Discrete Dynamics in Nature and SocietyVolume 2012 (2012), Article ID 864340, 8 pagesdoi:10.1155/2012/864340

Research Article

Nearly Derivations on Banach Algebras

M. Eshaghi Gordji,¹ H. Khodaei,¹ and G. H. Kim²

¹Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran²Department of Mathematics, Kangnam University, Yongin, Gyeonggi 446-702, Republic of Korea

Received 11 December 2011; Accepted 23 January 2012

Academic Editor: John Rassias

Copyright © 2012 M. Eshaghi Gordji et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Let be a fixed integer greater than 3 and let be a real number with . We investigate the Hyers-Ulam stability of derivations on Banach algebras related to the following generalized Cauchy functional inequality .

1. Introduction and Preliminaries

Let ba a Banach algebra and let be a Banach -bimodule. Then , the dual space of , is also a Banach -bimodule with module multiplications defined by A bounded linear operator is called a derivation if

Let . We define for all . is a derivation from into , which is called inner derivation. A Banach algebra is amenable if every derivation from into every dual -bimodule is inner. This definition was introduced by Johnson in [1]. A Banach algebra is weakly amenable if every derivation from into is inner. Bade et al. [2] have introduced the concept of weak amenability for commutative Banach algebras.

The stability problem of functional equations originated from a question of Ulam [3, 4] concerning the stability of group homomorphisms.

A famous talk presented by Ulam in 1940 triggered the study of stability problems for various functional equations.

We are given a group and a metric group with metric . Given , does there exist a such that if satisfies for all , then a homomorphism exists with for all ?

In the following year, Hyers was able to give a partial solution to Ulam’s question that was the first significant breakthrough and step toward more solutions in this area (see [5]). Since then, a large number of papers have been published in connection with various generalizations of Ulam’s problem and Hyers’ theorem.

Let be a fixed integer greater than 3 and let be a real number with . We investigate the Hyers-Ulam stability of derivations on Banach algebras related to the following generalized Cauchy functional inequality:

2. Main Results

Let be a Banach algebra and let be a Banach -module. From now on, the sum of and will be denoted by . Also, will be denoted by . In the following, we will use the Pascal formula: here, denotes Moreover, we assume that is a positive integer and suppose that .

Lemma 2.1. Let be a mapping such that for all . Then is Cauchy additive.

Proof. Substituting in the functional inequality (2.2), we get Since and , . Letting , and in (2.2) and using Pascal formula, we get for all . Hence for all . Letting , , and in (2.2), we get for all . Hence for all . Since , we obtain from (2.7) and (2.4) that for all . It follows from (2.5) and (2.8) that for all . By using (2.5) and (2.9), we get and so for all . Hence, we obtain from (2.7) that for all . Letting , , and in (2.2), we get for all . Next, notice that, using oddness of and , we have for all , as desired.

We can prove the following lemma by the same reasoning as in the proof of Theorem  2.2 of [6].

Lemma 2.2. Let be an additive mapping such that for all and all . Then the mapping is -linear.

Theorem 2.3. Let be a mapping satisfying and the inequality for some , for all and all . Then there exists a unique derivation such that for all .

Proof. Letting , , , and in (2.12), we get for all . Letting , , , , and in (2.12), we get for all . Letting , , , , and in (2.12), we get for all . It follows from (2.15) and (2.16) that for all . It follows from (2.14) and (2.17) that for all . It follows from (2.15) and (2.18) that for all . From the last two inequalities, we have for all . It follows from (2.18) and (2.20) that for all . Hence for all and integers . Thus it follows that a sequence is Cauchy in and so it converges. Therefore we can define a mapping by for all . In addition it is clear from (2.12) that the following inequality: holds for all and all . If we put in the last inequality, then is additive by Lemma 2.1. Letting , and in last inequality and using Lemma 2.1, we get So for all and all . Now by using Lemmas 2.1 and 2.2, we infer that the mapping is -linear. Taking the limit as in (2.22) with , we get (2.13).To prove the afore-mentioned uniqueness, we assume now that there is another -linear mapping which satisfies the inequality (2.13). Then we get for all and integers . Thus from , one establishes for all , completing the proof of uniqueness.Now, we have to show that is a derivation. To this end, let in (2.12), we get for all . It follows from linearity of and (2.27) that for all . This means that is a derivation from into . Therefore the mapping is a unique derivation satisfying (2.13), as desired.

Theorem 2.4. Let be an amenable Banach algebra and let be a mapping such that and (2.12). If then there exists such that for all .

Proof. Let . Then by (2.29), we have . By Theorem 2.3, there exists a derivation satisfying (2.13). Then we have This means that is bounded, and hence is continuous. On the other hand, is amenable. Then every continuous derivation from into is an inner derivation. It follows that is and an inner derivation. In the other words, there exists such that for all . This completes the proof.

We know that every nuclear -algebra is amenable (see [7]). Then we have the following result.

Corollary 2.5. Let be a nuclear -algebra and let be a mapping such that , and (2.12) and (2.29). Then there exists such that for all .

Theorem 2.6. Let be a -algebra and let be a mapping such that , and (2.12) and (2.29). Then there exists such that for all .

Proof. We know that every -algebra is weakly amenable (see, e.g., [7]). Then every continuous derivation from into is an inner derivation. By the same reasoning as in the proof of Theorem 2.4, there exists a such that for all , and for all . By definition of mudule actions of on , we have for all .

Corollary 2.7. Let be a commutative -algebra and let be a mapping such that , and (2.12) and (2.29). Then for all .

References

B. E. Johnson, “Cohomology in Banach algebras,” Memoirs of the American Mathematical Society, vol. 127, 1972.

W. G. Bade, P. G. Curtis, and H. G. Dales, “Amenability and weak amenability for Beurling and Lipschits algebra,” Proceedings London Mathematical Society, vol. 55, no. 3, pp. 359–377, 1987.

S. M. Ulam, A Collection ofMathematical Problems, Interscience Publisher, New York, NY, USA, 1960.

S. M. Ulam, Problems in Modern Mathematics, vol. 6, Wiley-Interscience, New York, 1964.

D.H. Hyers, “On the stability of the linear functional equation,” Proceedings of the National Academy of Sciences of the United States of America, vol. 27, pp. 222–224, 1941.

M. E. Gordji, “Nearly involutions onBanach algebras; A fixed point approach,” Fixed Point Theory. In press

H. G. Dales, Banach Algebras and Automatic Continuity, vol. 24 of London Mathematical Society Monographs, New Series, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, NY, USA, 2000.