Some additive mappings on Banach \({\ast}\)-algebras with derivations

Volume 11, Issue 3, pp 335--341 http://dx.doi.org/10.22436/jnsa.011.03.02

Publication Date: February 09, 2018 Submission Date: June 26, 2017 Revision Date: November 25, 2017 Accteptance Date: December 01, 2017

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Authors

Jae-Hyeong Bae - Humanitas College, Kyung Hee University, Yongin 17104, Republic of Korea. Ick-Soon Chang - Department of Mathematics, Chungnam National University, 99 Daehangno, Yuseong-gu, Daejeon 34134, Republic of Korea.

Abstract

We take into account some additive mappings in Banach \(\ast\)-algebras with derivations. We will first study the conditions for additive mappings with derivations on Banach \(\ast\)-algebras. Then we prove some theorems involving linear mappings on Banach $\ast$-algebras with derivations. So derivations on \(C^{\ast}\)-algebra are characterized.

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Keywords

• Banach \(\ast\)-algebra
• \(C^{\ast}\)-algebra
• additive mapping with involution
• derivation

MSC

•  16N60
•  16W80
•  39B72
•  39B82
•  46H40
•  46L57

References

• [1] R. P. Agarwal, R. Saadati, A. Salamati , Approximation of the multiplicatives on random multi-normed space , J. Inequal. Appl., 2017 (2017 ), 10 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [2] T. Aoki, On the stability of the linear transformation in Banach spaces , J. Math. Soc. Japan, 2 (1950), 64–66.
  • View Article
  • MathSciNet
  • Google Scholar


• [3] Z. Baderi, R. Saadati, Generalized stability of Euler-Lagrange quadratic functional equation in random normed spaces under arbitrary t-norms , Thai J. Math., 14 (2016), 585–590.
  • MathSciNet
  • Google Scholar
  • MATH


• [4] R. Badora , On approximate ring homomorphisms , J. Math. Anal. Appl., 276 (2002), 589–597.
  • View Article
  • MathSciNet
  • Google Scholar


• [5] R. Badora , On approximate derivations, Math. Inequal. Appl., 9 (2006), 167–173.


• [6] F. F. Bonsall, J. Duncan , Complete normed algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer- Verlag, New York-Heidelberg (1973)
  • View Article


• [7] D. G. Bourgin, Approximately isometric and multiplicative transformations on continuous function rings , Duke Math. J., 16 (1949), 385–397.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [8] M. Brešar, J. Vukman , On some additive mappings in rings with involution, Aequationes Math., 38 (1989), 178–185.
  • View Article
  • MathSciNet
  • Google Scholar


• [9] M. Brešar, J. Vukman , On left derivations and related mappings , Proc. Amer. Math. Soc., 110 (1990), 7–16.
  • View Article
  • MathSciNet
  • Google Scholar


• [10] Y. J. Cho, C.-K. Park, T. M. Rassias, R. Saadati, Stability of functional equations in Banach algebras, Springer, Cham (2015)
  • View Article
  • MathSciNet
  • Google Scholar


• [11] M. N. Daif, M. S. Tammam El-Sayiad , On generalized derivations of semiprime rings with involution, Int. J. Algebra, 1 (2007), 551–555.
  • MathSciNet
  • Google Scholar
  • MATH


• [12] P. Găvruţa, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184 (1994), 431–436.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [13] M. E. Gordji , Nearly involutions on Banach algebras, A fixed point approach, Fixed Point Theory, 14 (2013), 117–123.
  • MathSciNet
  • Google Scholar
  • MATH


• [14] D. H. Hyers, On the stability of the linear functional equation , Proc. Nat. Acad. Sci. U. S. A., 27 (1911), 222–224.
  • Google Scholar


• [15] B. E. Johnson, A. M. Sinclair , Continuity of derivations and a problem of Kaplansky , Amer. J. Math., 90 (1968), 1067– 1073.
  • View Article
  • MathSciNet
  • MATH


• [16] P. Kannappan, Functional equations and inequalities with applications, Springer Monographs in Mathematics, Springer, New York (2009)
  • View Article
  • MathSciNet


• [17] G. V. Milovanović(ed.), T. M. Rassias (ed.), Analytic number theory, approximation theory, and special functions, In honor of Hari M. Srivastava, Springer, New York (2014)
  • View Article
  • MathSciNet
  • Google Scholar


• [18] C.-K. Park, G. A. Anastassiou, R. Saadati, S.-S. Yun , Functional inequalities in fuzzy normed spaces, J. Comput. Anal. Appl., 22 (2017), 601–612.
  • Google Scholar


• [19] T. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297–300.
  • View Article
  • MathSciNet
  • Google Scholar


• [20] P. Šemrl, The functional equation of multiplicative derivation is superstable on standard operator algebras, Integral Equations Operator Theory, 18 (1994), 118–122.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [21] I. M. Singer, J. Wermer, Derivations on commutative normed algebras, Math. Ann., 129 (1955), 260–264.
  • View Article
  • MathSciNet
  • Google Scholar


• [22] M. P. Thomas, The image of a derivation is contained in the radical, Ann. of Math., 128 (1988), 435–460.
  • View Article
  • MathSciNet


• [23] S. M. Ulam , A collection of mathematical problems, Interscience Tracts in Pure and Applied Mathematics, Interscience Publishers, New York-London (1960)