Open Access

Stability of Homomorphisms and Generalized Derivations on Banach Algebras

Journal of Inequalities and Applications20092009:595439

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

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.

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

(1.1)

for all , then there is a homomorphism with

(1.2)

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).

Let be a mapping from a normed vector space into a Banach space subject to the inequality
(1.3)
for all , where and are constants with and . Then the limit
(1.4)
exists for all and is the unique additive mapping which satisfies
(1.5)

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 [715]. We also refer readers to the books in [1619].

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

(1.6)

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, 3335]).

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]).

Let be a complete generalized metric space and let be a strictly contractive mapping with Lipschitz constant . Then for each given element , either
(1.7)

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

(2.1)

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

(2.2)

for all .

Lemma 2.1.

Let and be linear spaces. A mapping with satisfies
(2.3)

for all if and only if is additive.

Proof.

Let satisfy (2.3). Letting in (2.3), we get
(2.4)
for all Hence
(2.5)
for all Letting in (2.3), we get for all Therefore by (2.5) we have for all This means that is odd. Letting in (2.3) and using the oddness of , we infer that for all Hence by (2.4) we have for all Therefore it follows from (2.3) that satisfies
(2.6)
for all Replacing and by and in (2.6), respectively, we get
(2.7)
for all Replacing by in (2.7) and using the oddness of , we get
(2.8)

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.

Let be a mapping with for which there exist functions such that
(2.9)
(2.10)
(2.11)
for all . If there exists a constant such that
(2.12)
for all , then there exists a unique (ring) homomorphism satisfying
(2.13)
(2.14)
for all , where
(2.15)

Proof.

By the assumption, we have
(2.16)
for all Letting in (2.10), we get
(2.17)
for all Hence
(2.18)
for all Letting in (2.10), we get
(2.19)
for all Therefore by (2.18) we have
(2.20)
for all Letting in (2.10), we get
(2.21)
for all Now, it follows from (2.20) and (2.21) that
(2.22)
for all Let We introduce a generalized metric on as follows:
(2.23)

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

Now we consider the mapping defined by
(2.24)
Let and let be an arbitrary constant with . From the definition of , we have
(2.25)
for all . By the assumption and the last inequality, we have
(2.26)
for all . So for any . It follows from (2.22) that . Therefore according to Theorem 1.2, the sequence converges to a fixed point of , that is,
(2.27)
and for all . Also is the unique fixed point of in the set and
(2.28)
that is, inequality (2.13) holds true for all . It follows from the definition of , (2.10), and (2.16) that for all Since by Lemma 2.1 the mapping is additive. So it follows from the definition of , (2.9), and (2.11) that
(2.29)
for all So is homomorphism. Similarly, we have from (2.9) and (2.11) that
(2.30)

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.

Let and be linear spaces. A mapping satisfies
(2.31)

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.

Let be a mapping for which there exist functions such that
(2.32)
for all and all . If there exists a constant such that
(2.33)
for all , then there exists a unique homomorphism satisfying
(2.34)

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.

Let be non-negative real numbers with . Suppose that is a mapping such that
(2.35)
for all and all . Then there exists a unique homomorphism satisfying
(2.36)

for all

Proof.

The proof follows from Theorem 2.2 by taking
(2.37)

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

Corollary 2.7.

Let be non-negative real numbers with and Suppose that is a mapping such that
(2.38)
for all and all . Then there exists a unique homomorphism satisfying
(2.39)

for all

Proof.

The proof follows from Theorem 2.5 by taking
(2.40)

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

(3.1)

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

Theorem 3.1.

Let be mappings with for which there exists a function such that
(3.2)
(3.3)
(3.4)
(3.5)
for all . If there exists a constants such
(3.6)
for all , then there exist a unique -Jordan derivation and a unique generalized -Jordan derivation satisfying
(3.7)

for all , where is defined as in Theorem 2.2.

Proof.

It follows from the assumptions that
(3.8)
for all By the proof of Theorem 2.5, there exist unique additive mappings satisfying (3.7) and
(3.9)
for all . It follows from the definitions of (3.3), and (3.8) that
(3.10)
for all Hence
(3.11)

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

Remark 3.2.

Applying Theorem 3.1 for the case , there exist a unique -Jordan derivation and a unique generalized -Jordan derivation satisfying
(3.12)

for all .

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

Theorem 3.3.

Let be mappings with for which there exists a function satisfying (3.2)–(3.5). If there exists a constant such
(3.13)
for all , then there exist a unique -Jordan derivation and a unique generalized -Jordan derivation satisfying
(3.14)

for all , where is defined as in Theorem 2.2.

Remark 3.4.

Applying Theorem 3.3 for the case , there exist a unique -Jordan derivation and a unique generalized -Jordan derivation satisfying
(3.15)

for all .

Declarations

Acknowledgment

The second author was supported by Hanyang University in 2009.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili
(2)
Department of Mathematics, Hanyang University

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© A. Najati and C. Park 2009

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