# 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

**DOI: **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.

## 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

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