- Research Article
- Open Access
Superstability for Generalized Module Left Derivations and Generalized Module Derivations on a Banach Module (I)
© Huai-Xin Cao et al. 2009
- Received: 23 January 2009
- Accepted: 3 July 2009
- Published: 18 August 2009
We discuss the superstability of generalized module left derivations and generalized module derivations on a Banach module. Let be a Banach algebra and a Banach -module, and . The mappings , and are defined and it is proved that if (resp., is dominated by then is a generalized (resp., linear) module- left derivation and is a (resp., linear) module- left derivation. It is also shown that if (resp., is dominated by then is a generalized (resp., linear) module- derivation and is a (resp., linear) module- derivation.
- Generalize Module
- Additive Mapping
- Banach Algebra
- Generalize Derivation
- Jacobson Radical
The study of stability problems had been formulated by Ulam in  during a talk in 1940: under what condition does there exist a homomorphism near an approximate homomorphism? In the following year 1941, Hyers in  has answered affirmatively the question of Ulam for Banach spaces, which states that if and is a map with , a normed space, , a Banach space, such that
for all in . In addition, if the mapping is continuous in for each fixed in , then the mapping is real linear. This stability phenomenon is called the Hyers-Ulam stability of the additive functional equation . A generalized version of the theorem of Hyers for approximately additive mappings was given by Aoki in  and for approximate linear mappings was presented by Rassias in  by considering the case when the left-hand side of (1.1) is controlled by a sum of powers of norms. The stability result concerning derivations between operator algebras was first obtained by Šemrl in , Badora in  gave a generalization of Bourgin's result . He also discussed the Hyers-Ulam stability and the Bourgin-type superstability of derivations in .
Singer and Wermer in  obtained a fundamental result which started investigation into the ranges of linear derivations on Banach algebras. The result, which is called the Singer-Wermer theorem, states that any continuous linear derivation on a commutative Banach algebra maps into the Jacobson radical. They also made a very insightful conjecture, namely, that the assumption of continuity is unnecessary. This was known as the Singer- Wermer conjecture and was proved in 1988 by Thomas in . The Singer-Wermer conjecture implies that any linear derivation on a commutative semisimple Banach algebra is identically zero . After then, Hatori and Wada in  proved that the zero operator is the only derivation on a commutative semisimple Banach algebra with the maximal ideal space without isolated points. Based on these facts and a private communication with Watanabe , Miura et al. proved the Hyers-Ulam-Rassias stability and Bourgin-type superstability of derivations on Banach algebras in . Various stability results on derivations and left derivations can be found in [14–20]. More results on stability and superstability of homomorphisms, special functionals, and equations can be found in [21–30].
Recently, Kang and Chang in  discussed the superstability of generalized left derivations and generalized derivations. Indeed, these superstabilities are the so-called "Hyers-Ulam superstabilities." In the present paper, we will discuss the superstability of generalized module left derivations and generalized module derivations on a Banach module.
Let and be one of the following cases: (a) a unital algebra; (b) a Banach algebra with an approximate unit; (c) a -algebra. Then module- left derivations, module- derivations, generalized module- left derivations, and generalized module- derivations on become left derivations, derivations, generalized left derivations, and generalized derivations on discussed in .
It is easy to check that the functional satisfies the conditions (a), (b), and (c) in Theorem 2.1, where , . Especially, if has a unit and are mappings with such that for all , then is a generalized left derivation and is a left derivation.
then is a generalized module- derivation and is a module- derivation. Especially, if has a unit and are mappings with such that for all and some constants , then is a generalized derivation and is a derivation.
then is a linear generalized module- derivation on and is a linear module- derivation on . Especially, if is a unital commutative Banach algebra and are mappings with such that for all , all and some constants , then is a linear generalized derivation and is a linear derivation which maps into the Jacobson radical rad of .
This subject is supported by the NNSFs of China (no: 10571113,10871224).
- Ulam SM: Problems in Modern Mathematics. John Wiley & Sons, New York, NY, USA; 1964:xvii+150.MATHGoogle Scholar
- Hyers DH: On the stability of the linear functional equation. Proceedings of the National Academy of Sciences 1941, 27: 222–224. 10.1073/pnas.27.4.222MathSciNetView ArticleGoogle Scholar
- 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/00210064MathSciNetView ArticleMATHGoogle Scholar
- Rassias TM: On the stability of the linear mapping in Banach spaces. Proceedings of the American Mathematical Society 1978,72(2):297–300. 10.1090/S0002-9939-1978-0507327-1MathSciNetView ArticleMATHGoogle Scholar
- Šemrl P: The functional equation of multiplicative derivation is superstable on standard operator algebras. Integral Equations and Operator Theory 1994,18(1):118–122. 10.1007/BF01225216MathSciNetView ArticleMATHGoogle Scholar
- Badora R: On approximate ring homomorphisms. Journal of Mathematical Analysis and Applications 2002,276(2):589–597. 10.1016/S0022-247X(02)00293-7MathSciNetView ArticleMATHGoogle Scholar
- Bourgin DG: Approximately isometric and multiplicative transformations on continuous function rings. Duke Mathematical Journal 1949, 16: 385–397. 10.1215/S0012-7094-49-01639-7MathSciNetView ArticleMATHGoogle Scholar
- Badora R: On approximate derivations. Mathematical Inequalities & Applications 2006,9(1):167–173.MathSciNetView ArticleMATHGoogle Scholar
- Singer IM, Wermer J: Derivations on commutative normed algebras. Mathematische Annalen 1955, 129: 260–264. 10.1007/BF01362370MathSciNetView ArticleMATHGoogle Scholar
- Thomas MP: The image of a derivation is contained in the radical. Annals of Mathematics 1988,128(3):435–460. 10.2307/1971432MathSciNetView ArticleMATHGoogle Scholar
- Johnson BE: Continuity of derivations on commutative algebras. American Journal of Mathematics 1969, 91: 1–10. 10.2307/2373262MathSciNetView ArticleMATHGoogle Scholar
- Hatori O, Wada J: Ring derivations on semi-simple commutative Banach algebras. Tokyo Journal of Mathematics 1992,15(1):223–229. 10.3836/tjm/1270130262MathSciNetView ArticleMATHGoogle Scholar
- Miura T, Hirasawa G, Takahasi S-E: A perturbation of ring derivations on Banach algebras. Journal of Mathematical Analysis and Applications 2006,319(2):522–530. 10.1016/j.jmaa.2005.06.060MathSciNetView ArticleMATHGoogle Scholar
- Amyari M, Baak C, Moslehian MS: Nearly ternary derivations. Taiwanese Journal of Mathematics 2007,11(5):1417–1424.MathSciNetMATHGoogle Scholar
- Moslehian MS: Ternary derivations, stability and physical aspects. Acta Applicandae Mathematicae 2008,100(2):187–199. 10.1007/s10440-007-9179-xMathSciNetView ArticleMATHGoogle Scholar
- Moslehian MS: Hyers-Ulam-Rassias stability of generalized derivations. International Journal of Mathematics and Mathematical Sciences 2006, 2006:-8.Google Scholar
- Park C-G: Homomorphisms between -algebras, -derivations on a -algebra and the Cauchy-Rassias stability. Nonlinear Functional Analysis and Applications 2005,10(5):751–776.MathSciNetMATHGoogle Scholar
- Park C-G: Linear derivations on Banach algebras. Nonlinear Functional Analysis and Applications 2004,9(3):359–368.MathSciNetMATHGoogle Scholar
- Amyari M, Rahbarnia F, Sadeghi Gh: Some results on stability of extended derivations. Journal of Mathematical Analysis and Applications 2007,329(2):753–758. 10.1016/j.jmaa.2006.07.027MathSciNetView ArticleMATHGoogle Scholar
- Brešar M, Vukman J: On left derivations and related mappings. Proceedings of the American Mathematical Society 1990,110(1):7–16.MathSciNetView ArticleMATHGoogle Scholar
- 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.1994.1211MathSciNetView ArticleMATHGoogle Scholar
- Ulam SM: Problems in Modern Mathematics. John Wiley & Sons, New York, NY, USA; 1960:xiii+150.Google Scholar
- Baker JA: The stability of the cosine equation. Proceedings of the American Mathematical Society 1980,80(3):411–416. 10.1090/S0002-9939-1980-0580995-3MathSciNetView ArticleMATHGoogle Scholar
- Hyers DH, Rassias TM: Approximate homomorphisms. Aequationes Mathematicae 1992,44(2–3):125–153. 10.1007/BF01830975MathSciNetView ArticleMATHGoogle Scholar
- Rassias TM, Tabo J (Eds): Stability of Mappings of Hyers-Ulam Type, Hadronic Press Collection of Original Articles. Hadronic Press, Palm Harbor, Fla, USA; 1994:vi+173.Google Scholar
- Hyers DH, Isac G, Rassias TM: Stability of Functional Equations in Several Variables, Progress in Nonlinear Differential Equations and Their Applications, 34. Birkhäuser, Boston, Mass, USA; 1998:vi+313.View ArticleGoogle Scholar
- Isac G, Rassias TM: On the Hyers-Ulam stability of -additive mappings. Journal of Approximation Theory 1993,72(2):131–137. 10.1006/jath.1993.1010MathSciNetView ArticleMATHGoogle Scholar
- Rassias JM, Rassias MJ: Refined Ulam stability for Euler-Lagrange type mappings in Hilbert spaces. International Journal of Applied Mathematics & Statistics 2007, 7: 126–132.MathSciNetGoogle Scholar
- Rassias JM, Rassias MJ: The Ulam problem for 3-dimensional quadratic mappings. Tatra Mountains Mathematical Publications 2006, 34, part 2: 333–337.Google Scholar
- Rassias JM, Rassias MJ: On the Hyers-Ulam stability of quadratic mappings. The Journal of the Indian Mathematical Society 2000,67(1–4):133–136.MATHGoogle Scholar
- Kang S-Y, Chang I-S: Approximation of generalized left derivations. Abstract and Applied Analysis 2008, -8.Google Scholar
- Forti GL: Comments on the core of the direct method for proving Hyers-Ulam stability of functional equations. Journal of Mathematical Analysis and Applications 2004,295(1):127–133. 10.1016/j.jmaa.2004.03.011MathSciNetView ArticleMATHGoogle Scholar
- Brzdęk J, Pietrzyk A: A note on stability of the general linear equation. Aequationes Mathematicae 2008,75(3):267–270. 10.1007/s00010-007-2894-6MathSciNetView ArticleMATHGoogle Scholar
- Pietrzyk A: Stability of the Euler-Lagrange-Rassias functional equation. Demonstratio Mathematica 2006,39(3):523–530.MathSciNetMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.