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Approximate ∗-derivations on fuzzy Banach ∗-algebras
© Jang; licensee Springer 2012
Received: 16 March 2012
Accepted: 20 July 2012
Published: 3 August 2012
In this paper, we establish functional equations of ∗-derivations and prove the stability of ∗-derivations on fuzzy Banach ∗-algebras. We also prove the superstability of ∗-derivations on fuzzy Banach ∗-algebras.
MSC:39B52, 47B47, 46L05, 39B72.
Let be a 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 A, then the ∗-derivation δ is bounded. For several reasons, the theory of bounded derivations of -algebras is very important in the theory of quantum mechanics and operator algebras [3, 4].
A functional equation is called stable if any function satisfying a functional equation “approximately” is near to a true solution of the functional equation. We say that a functional equation is superstable if every approximate solution is an exact solution of it.
which are called the Cauchy equation and the Jensen equation, respectively. Every solution of the functional equations (1.1) and (1.2) is said to be an additive mapping.
Since Katsaras  introduced the idea of fuzzy norm on a linear space, several definitions for a fuzzy norm on a linear space have been introduced and discussed from different points of view [5–7]. We use the definition of fuzzy normed spaces given in [5, 17] to investigate the stability of derivation in the fuzzy Banach ∗-algebra setting. The stability of functional equations in fuzzy normed spaces was begun by , after then lots of results of fuzzy stability were investigated [11, 13, 16, 18].
Let X be a real vector space. A function is called a fuzzy norm on X if for all and all ,
() for ;
() if and only if for all ;
() if ;
() is a non-decreasing function of and ;
() for , is continuous on .
The pair is called a fuzzy normed vector space.
Furthermore, we can make a fuzzy normed ∗-algebra if we add () and () as follows:
Let be a fuzzy normed vector space. A sequence in X is said to be convergent or converge if there exists an such that for all . In this case, x is called the limit of the sequence and we denote it by N-.
Let be a fuzzy normed vector space. A sequence in X is called Cauchy if for each and each there exists an such that for all and all , we have .
It is well known that every convergent sequence in a fuzzy normed vector space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed vector space is called a fuzzy Banach space.
We say that a mapping between fuzzy normed vector spaces X and Y is continuous at a point if for each sequence converging to in X, then the sequence converges to . If is continuous at each , then is said to be continuous on X.
introduced in  we prove fuzzy version of the stability of ∗-derivations associated to the Cauchy functional equation and the Jensen functional equation. We also prove the superstability of ∗-derivations on fuzzy Banach ∗-algebras.
2 Stability of ∗-derivations on fuzzy Banach ∗-algebras
In this section, let be a fuzzy Banach ∗-algebra.
for all .
which is greater than or equal to . So, we have that for all . It follows from that for all . So, δ is a *-derivation on . □
for all .
3 Stability of ∗-derivations associated to the Jensen equation
The stability of the Jensen equation has been studied first by Kominek and then by several other mathematicians: (). In this section, we study the stability of ∗-derivation associated to the Jensen equation in a fuzzy Banach ∗-algebra .
for all .
for all .
which is greater than or equal to .
for all and all . Fix temporarily. Since , there exists such that for all .
Hence, δ is the ∗-derivation on that we want. □
4 Superstability of ∗-derivations
In this section, we prove the superstability of ∗-derivations on a fuzzy Banach ∗-algebras. More precisely, we introduce the concept of -approximate ∗-derivation and show that any -approximate ∗-derivation is just a ∗-derivation.
for all . Then δ is called a -approximate ∗-derivation on .
Theorem 4.2 Let be a fuzzy Banach ∗-algebra with approximate unit. Then any -approximate ∗-derivation δ on is a ∗-derivation.
for all . Fix temporarily. Since , there exists such that and for all and .
for all . Fix temporarily. By (4.1), we can find such that , , and for all .
we can have for all . By () and using approximate unit for all . Thus, δ is a ∗-derivation on . □
The author would like to thank the editor Prof. Wong and two referees for their valuable comments. And the author was partially supported by the Research Fund, University of Ulsan 2011.
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