- Research Article
- Open Access

# Stability of Homomorphisms and Generalized Derivations on Banach Algebras

- Abbas Najati
^{1}and - Choonkil Park
^{2}Email author

**2009**:595439

https://doi.org/10.1155/2009/595439

© A. Najati and C. Park 2009

**Received:**14 June 2009**Accepted:**18 November 2009**Published:**19 November 2009

## Abstract

We prove the generalized Hyers-Ulam stability of homomorphisms and generalized derivations associated to the following functional equation on Banach algebras.

## Keywords

- Banach Space
- Functional Equation
- Linear Space
- Stability Problem
- Banach Algebra

## 1. Introduction

The first stability problem concerning group homomorphisms was raised from a question of Ulam [1]. *Let*
*be a group and let*
*be a metric group with the metric*
*. Given*
*, does there exist*
*such that if a mapping*
*satisfies the inequality*

*for all*
, *then there is a homomorphism*
*with*

*for all*

Hyers [2] gave a first affirmative answer to the question of Ulam for Banach spaces. Aoki [3] and Rassias [4] provided a generalization of the Hyers' theorem for additive and linear mappings, respectively, by allowing the Cauchy difference to be unbounded (see also [5]).

Theorem 1.1 (Rassias).

for all . If then inequality (1.3) holds for and (1.5) for . Also, if for each the mapping is continuous in , then is linear.

In 1994, a generalization of the Rassias' theorem was obtained by G vruţa [6], who replaced the bound by a general control function For the stability problems of various functional equations and mappings and their Pexiderized versions, we refer the readers to [7–15]. We also refer readers to the books in [16–19].

Let
be a real or complex algebra. A mapping
is said to be a*(ring) derivation* if

for all
If, in addition,
for all
and all
then
is called a *linear derivation,* where
denotes the scalar field of
. Singer and Wermer [20] proved that if
is a commutative Banach algebra and
is a continuous linear derivation, then
They also conjectured that the same result holds even
is a discontinuous linear derivation. Thomas [21] proved the conjecture. As a direct consequence, we see that there are no nonzero linear derivations on a semisimple commutative Banach algebra, which had been proved by Johnson [22]. On the other hand, it is not the case for ring derivations. Hatori and Wada [23] determined a representation of ring derivations on a semi-simple commutative Banach algebra (see also [24]) and they proved that only the zero operator is a ring derivation on a semi-simple commutative Banach algebra with the maximal ideal space without isolated points. The stability of derivations between operator algebras was first obtained by
emrl [25]. Badora [26] and Miura et al. [8] proved the Hyers-Ulam-Rassias stability of ring derivations on Banach algebras. An additive mapping
is called a *Jordan derivation* in case
is fulfilled for all
Every derivation is a Jordan derivation. The converse is in general not true (see [27, 28]). The concept of generalized derivation has been introduced by M. Brešar [29]. Hvala [30] and Lee [31] introduced a concept of
-derivation (see also [32]). Let
be automorphisms of
An additive mapping
is called a
*-derivation* in case
holds for all pairs
An additive mapping
is called a
*-Jordan derivation* in case
holds for all
An additive mapping
is called a *generalized*
*-derivation* in case
holds for all pairs
where
is a
-derivation. An additive mapping
is called a *generalized*
*-Jordan derivation* in case
holds for all
where
is a
-Jordan derivation. It is clear that every generalized
-derivation is a generalized
-Jordan derivation.

The aim of the present paper is to establish the stability problem of homomorphisms and generalized -derivations by using the fixed point method (see [7, 33–35]).

Let
be a set. A function
is called a *generalized metric* on
if
satisfies

(i) if and only if ;

(ii) for all ;

(iii) for all

We recall the following theorem by Margolis and Diaz.

Theorem 1.2 (See [36]).

for all nonnegative integers or there exists a nonnegative integer such that

(1) for all ;

(2)the sequence converges to a fixed point of ;

(3) is the unique fixed point of in the set ;

(4) for all .

## 2. Stability of Homomorphisms

Daróczy et al. [37] have studied the functional equation

where is a fixed parameter and is unknown, is a nonvoid open interval and (2.1) holds for all They characterized the equivalence of (2.1) and Jensen's functional equation in terms of the algebraic properties of the parameter For in (2.1), we get the Jensen's functional equation. In the present paper, we establish the general solution and some stability results concerning the functional equation (2.1) in normed spaces for This applied to investigate and prove the generalized Hyers-Ulam stability of homomorphisms and generalized derivations in real Banach algebras. In this section, we assume that is a normed algebra and is a Banach algebra. For convenience, we use the following abbreviation for a given mapping

for all .

Lemma 2.1.

for all if and only if is additive.

Proof.

for all Adding (2.6) to (2.8), we get for all Using the identity and replacing by in the last identity, we infer that for all Hence is additive. The converse is obvious.

Theorem 2.2.

Proof.

It is easy to show that is a generalized complete metric space [34].

for all Since is homomorphism, we get (2.14) from (2.30).

Finally it remains to prove the uniqueness of . Let another homomorphism satisfying (2.13). Since and is additive, we get and for all , that is, is a fixed point of . Since is the unique fixed point of in , we get

We need the following lemma in the proof of the next theorem.

Lemma 2.3 (See [38]).

Let and be linear spaces and be an additive mapping such that for all and all Then the mapping is -linear.

Lemma 2.4.

for all and all if and only if is -linear.

Proof.

Let satisfy (2.31). Letting in (2.31), we get By Lemma 2.1, the mapping is additive. Letting in (2.31) and using the additivity of we get that for all and all So by Lemma 2.4, the mapping is -linear. The converse is obvious.

The following theorem is an alternative result of Theorem 2.2 with similar proof.

Theorem 2.5.

for all , where is defined as in Theorem 2.2.

Proof.

It follows from the assumptions that and so The rest of the proof is similar to the proof of Theorem 2.2 and we omit the details.

Corollary 2.6.

for all

Proof.

for all . Then we can choose and we get the desired results.

Corollary 2.7.

for all

Proof.

for all . Then we can choose and we get the desired results.

## 3. Stability of Generalized -Derivations

In this section, we assume that is a Banach algebra, and are automorphisms of For convenience, we use the following abbreviation for given mappings

for all . Now we prove the generalized Hyers-Ulam stability of generalized -derivations and generalized -Jordan derivations in Banach algebras.

Theorem 3.1.

for all , where is defined as in Theorem 2.2.

Proof.

for all Hence is a -Jordan derivation and is a generalized -Jordan derivation.

Remark 3.2.

for all .

The following theorem is an alternative result of Theorem 3.1 with similar proof.

Theorem 3.3.

for all , where is defined as in Theorem 2.2.

Remark 3.4.

for all .

## Declarations

### Acknowledgment

The second author was supported by Hanyang University in 2009.

## Authors’ Affiliations

## References

- Ulam SM:
*A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8*. Interscience, New York, NY, USA; 1960:xiii+150.Google Scholar - Hyers DH:
**On the stability of the linear functional equation.***Proceedings of the National Academy of Sciences of the United States of America*1941,**27:**222–224. 10.1073/pnas.27.4.222MathSciNetView ArticleMATHGoogle 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 ThM:
**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 - Bourgin DG:
**Classes of transformations and bordering transformations.***Bulletin of the American Mathematical Society*1951,**57:**223–237. 10.1090/S0002-9904-1951-09511-7MathSciNetView 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 - Mirzavaziri M, Moslehian MS:
**A fixed point approach to stability of a quadratic equation.***Bulletin of the Brazilian Mathematical Society*2006,**37**(3):361–376. 10.1007/s00574-006-0016-zMathSciNetView 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 - Moslehian MS:
**Hyers-Ulam-Rassias stability of generalized derivations.***International Journal of Mathematics and Mathematical Sciences*2006,**2006:**-8.Google Scholar - Najati A:
**Hyers-Ulam stability of an****-Apollonius type quadratic mapping.***Bulletin of the Belgian Mathematical Society. Simon Stevin*2007,**14**(4):755–774.MathSciNetMATHGoogle Scholar - Najati A:
**On the stability of a quartic functional equation.***Journal of Mathematical Analysis and Applications*2008,**340**(1):569–574. 10.1016/j.jmaa.2007.08.048MathSciNetView ArticleMATHGoogle Scholar - Najati A, Moghimi MB:
**Stability of a functional equation deriving from quadratic and additive functions in quasi-Banach spaces.***Journal of Mathematical Analysis and Applications*2008,**337**(1):399–415. 10.1016/j.jmaa.2007.03.104MathSciNetView ArticleMATHGoogle Scholar - Najati A, Park C:
**Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras associated to the Pexiderized Cauchy functional equation.***Journal of Mathematical Analysis and Applications*2007,**335**(2):763–778. 10.1016/j.jmaa.2007.02.009MathSciNetView ArticleMATHGoogle Scholar - Najati A, Park C:
**On the stability of an****-dimensional functional equation originating from quadratic forms.***Taiwanese Journal of Mathematics*2008,**12**(7):1609–1624.MathSciNetMATHGoogle Scholar - Park C:
**On the stability of the linear mapping in Banach modules.***Journal of Mathematical Analysis and Applications*2002,**275**(2):711–720. 10.1016/S0022-247X(02)00386-4MathSciNetView ArticleMATHGoogle Scholar - Czerwik P:
*Functional Equations and Inequalities in Several Variables*. World Scientific, River Edge, NJ, USA; 2002:x+410.View ArticleMATHGoogle 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 ArticleMATHGoogle Scholar - Jung S-M:
*Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis*. Hadronic Press, Palm Harbor, Fla, USA; 2001:ix+256.MATHGoogle Scholar - Rassias ThM:
*Functional Equations, Inequalities and Applications*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2003:x+224.View 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 - Šemrl P:
**On ring derivations and quadratic functionals.***Aequationes Mathematicae*1991,**42**(1):80–84. 10.1007/BF01818480MathSciNetView 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 derivations.***Mathematical Inequalities & Applications*2006,**9**(1):167–173.MathSciNetView ArticleMATHGoogle Scholar - Cusack JM:
**Jordan derivations on rings.***Proceedings of the American Mathematical Society*1975,**53**(2):321–324. 10.1090/S0002-9939-1975-0399182-5MathSciNetView ArticleMATHGoogle Scholar - Herstein IN:
**Jordan derivations of prime rings.***Proceedings of the American Mathematical Society*1957,**8:**1104–1110. 10.1090/S0002-9939-1957-0095864-2MathSciNetView ArticleMATHGoogle Scholar - Brešar M:
**Jordan derivations on semiprime rings.***Proceedings of the American Mathematical Society*1988,**104**(4):1003–1006.MathSciNetView ArticleMATHGoogle Scholar - Hvala B:
**Generalized derivations in rings.***Communications in Algebra*1998,**26**(4):1147–1166. 10.1080/00927879808826190MathSciNetView ArticleMATHGoogle Scholar - Lee T-K:
**Generalized derivations of left faithful rings.***Communications in Algebra*1999,**27**(8):4057–4073. 10.1080/00927879908826682MathSciNetView ArticleMATHGoogle Scholar - Liu C-K, Shiue W-K:
**Generalized Jordan triple****-derivations on semiprime rings.***Taiwanese Journal of Mathematics*2007,**11**(5):1397–1406.MathSciNetMATHGoogle Scholar - Amyari M, Moslehian MS:
**Hyers-Ulam-Rassias stability of derivations on Hilbert****-modules.**In*Topological Algebras and Applications, Contemporary Mathematics*.*Volume 427*. American Mathematical Society, Providence, RI, USA; 2007:31–39.View ArticleGoogle Scholar - Cădariu L, Radu V:
**On the stability of the Cauchy functional equation: a fixed point approach.**In*Iteration Theory, Grazer Mathematische Berichte*.*Volume 346*. Karl-Franzens-Universitaet Graz, Graz, Austria; 2004:43–52.Google Scholar - Jung S-M, Kim T-S:
**A fixed point approach to the stability of the cubic functional equation.***Boletín de la Sociedad Matemática Mexicana*2006,**12**(1):51–57.MathSciNetMATHGoogle Scholar - Diaz JB, Margolis B:
**A fixed point theorem of the alternative, for contractions on a generalized complete metric space.***Bulletin of the American Mathematical Society*1968,**74:**305–309. 10.1090/S0002-9904-1968-11933-0MathSciNetView ArticleMATHGoogle Scholar - Daróczy Z, Lajkó K, Lovas RL, Maksa Gy, Páles Zs:
**Functional equations involving means.***Acta Mathematica Hungarica*2007,**116**(1–2):79–87. 10.1007/s10474-007-5296-2MathSciNetView ArticleMATHGoogle Scholar - Park C:
**Homomorphisms between Poisson****-algebras.***Bulletin of the Brazilian Mathematical Society*2005,**36**(1):79–97. 10.1007/s00574-005-0029-zMathSciNetView ArticleMATHGoogle Scholar

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