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Abstract and Applied Analysis
Volume 2008, Article ID 915292, 8 pages
http://dx.doi.org/10.1155/2008/915292
Research Article

Approximation of Generalized Left Derivations

1Department of Industrial Mathematics, National Institute for Mathematical Sciences, Daejeon 305-340, South Korea
2Department of Mathematics, Mokwon University, Daejeon 302-729, South Korea

Received 26 February 2008; Accepted 15 April 2008

Academic Editor: Paul Eloe

Copyright © 2008 Sheon-Young Kang and Ick-Soon Chang. 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.

Abstract

We need to take account of the superstability for generalized left derivations (resp., generalized derivations) associated with a Jensen-type functional equation, and we also deal with problems for the Jacobson radical ranges of left derivations (resp., derivations).

1. Introduction

Let be an algebra over the real or complex field An additive mapping is said to be a left derivation (resp., derivation) if the functional equation (resp., ) holds for all Furthermore, if the functional equation is valid for all and all then is a linear left derivation (resp., linear derivation). An additive mapping is called a generalized left derivation (resp., generalized derivation) if there exists a left derivation (resp., derivation) such that the functional equation (resp., ) is fulfilled for all In addition, if the functional equations and hold for all and all then is a linear generalized left derivation (resp., linear generalized derivation).

It is of interest to consider the concept of stability for a functional equation arising when we replace the functional equation by an inequality which acts as a perturbation of the equation. The study of stability problems had been formulated by Ulam [1] during a talk in 1940: “Under what condition does there exists a homomorphism near an approximate homomorphism?” In the following year 1941, Hyers [2] was 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 thatfor all , then there exists a unique additive mapsuch thatfor all Moreover, if is continuous in for each fixed in where denotes the set of real numbers, then is 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 [3] and for approximate linear mappings was presented by Rassias [4] in 1978 by considering the case when the inequality (1.1) is unbounded. Due to the fact that the additive functional equation is said to have the Hyers-Ulam-Rassias stability property. The stability result concerning derivations between operator algebras was first obtained by Šemrl [5]. Recently, Badora [6] gave a generalization of the Bourgin's result [7]. He also dealt with the Hyers-Ulam stability and the Bourgin-type superstability of derivations in [8].

In 1955, Singer and Wermer [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 [10]. The Singer-Wermer conjecture implies that any linear derivation on a commutative semisimple Banach algebra is identically zero which is the result of Johnson [11]. After then, Hatori and Wada [12] showed that a zero operator is the only derivation on a commutative semisimple Banach algebra with the maximal ideal space without isolated points. Note that this differs from the above result of B.E. Johnson. 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 are given by Moslehian and Park, see, for example, [1418].

The main purpose of the present paper is to consider the superstability of generalized left derivations (resp., generalized derivations) on Banach algebras associated to the following Jensen type functional equation:where is an integer. This functional equation is introduced in [19]. Moreover, we will investigate the problems for the Jacobson radical ranges of left derivations (resp., derivations) on Banach algebras. We use the direct method and some ideas of Amyari et al. [19].

2. Main Results

Throughout this paper, the element of an algebra will denote a unit. We now establish the superstability of a generalized left derivation associated with the Jensen type functional equation as follows.

Theorem 2.1. Let be a Banach algebra with unit. Suppose that is a mapping with for which there exists a mapping such that the functional inequality:for all Then, is a generalized left derivation, and is a left derivation.

Proof. Substituting in (2.1), we getfor all Let us take and replace by in the above relation. Then, it becomesfor all An induction implies thatfor all By virtue of (2.4), one can easily check that for for all So, the sequence is Cauchy. Since is complete, converges. Let be the mapping defined by ()By letting in (2.4), we getfor all
Now, we assert that is additive. Replacing and by and in (2.2), respectively, we havefor all taking the limit as we obtainfor all Letting in the previous identity yields for all So, (2.9) becomes for all namely, is additive.
To demonstrate the uniqueness of the additive mapping subject to (2.7), we assume that there exists another additive mapping satisfying the inequality (2.7), for all Since and we see thatfor all By letting in this inequality, we conclude that that is, is unique.
Next, we are going to prove that is a generalized left derivation. If we take in (2.1), we also havefor all Moreover, if we replace and with and in (2.11), respectively, and then divide both sides by we getfor all Letting we obtainfor all Suppose that in the above equation. Then, it followsfor all Thus, if then by the additivity of we getfor all Hence, is additive.
Let for all Since, and satisfy the inequality given in (2.11), thenfor all We note thatfor all Since is additive, we can rewrite (2.17) asfor all Based on the above relation, one has , for all Moreover, we can obtain for all as If we also have that Therefore, we getfor all
We now want to verify that is a left derivation using the equations developed in the previous part. Indeed, using the facts that satisfies (2.19), we havefor all which means that is a generalized left derivation.
We finally need to show that is a left derivation. Let us replace by in (2.11). Then,for all Passing the limit as we getfor all This implies that for all and thus if we deduce that for all Hence, we get for all Since, is a left derivation, we can conclude that is a left derivation as well. This completes the proof of the theorem.

Employing the similar way as in the proof of Theorem 2.1, we get the following result for a generalized derivation.

Theorem 2.2. Let be a Banach algebra with unit. Suppose that is a mapping with for which there exists a mapping such thatfor all Then, is a generalized derivation, and is a derivation.

In view of the Thomas' result [10], derivations on Banach algebras now belong to the noncommutative setting. Among various noncommutative version of the Singer-Wermer theorem, Brešar and Vukman [20] proved the following. Any continuous linear left derivation on a Banach algebra maps into its Jacobson radical and also any left derivation on a semiprime ring is a derivation which maps into its center.

The following is the functional inequality with the problem as in the above Brešar and Vukman's result.

Theorem 2.3. Let be a semiprime Banach algebra with unit. Suppose that is a mapping with for which there exists a mapping such that the functional inequality:for all and all . Then, is a linear generalized left derivation. In this case, is a linear derivation which maps into the intersection of its center and its Jacobson radical

Proof. We consider in (2.24) and then satisfies the inequality (2.1). It follows from Theorem 2.1 that is a generalized left derivation, and is a left derivation, wherefor all Letting in (2.24), we havefor all and all If we also replace and with and in (2.26), respectively, and then divide both sides by we see thatfor all and all as So, we getfor all and all From the additivity of we find thatfor all and all Let us now assume that is a nonzero complex number and that a positive integer greater than Then by applying a geometric argument, there exist such that . In particular, due to the additivity of we obtain for all Thus, we havefor all Also, it is obvious that for all that is, is -linear. Therefore, is a linear generalized left derivation, and so is also a linear left derivation. According to the Brešar and Vukman's result which tells us that is a linear derivation which maps into its center Since is a commutative Banach algebra, the Singer-Wermer conjecture tells us that maps into and thus Using the semiprimeness of as well as the identity, we havefor all we have that is, is a linear derivation which maps into the intersection of its center and its Jacobson radical The proof of the theorem is ended.

The next corollary is the Brešar and Vukman's result.

Corollary 2.4. Let be a Banach algebra with unit. Suppose that is a continuous mapping with for which there exists a mapping such that the functional inequality (2.26). Then, is a linear generalized left derivation. In this case, maps into its Jacobson radical

Proof. On account of Theorem 2.3, is a linear left derivation on Hence, maps into its Jacobson radical by the Brešar and Vukman's result, which completes the proof.

With the help of Theorem 2.2, the following property can be derived along the same argument in the proof of Theorem 2.3.

Theorem 2.5. Let be a commutative Banach algebra with unit. Suppose that is a mapping with for which there exists a mapping such that the functional inequality:for all and all . Then, is a linear generalized derivation. In this case, maps into its Jacobson radical

Remark 2.6. We can generalize our results by substituting another functions or another forms satisfying suitable conditions (see, e.g., [19, 21]) for the bound of the functional inequalities connected to the Jensen type functional equation.

Acknowledgments

The authors would like to thank referees for their valuable comments regarding a previous version of this paper. The corresponding author dedicates this paper to his late father.

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