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Superstability for Generalized Module Left Derivations and Generalized Module Derivations on a Banach Module (I)
Journal of Inequalities and Applications volume 2009, Article number: 718020 (2009)
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.
The study of stability problems had been formulated by Ulam in  during a talk in 1940: under what condition does there exist a homomorphism near an approximate homomorphism? In the following year 1941, Hyers in  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 , then there exists a unique additive mapping 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  and for approximate linear mappings was presented by Rassias in  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 , Badora in  gave a generalization of Bourgin's result . He also discussed the Hyers-Ulam stability and the Bourgin-type superstability of derivations in .
Singer and Wermer in  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 . The Singer-Wermer conjecture implies that any linear derivation on a commutative semisimple Banach algebra is identically zero . After then, Hatori and Wada in  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 , Miura et al. proved the Hyers-Ulam-Rassias stability and Bourgin-type superstability of derivations on Banach algebras in . 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  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.
A mapping is said to be module-additive if
A module- additive mapping is said to be a module- left derivation (resp., module- derivation) if the functional equation
A mapping is said to be module- additive if
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
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).
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 .
2. Main Results
Let be a Banach algebra, a Banach -bimodule, and integers greater than 1, and satisfy the following conditions:
Suppose that and are mappings such that , exists for all and
for all and where
Then is a generalized module- left derivation and is a module- left derivation.
By taking , we see from (2.1) that
for all . Letting and replacing by in (2.3) yield that
Next, we show that the mapping is additive. To do this, let us replace by in (2.3), respectively. Then
for all . If we let in the above inequality, then the condition (a) yields that
for all . Since , taking and , respectively, we see that and for all . Now, for all , put . Then by (2.7), we get that
This shows that is additive.
Now, we are going to prove that is a generalized module- left derivation. Letting in (2.1) gives that
for all and . By replacing with in (2.10), respectively, we deduce that
for all and . Letting , the condition (b) yields that
for all and . Since is additive, is module- additive. Put . Then by (2.10) we see from the condition (a) that
for all and . Hence
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)
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 . By letting , we get from the condition (a) that
for all . Now, (2.12) implies that for all and all . Hence, is a module- left derivation on . This completes the proof.
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.
In Theorem 2.1, if the condition (2.1) is replaced with
for all and where
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.
Let be complex vector spaces. Then a mapping is linear if and only if
for all and all .
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,
for all . This shows that is linear. The proof is completed.
Let be a Banach algebra, a Banach -bimodule, and integers greater than , and satisfy the following conditions:
Suppose that and are mappings such that , exists for all and
for all , and all , where stands for
Then is a linear generalized module- left derivation and is a linear module- left derivation.
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
for every . Taking in (2.22) yields that
for all and all . If we replace and with and in (2.25), respectively, then we see that
as for all and all . Hence,
for all and all . Since is additive, taking in (2.27) implies that
for all and all . Lemma 2.4 yields that is linear and so is . This completes the proof.
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
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 .
In Theorem 2.5, if the condition (2.22) is replaced with
for all , and where stands for
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 .
The controlling function
consists of the "mixed sum-product of powers of norms," introduced by Rassias (in 2007)  and applied afterwards by Ravi et al. (2007-2008). Moreover, it is easy to check that the functional
satisfies theconditions (a), (b), and (c) in Theorems 2.1 and 2.5, where and are all constants.
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This subject is supported by the NNSFs of China (no: 10571113,10871224).
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Cao, H., Lv, J. & Rassias, J.M. Superstability for Generalized Module Left Derivations and Generalized Module Derivations on a Banach Module (I). J Inequal Appl 2009, 718020 (2009). https://doi.org/10.1155/2009/718020
- Generalize Module
- Additive Mapping
- Banach Algebra
- Generalize Derivation
- Jacobson Radical