- Research Article
- Open Access
Superstability for Generalized Module Left Derivations and Generalized Module Derivations on a Banach Module (I)
© Huai-Xin Cao et al. 2009
- Received: 23 January 2009
- Accepted: 3 July 2009
- Published: 18 August 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.
- Generalize Module
- Additive Mapping
- Banach Algebra
- Generalize Derivation
- Jacobson Radical
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 . 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.
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 .
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.
This shows that is additive.
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.
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.
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.
for all and all .
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:
Then is a linear generalized module- left derivation and is a linear module- left derivation.
for all and all . Lemma 2.4 yields that is linear and so is . This completes the proof.
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 .
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 .
satisfies theconditions (a), (b), and (c) in Theorems 2.1 and 2.5, where and are all constants.
This subject is supported by the NNSFs of China (no: 10571113,10871224).
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