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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,
δ is quadratic homogeneous, that is, for all and all ,
for all ,
for all .
for all . Then it is easy to show that is a quadratic ∗-derivation on .
for all .
for all and all (). This means that δ is quadratic homogeneous.
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.
- Jang S, Park C:Approximate ∗-derivations and approximate quadratic ∗-derivations on -algebras. J. Inequal. Appl. 2011., 2011: Article ID 55Google Scholar
- Bratteli O Lecture Notes in Mathematics 1229. In Derivation, Dissipation and Group Actions on C∗-Algebras. Springer, Berlin; 1986.Google Scholar
- Bratteli O, Goodman FM, Jørgensen PET: Unbounded derivations tangential to compact groups of automorphisms II. J. Funct. Anal. 1985, 61: 247–289. 10.1016/0022-1236(85)90022-9MathSciNetView ArticleMATHGoogle Scholar
- Lee S, Jang S:Unbounded derivations on compact actions of -algebras. Commun. Korean Math. Soc. 1990, 5: 79–86.Google Scholar
- Ulam SM Science Edn. In Problems in Modern Mathematics. Wiley, New York; 1940. Chapter VIGoogle Scholar
- Hyers DH: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 1941, 27: 222–224. 10.1073/pnas.27.4.222MathSciNetView ArticleMATHGoogle Scholar
- Rassias TM: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 1978, 72: 297–300. 10.1090/S0002-9939-1978-0507327-1View ArticleMathSciNetMATHGoogle Scholar
- Aczl J, Dhombres J: Functional Equations in Several Variables. Cambridge University Press, Cambridge; 1989.View ArticleGoogle Scholar
- Czerwik S: On the stability of the quadratic mapping in normed spaces. Abh. Math. Semin. Univ. Hamb. 1992, 62: 59–64. 10.1007/BF02941618MathSciNetView ArticleMATHGoogle Scholar
- Gajda Z: On stability of additive mappings. Int. J. Math. Math. Sci. 1991, 14: 431–434. 10.1155/S016117129100056XMathSciNetView ArticleMATHGoogle Scholar
- Hyers DH, Isac G, Rassias TM: Stability of Functional Equations in Several Variables. Birkhäuser, Basel; 1998.View ArticleMATHGoogle Scholar
- Jun K, Kim H: Approximate derivations mapping into the radicals of Banach algebras. Taiwan. J. Math. 2007, 11: 277–288.MathSciNetMATHGoogle Scholar
- Kannappan P: Quadratic functional equation and inner product spaces. Results Math. 1995, 27: 368–372.MathSciNetView ArticleMATHGoogle Scholar
- Găvruta P: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 1994, 184: 431–436. 10.1006/jmaa.1994.1211MathSciNetView ArticleMATHGoogle Scholar
- Jung S Springer Optimization and Its Applications 48. In Hyers-Ulam-Rassias Stability of Functional Equations in Non-Linear Analysis. Springer, New York; 2011.View ArticleGoogle Scholar
- Skof F: Propriet locali e approssimazione di operatori. Rend. Semin. Mat. Fis. Milano 1983, 53: 113–129. 10.1007/BF02924890MathSciNetView ArticleGoogle Scholar
- Lee J, An J, Park C: On the stability of quadratic functional equations. Abstr. Appl. Anal. 2008., 2008: Article ID 628178Google Scholar
- Gharetapeh SK, Eshaghi Gordji M, Ghaemi MB, Rashidi E:Ternary Jordan homomorphisms in -ternary algebras. J. Nonlinear Sci. Appl. 2011, 4: 1–10.MathSciNetMATHGoogle Scholar
- Park C, Boo D:Isomorphisms and generalized derivations in proper -algebras. J. Nonlinear Sci. Appl. 2011, 4: 19–36.MathSciNetMATHGoogle Scholar
- Javadian A, Eshaghi Gordji M, Savadkouhi MB: Approximately partial ternary quadratic derivations on Banach ternary algebras. J. Nonlinear Sci. Appl. 2011, 4: 60–69.MathSciNetMATHGoogle Scholar
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