- Research Article
- Open Access

# Superstability for Generalized Module Left Derivations and Generalized Module Derivations on a Banach Module (I)

- Huai-Xin Cao
^{1}Email author, - Ji-Rong Lv
^{1}and - J. M. Rassias
^{2}

**2009**:718020

https://doi.org/10.1155/2009/718020

© Huai-Xin Cao et al. 2009

**Received: **23 January 2009

**Accepted: **3 July 2009

**Published: **18 August 2009

## Abstract

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.

## Keywords

## 1. Introduction

The study of stability problems had been formulated by Ulam in [1] during a talk in 1940: under what condition does there exist a homomorphism near an approximate homomorphism? In the following year 1941, Hyers in [2] 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 [3] and for approximate linear mappings was presented by Rassias in [4] 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 [5], Badora in [6] gave a generalization of Bourgin's result [7]. He also discussed the Hyers-Ulam stability and the Bourgin-type superstability of derivations in [8].

Singer and Wermer in [9] 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 [10]. The Singer-Wermer conjecture implies that any linear derivation on a commutative semisimple Banach algebra is identically zero [11]. After then, Hatori and Wada in [12] 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 [13], Miura et al. proved the Hyers-Ulam-Rassias stability and Bourgin-type superstability of derivations on Banach algebras in [13]. 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 [31] 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.

To give our results, let us give some notations. Let be an algebra over the real or complex field and an -bimodule.

Definition 1.1.

*module-*

*left derivation*(resp.,

*module-*

*derivation*) if the functional equation

holds.

Definition 1.2.

*generalized module-*

*left derivation*(resp.,

*generalized module-*

*derivation*) if there exists a module- left derivation (resp., module- derivation) such that

In addition, if the mappings
and
are all linear, then the mapping
is called a *linear generalized module-*
* left derivation* (resp., *linear generalized module-*
* derivation*).

Remark 1.3.

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 [31].

## 2. Main Results

Theorem 2.1.

Let be a Banach algebra, a Banach -bimodule, and integers greater than 1, and satisfy the following conditions:

Then is a generalized module- left derivation and is a module- left derivation.

Proof.

Now, we are going to prove that is a generalized module- left derivation. Letting in (2.1) gives that

This shows that is a module- left derivation on and then is a generalized module- left derivation on .

Lastly, we prove that is a module- left derivation on . To do this, we compute from (2.10) that

for all . Now, (2.12) implies that for all and all . Hence, is a module- left derivation on . This completes the proof.

Remark 2.2.

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.

Remark 2.3.

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.

Lemma 2.4.

Proof.

for all . This shows that is linear. The proof is completed.

Theorem 2.5.

Let be a Banach algebra, a Banach -bimodule, and integers greater than , and satisfy the following conditions:

Then is a linear generalized module- left derivation and is a linear module- left derivation.

Proof.

for all and all . Lemma 2.4 yields that is linear and so is . This completes the proof.

Remark 2.6.

for all . Then is a linear generalized left derivation and is a linear derivation which maps into the intersection of the center and the Jacobson radical of .

Remark 2.7.

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 .

Remark 2.8.

satisfies theconditions (a), (b), and (c) in Theorems 2.1 and 2.5, where and are all constants.

## Declarations

### Acknowledgment

This subject is supported by the NNSFs of China (no: 10571113,10871224).

## Authors’ Affiliations

## References

- 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

## Copyright

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.