triples,, J. Math. Anal. Appl., 337 (2008), 1404. doi: 10.1016/j.jmaa.2007.04.063. Google Scholar [27] J. C. Parnami and H. L. Vasudeva, On Jensen’s functional equation,, Aequationes Math., 43 (1992), 211. doi: 10.1007/BF01835703. Google Scholar [28] Th. M. Rassias, On the stability of the linear mapping in Banach spaces,, Proc. Amer. Math. Soc., 72 (1978), 297. doi: 10.1090/S0002-9939-1978-0507327-1. Google Scholar [29] Th. M. Rassias, On a modified Hyers-Ulam sequence,, J. Math. Anal. Appl., 158 (1991), 106. doi: 10.1016/0022-247X(91)90270-A. Google Scholar [30] Th. M. Rassias, On the stability of functional equations and a problem of Ulam,, Acta Applicandae Mathematicae, 62 (2000), 23. doi: 10.1023/A:1006499223572. Google Scholar [31] Th. M. Rassias and P. Šemrl, On the Hyers-Ulam stability of linear mappings,, J. Math. Anal. Appl., 173 (1993), 325. doi: 10.1006/jmaa.1993.1070. Google Scholar [32] G. L. Sewell, "Quantum Mechanics and its Emergent Macrophysics,", Princeton Univ. Press, (2002). Google Scholar [33] C. Trapani, Quasi-*-algebras of operators and their applications,, Rev. Math. Phys., 7 (1995), 1303. doi: 10.1142/S0129055X95000475. Google Scholar [34] S. M. Ulam, "A Collection of the Mathematical Problems,", Interscience Tracts in Pure and Applied Mathematics, 8 (1960). Google Scholar

show all references

##### References:
 [1] J.-P. Antoine, A. Inoue and C. Trapani, "Partial *-Algebras and Their Operator Realizations,", Mathematics and its Applications, 553 (2002). Google Scholar [2] T. Aoki, On the stability of the linear transformation in Banach spaces,, J. Math. Soc. Japan, 2 (1950), 64. Google Scholar [3] F. Bagarello, A. Inoue and C. Trapani, Some classes of topological quasi *-algebras,, Proc. Amer. Math. Soc., 129 (2001), 2973. doi: 10.1090/S0002-9939-01-06019-1. Google Scholar [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. doi: 10.1007/BF01055705. Google Scholar [5] F. Bagarello and C. Trapani, States and representations of $CQ$*-algebras,, Ann. Inst. H. Poincaré Phys. Théor., 61 (1994), 103. Google Scholar [6] F. Bagarello and C. Trapani, $CQ$*-algebras: Structure properties,, Publ. Res. Inst. Math. Sci., 32 (1996), 85. doi: 10.2977/prims/1195163181. Google Scholar [7] F. Bagarello and C. Trapani, Morphisms of certain Banach $C$*-modules,, Publ. Res. Inst. Math. Sci., 36 (2000), 681. doi: 10.2977/prims/1195139642. Google Scholar [8] S. Czerwik, "Stability of Functional Equations of Ulam-Hyers-Rassias Type,", Hadronic Press, (). Google Scholar [9] S. Czerwik, "Functional Equations and Inequalities in Several Variables,", World Scientific Publishing Co., (2002). Google Scholar [10] G. O. S. Ekhaguere, Partial $W$*-dynamical systems,, in, (1991), 202. Google Scholar [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. doi: 10.1016/j.jmaa.2008.03.039. Google Scholar [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. Google Scholar [13] P. Găvruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings,, J. Math. Anal. Appl., 184 (1994), 431. doi: 10.1006/jmaa.1994.1211. Google Scholar [14] R. Haag and D. Kastler, An algebraic approach to quantum field theory,, J. Math. Phys., 5 (1964), 848. doi: 10.1063/1.1704187. Google Scholar [15] D. H. Hyers, On the stability of the linear functional equation,, Proc. Nat. Acad. Sci. USA, 27 (1941), 222. doi: 10.1073/pnas.27.4.222. Google Scholar [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 (1998). Google Scholar [17] D. H. Hyers and Th. M. Rassias, Approximate homomorphisms,, Aequationes Math., 44 (1992), 125. doi: 10.1007/BF01830975. Google Scholar [18] S.-M. Jung, "Hyers-Ulam-Rassias Stability of Functional Equations in Mathimatical Analysis,", Hadronic Press, (2001). Google Scholar [19] Z. Kominek, On a local stability of the Jensen functional equation,, Demonstratio Math., 22 (1989), 499. Google Scholar [20] G. Lassner, Topological algebras and their applications in quantum statistics,, Wiss. Z. KMU, 30 (1981), 572. Google Scholar [21] G. Lassner and G. A. Lassner, Quasi* -algebras and twisted product,, Publ. RIMS, 25 (1989), 279. doi: 10.2977/prims/1195173612. Google Scholar [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 (6726). doi: 10.1155/2008/672618. Google Scholar [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. Google Scholar [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. doi: 10.1016/j.jmaa.2007.12.039. Google Scholar [25] C. Park, Homomorphisms between Poisson $JC$*-algebras,, Bull. Braz. Math. Soc., 36 (2005), 79. doi: 10.1007/s00574-005-0029-z. Google Scholar [26] C. Park and Th. M. Rassias, Homomorphisms and derivations in proper $JCQ$*-triples,, J. Math. Anal. Appl., 337 (2008), 1404. doi: 10.1016/j.jmaa.2007.04.063. Google Scholar [27] J. C. Parnami and H. L. Vasudeva, On Jensen’s functional equation,, Aequationes Math., 43 (1992), 211. doi: 10.1007/BF01835703. Google Scholar [28] Th. M. Rassias, On the stability of the linear mapping in Banach spaces,, Proc. Amer. Math. Soc., 72 (1978), 297. doi: 10.1090/S0002-9939-1978-0507327-1. Google Scholar [29] Th. M. Rassias, On a modified Hyers-Ulam sequence,, J. Math. Anal. Appl., 158 (1991), 106. doi: 10.1016/0022-247X(91)90270-A. Google Scholar [30] Th. M. Rassias, On the stability of functional equations and a problem of Ulam,, Acta Applicandae Mathematicae, 62 (2000), 23. doi: 10.1023/A:1006499223572. Google Scholar [31] Th. M. Rassias and P. Šemrl, On the Hyers-Ulam stability of linear mappings,, J. Math. Anal. Appl., 173 (1993), 325. doi: 10.1006/jmaa.1993.1070. Google Scholar [32] G. L. Sewell, "Quantum Mechanics and its Emergent Macrophysics,", Princeton Univ. Press, (2002). Google Scholar [33] C. Trapani, Quasi-*-algebras of operators and their applications,, Rev. Math. Phys., 7 (1995), 1303. doi: 10.1142/S0129055X95000475. Google Scholar [34] S. M. Ulam, "A Collection of the Mathematical Problems,", Interscience Tracts in Pure and Applied Mathematics, 8 (1960). Google Scholar
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