Open Access

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

Journal of Inequalities and Applications20092009:718020

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

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

(11)
for all in , then there exists a unique additive mapping such that
(12)

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 [1420]. More results on stability and superstability of homomorphisms, special functionals, and equations can be found in [2130].

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.

A mapping is said to be module- additive if
(13)
A module- additive mapping is said to be a module-   left derivation (resp., module-   derivation) if the functional equation
(14)
respectively,
(15)

holds.

Definition 1.2.

A mapping is said to be module-   additive if
(16)
A module- additive mapping is called a generalized module-   left derivation (resp., generalized module-   derivation) if there exists a module- left derivation (resp., module- derivation) such that
(17)
respectively,
(18)

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:

(a)

(b)

(c)

Suppose that and are mappings such that , exists for all and
(21)
for all and where
(22)

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

Proof.

By taking , we see from (2.1) that
(23)
for all . Letting and replacing by in (2.3) yield that
(24)
for all . From [32, Theorem 1] (analogously as in [33, the proof of Theorem 1] or [34]), one can easily deduce that the limit exists for every , and
(25)
Next, we show that the mapping is additive. To do this, let us replace by in (2.3), respectively. Then
(26)
for all . If we let in the above inequality, then the condition (a) yields that
(27)
for all . Since , taking and , respectively, we see that and for all . Now, for all , put . Then by (2.7), we get that
(28)

This shows that is additive.

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

(29)
that is,
(210)
for all and . By replacing with in (2.10), respectively, we deduce that
(211)
for all and . Letting , the condition (b) yields that
(212)
for all and . Since is additive, is module- additive. Put . Then by (2.10) we see from the condition (a) that
(213)
for all and . Hence
(214)
for all and . It follows from (2.12) that for all and , and then for all . Since is additive, is module- additive. So, for all and by (2.12)
(215)

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

(216)
for all . By letting , we get from the condition (a) that
(217)

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.

In Theorem 2.1, if the condition (2.1) is replaced with
(218)
for all and where
(219)

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.

Let     be complex vector spaces. Then a mapping  is linear if and only if
(220)

for all and all .

Proof.

It suffices to prove the sufficiency. Suppose that for all and all . Then is additive and for all and all Let be any nonzero complex number. Take a positive integer such that . Take a real number such that . Put . Then and, therefore,
(221)

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:

(a)

(b)

(c) .

Suppose that and are mappings such that , exists for all and
(222)
for all , and all , where stands for
(223)

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

Proof.

Clearly, the inequality (2.1) is satisfied. Hence, Theorem 2.1 and its proof show that is a generalized left derivation and is a left derivation on with
(224)
for every . Taking in (2.22) yields that
(225)
for all and all . If we replace and with and in (2.25), respectively, then we see that
(226)
as for all and all . Hence,
(227)
for all and all . Since is additive, taking in (2.27) implies that
(228)

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

Remark 2.6.

It is easy to check that the functional satisfies the conditions (a), (b), and (c) in Theorem 2.5, where , are constants. Especially, if is a complex semiprime Banach algebra with unit and are mappings with such that
(229)

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.

In Theorem 2.5, if the condition (2.22) is replaced with
(230)
for all , and where stands for
(231)

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.

The controlling function
(232)
consists of the "mixed sum-product of powers of norms," introduced by Rassias (in 2007) [28] and applied afterwards by Ravi et al. (2007-2008). Moreover, it is easy to check that the functional
(233)

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

(1)
College of Mathematics and Information Science, Shaanxi Normal University
(2)
Pedagogical Department, Section of Mathematics and Informatics, National and Capodistrian University of Athens

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© Huai-Xin Cao et al. 2009

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