- Research Article
- Open Access
Stability of Homomorphisms and Generalized Derivations on Banach Algebras
© A. Najati and C. Park 2009
- Received: 14 June 2009
- Accepted: 18 November 2009
- Published: 19 November 2009
- Banach Space
- Functional Equation
- Linear Space
- Stability Problem
- Banach Algebra
The first stability problem concerning group homomorphisms was raised from a question of Ulam . Let be a group and let be a metric group with the metric . Given , does there exist such that if a mapping satisfies the inequality
Hyers  gave a first affirmative answer to the question of Ulam for Banach spaces. Aoki  and Rassias  provided a generalization of the Hyers' theorem for additive and linear mappings, respectively, by allowing the Cauchy difference to be unbounded (see also ).
Theorem 1.1 (Rassias).
In 1994, a generalization of the Rassias' theorem was obtained by G vruţa , who replaced the bound by a general control function For the stability problems of various functional equations and mappings and their Pexiderized versions, we refer the readers to [7–15]. We also refer readers to the books in [16–19].
for all If, in addition, for all and all then is called a linear derivation, where denotes the scalar field of . Singer and Wermer  proved that if is a commutative Banach algebra and is a continuous linear derivation, then They also conjectured that the same result holds even is a discontinuous linear derivation. Thomas  proved the conjecture. As a direct consequence, we see that there are no nonzero linear derivations on a semisimple commutative Banach algebra, which had been proved by Johnson . On the other hand, it is not the case for ring derivations. Hatori and Wada  determined a representation of ring derivations on a semi-simple commutative Banach algebra (see also ) and they proved that only the zero operator is a ring derivation on a semi-simple commutative Banach algebra with the maximal ideal space without isolated points. The stability of derivations between operator algebras was first obtained by emrl . Badora  and Miura et al.  proved the Hyers-Ulam-Rassias stability of ring derivations on Banach algebras. An additive mapping is called a Jordan derivation in case is fulfilled for all Every derivation is a Jordan derivation. The converse is in general not true (see [27, 28]). The concept of generalized derivation has been introduced by M. Brešar . Hvala  and Lee  introduced a concept of -derivation (see also ). Let be automorphisms of An additive mapping is called a -derivation in case holds for all pairs An additive mapping is called a -Jordan derivation in case holds for all An additive mapping is called a generalized -derivation in case holds for all pairs where is a -derivation. An additive mapping is called a generalized -Jordan derivation in case holds for all where is a -Jordan derivation. It is clear that every generalized -derivation is a generalized -Jordan derivation.
We recall the following theorem by Margolis and Diaz.
Theorem 1.2 (See ).
Daróczy et al.  have studied the functional equation
where is a fixed parameter and is unknown, is a nonvoid open interval and (2.1) holds for all They characterized the equivalence of (2.1) and Jensen's functional equation in terms of the algebraic properties of the parameter For in (2.1), we get the Jensen's functional equation. In the present paper, we establish the general solution and some stability results concerning the functional equation (2.1) in normed spaces for This applied to investigate and prove the generalized Hyers-Ulam stability of homomorphisms and generalized derivations in real Banach algebras. In this section, we assume that is a normed algebra and is a Banach algebra. For convenience, we use the following abbreviation for a given mapping
It is easy to show that is a generalized complete metric space .
Finally it remains to prove the uniqueness of . Let another homomorphism satisfying (2.13). Since and is additive, we get and for all , that is, is a fixed point of . Since is the unique fixed point of in , we get
We need the following lemma in the proof of the next theorem.
Lemma 2.3 (See ).
Let satisfy (2.31). Letting in (2.31), we get By Lemma 2.1, the mapping is additive. Letting in (2.31) and using the additivity of we get that for all and all So by Lemma 2.4, the mapping is -linear. The converse is obvious.
The following theorem is an alternative result of Theorem 2.2 with similar proof.
The following theorem is an alternative result of Theorem 3.1 with similar proof.
The second author was supported by Hanyang University in 2009.
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