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

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 [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.

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.