## Banach Journal of Mathematical Analysis

### Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras

Chun-Gil Park

#### Abstract

Let $q$ be a positive rational number and $n$ be a nonnegative integer. We prove the Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras and of generalized derivations on quasi-Banach algebras for the following functional equation: \begin{eqnarray*} \sum_{i=1}^{n} f \left( \sum_{j=1}^{n}q (x_i-x_j) \right) + n f \left(\sum_{i=1}^{n} q x_i \right) = nq \sum_{i=1}^{n} f(x_i) . \end{eqnarray*} This is applied to investigate isomorphisms between quasi-Banach algebras.~The concept of Hyers-Ulam-Rassias stability originated from the Th.M. Rassias' stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300.

#### Article information

Source
Banach J. Math. Anal., Volume 1, Number 1 (2007), 23-32.

Dates
First available in Project Euclid: 21 April 2009

https://projecteuclid.org/euclid.bjma/1240321552

Digital Object Identifier
doi:10.15352/bjma/1240321552

Mathematical Reviews number (MathSciNet)
MR2350191

Zentralblatt MATH identifier
1135.39017

#### Citation

Park, Chun-Gil. Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras. Banach J. Math. Anal. 1 (2007), no. 1, 23--32. doi:10.15352/bjma/1240321552. https://projecteuclid.org/euclid.bjma/1240321552

#### References

• M. Amyari, C. Baak and M.S. Moslehian, Nearly ternary derivations, Taiwanese J. Math. (to appear).
• J.M. Almira and U. Luther, Inverse closedness of approximation algebras, J. Math. Anal. Appl. 314 (2006), 30–44.
• P. Ara and M. Mathieu, Local Multipliers of $C^*$-Algebras, Springer-Verlag, London, 2003.
• C. Baak and M.S. Moslehian, On the stability of $J^*$-homomorphisms, Nonlinear Anal.–TMA 63 (2005), 42–48.
• C. Baak and M.S. Moslehian, On the stability of $\theta$-derivations on $JB^*$-triples, Bull. Braz. Math. Soc. 38 (2007), 115–127.
• Y. Benyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis, Vol. 1, Colloq. Publ. 48, Amer. Math. Soc., Providence, 2000.
• P. Czerwik, On stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Hamburg 62 (1992), 59–64.
• P. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific Publishing Company, New Jersey, Hong Kong, Singapore and London, 2002.
• Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991), 431–434.
• P. G\v avruta, A generalization of the Hyers–Ulam–Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436.
• D.H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222–224.
• D.H. Hyers, G. Isac and Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkh$\ddota$user, Basel, 1998.
• K. Jun, S. Jung and Y. Lee, A generalization of the Hyers–Ulam–Rassias stability of a functional equation of Davison, J. Korean Math. Soc. 41 (2004), 501–511.
• K. Jun and H. Kim, Ulam stability problem for quadratic mappings of Euler–Lagrange, Nonlinear Anal.–TMA 61 (2005), 1093–1104.
• K. Jun and H. Kim, On the generalized $A$-quadratic mappings associated with the variance of a discrete type distribution Nonlinear Anal.–TMA 62 (2005), 975–987.
• S. Jung, Hyers–Ulam–Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, Florida, 2001.
• Y. Lee and K. Jun, A note on the Hyers–Ulam–Rassias stability of Pexider equation, J. Korean Math. Soc. 37 (2000), 111–124.
• M. Mirzavaziri and M.S. Moslehian, Automatic continuity of $\sigma$-derivations in $C^*$-algebras, Proc. Amer. Math. Soc. 134 (2006), 3319–3327.
• C. Park, On the stability of the linear mapping in Banach modules, J. Math. Anal. Appl. 275 (2002), 711–720.
• C. Park, Linear derivations on Banach algebras, Nonlinear Funct. Anal. Appl. 9 (2004), 359–368.
• C. Park, Hyers–Ulam–Rassias stability of a generalized Euler–Lagrange type additive mapping and isomorphisms between $C^*$-algebras, Bull. Belgian Math. Soc.–Simon Stevin 13 (2006), 619–631.
• C. Park and J. Hou, Homomorphisms between $C^*$-algebras associated with the Trif functional equation and linear derivations on $C^*$-algebras, J. Korean Math. Soc. 41 (2004), 461–477.
• C. Park and Th.M. Rassias, On a generalized Trif's mapping in Banach modules over a $C^*$-algebra, J. Korean Math. Soc. 43 (2006), 323–356.
• Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300.
• Th.M. Rassias, Problem 16; 2, Report of the 27$^\textth$ International Symp. on Functional Equations, Aequationes Math. 39 (1990), 292–293; 309.
• Th.M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. 251 (2000), 264–284.
• Th.M. Rassias, Functional Equations, Inequalities and Applications, Kluwer Academic Publishers, Dordrecht, Boston and London, 2003.
• Th.M. Rassias and P. Šemrl, On the behaviour of mappings which do not satisfy Hyers–Ulam stability, Proc. Amer. Math. Soc. 114 (1992), 989–993.
• S. Rolewicz, Metric Linear Spaces, PWN-Polish Sci. Publ., Reidel and Dordrecht, 1984.
• S.M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960.