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Comment on ‘Approximate ∗-derivations and approximate quadratic ∗-derivations on -algebras’ [Jang, Park, J. Inequal. Appl. 2011 (2011), Article ID 55]
© Park et al.; licensee Springer 2012
- Received: 1 March 2012
- Accepted: 31 May 2012
- Published: 30 August 2012
In (J. Inequal. Appl. 2011:Article ID 55, Section 4, 2011), Jang and Park proved the Hyers-Ulam stability of quadratic ∗-derivations on Banach ∗-algebras. One can easily show that all the quadratic ∗-derivations δ in Section 4 must be trivial. So the results are trivial. In this paper, we correct the statements and prove the corrected results.
MSC:39B52, 47B47, 39B72.
- quadratic ∗-derivation
- Banach ∗-algebra
- Hyers-Ulam stability
Suppose that is a complex Banach ∗-algebra. A -linear mapping is said to be a derivation on if for all , where is a domain of δ and is dense in . If δ satisfies the additional condition for all , then δ is called a ∗-derivation on . It is well known that if is a -algebra and is , then the derivation δ is bounded.
A functional equation is called stable if any function satisfying the functional equation ‘approximately’ is near to a true solution of the functional equation.
In 1940, Ulam  proposed the following question concerning the stability of group homomorphisms: Under what condition does there exist an additive mapping near an approximately additive mapping? Hyers  answered the problem of Ulam for the case where and are Banach spaces. A generalized version of the theorem of Hyers for an approximately linear mapping was given by Rassias . Since then, the stability problems of various functional equations have been extensively investigated by a number of authors (see [8–20]).
Jang and Park [, Section 4] proved the Hyers-Ulam stability of quadratic ∗-derivations on Banach ∗-algebras.
Theorem 1.1 ([, Theorem 4.2])
for all .
for all . Thus for some . Since δ is quadratic, and so . Letting in the last equality, we get . So δ must be zero. Thus the results are trivial.
In this paper, we correct the wrong statements in  and prove the corrected results.
In this section, we correct the statements of [, Section 4] and prove the Hyers-Ulam stability of the corrected results.
δ is a quadratic mapping,
for all ,
for all .
for all . Then it is easy to show that is a quadratic ∗-derivation on .
for all .
for all .
Passing to the limit as , we get the for all . So δ is a quadratic ∗-derivation on , as desired. □
for all .
Proof Putting in Theorem 2.3, we get the desired result. □
Similarly, we can obtain the following. We will omit the proof.
for all .
Proof Putting in Theorem 2.5, we get the desired result. □
The first author, the second author and the third author were supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2009-0070788), (NRF-2010-0021792) and (NRF-2010-0009232), respectively.
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