On the stability of ∗-derivations on Banach ∗-algebras
© Park and Bodaghi; licensee Springer 2012
Received: 12 June 2012
Accepted: 30 July 2012
Published: 7 August 2012
In the current paper, we study the stability and the superstability of ∗-derivations associated with the Cauchy functional equation and the Jensen functional equation. We also prove the stability and the superstability of Jordan ∗-derivations on Banach ∗-algebras.
MSC:39B52, 47B47, 39B72, 47H10, 46H25.
Keywords∗-derivation Banach ∗-algebra Jordan ∗-derivation stability superstability
The basic problem of the stability of functional equations asks whether an approximate solution of the Cauchy functional equation can be approximated by a solution of this equation . A functional equation is called stable if any approximately solution to the functional equation is near to a true solution of that functional equation, and is superstable if every approximately solution is an exact solution of it.
The study of stability problems for functional equations which had been proposed by Ulam  concerning the stability of group homomorphisms, affirmatively answered for Banach spaces by Hyers . The Hyers’ theorem was generalized by Aoki  and Bourgin  for additive mappings by considering an unbounded Cauchy difference. In , Th. M. Rassias succeeded in extending the result of the Hyers’ theorem by weakening the condition for the Cauchy difference controlled by , to be unbounded. Gǎvruta generalized the Rassias’ result in  for the unbounded Cauchy difference. And then, the stability problems of various functional equation have been extensively investigated by a number of authors and there are many interesting results concerning this problem (for instances,  and ). The stability of ∗-derivations and of quadratic ∗-derivations with the Cauchy functional equation and the Jensen functional equation on Banach ∗-algebras was investigated in . Jang and Park  proved the superstability of ∗-derivations and of quadratic ∗-derivations on -algebras. In , An, Cui, and Park investigated Jordan ∗-derivations on -algebras and Jordan ∗-derivations on -algebras associated with a special functional inequality.
In 2003, Cǎdariu and Radu employed the fixed-point method to the investigation of the Jensen functional equation. They presented a short and a simple proof (different from the ‘direct method,’ initiated by Hyers in 1941) for the Cauchy functional equation  and for the quadratic functional equation . The Hyers-Ulam stability of the Jensen functional equation was studied by this method in . Also, this method is applied to prove the stability and the superstability for cubic and quartic functional equation under certain conditions on Banach algebras in [5, 6] (for the stability of ternary quadratic derivations on ternary Banach algebras and -ternary rings, see ).
The stability and the superstability of homomorphisms on -algebras by using the fixed point alternative (Theorem 2.1) were proved in . The Hyers-Ulam stability of ∗-homomorphisms in unital -algebras associated with the Trif functional equation, and of linear ∗-derivations on unital -algebras have earlier been established by Park and Hou in .
In this paper, we prove the stability of ∗-derivations associated with the Cauchy functional equation and the Jensen functional equation on Banach ∗-algebras. We also show that these functional equations under some mild conditions are superstable. We indicate a more accurate approximation than the results of Jang and Park which are obtained in . In fact, we obtain an extension and refinement of their results on Banach ∗-algebras. So the condition of being -algebra for A in  can be redundant.
2 Stability of ∗-derivations
Throughout this paper, assume that B is a Banach ∗-algebra and that A is a Banach ∗-subalgebra of B. A bounded -linear mapping is said to be derivation on A if for all . In addition, if D satisfies the additional condition for all , then it is called a ∗-derivation.
Before proceeding to the main results, we will state the following theorem which is useful to our purpose (an extension of the result was given in ).
Theorem 2.1 (The fixed-point alternative )
for all ;
the sequence is convergent to a fixed-point of ;
is the unique fixed point of in the set ;
for all .
To achieve our maim in this section, we shall use the following lemma which is proved in .
Lemma 2.2 Letbe a positive integer and let X, Y be complex vector spaces. Suppose thatis an additive mapping. Then f is-linear if and only iffor all x in X and μ in.
We establish the Hyers-Ulam stability of ∗-derivations as follows:
for all . Passing to the limit as in (9), we conclude that for all . Thus, D is a ∗-derivation. □
for all and . □
In the following corollaries, we show that under some conditions the superstability for the inequality (1) is valid.
Corollary 2.5 Letbe an additive mapping satisfying (1) andbe a function satisfying (2). Then f is a ∗-derivation.
Proof It follows immediately from additivity of f that . Thus, for all . Now, by the proof of Theorem 2.3, f is a ∗-derivation. □
for alland all. Then f is a ∗-derivation on A.
Proof If we put and in (10), we get . Again, putting , , and in (10), we conclude that , and by induction we have for all and . Now, we can obtain the desired result by Theorem 2.3. □
A bounded -linear mapping is said to be Jordan derivation on A if for all . Note that the mapping , where a is a fixed element in A, is a Jordan ∗-derivation. For the first time, Jordan ∗-derivations were introduced in [23, 24] and the structure of such derivations has investigated in . The reason for introducing these mappings was the fact that the problem of representing quadratic forms by sesquilinear ones is closely connected with the structure of Jordan ∗-derivations.
The next theorem is in analogy with Theorem 2.3 for Jordan ∗-derivations. Since the proof is similar, it is omitted.
The following corollaries are analogous to Corollaries 2.4, 2.5, and 2.6, respectively. The proofs are similar and so we omit them.
Corollary 2.9 Letbe an additive mapping satisfying (11) and letbe a function satisfying (12). Then f is a Jordan ∗-derivation.
for alland for all. Then f is a Jordan ∗-derivation on A.
3 Stability of ∗-derivations associated with the Jensen functional equation
In this section, we investigate the stability and the superstability of ∗-derivations associated with the Jensen functional equation in Banach ∗-algebra.
for all and all . By Lemma 2.2, D is -linear.
The rest of the proof is similar to the proof of Theorem 2.3. □
Proof The proof follows from Theorem 3.1 by taking for all and . □
In the following corollary, we show that when f is an additive mapping, the superstability for the inequality (13) holds.
Corollary 3.3 Letbe an additive mapping satisfying (13) and letbe a function satisfying (14). Then f is a ∗-derivation.
Proof The proof is similar to the proof of Corollary 2.5. □
for alland all. If (), δ are nonnegative real numbers such that, then f is a ∗-derivation.
Proof Putting and in (21), we obtain . Replacing a by 2a and setting and in (21), we have , and thus for all and . Now, Theorem 3.1 shows that f is a ∗-derivation on A. □
for all . Now, Lemma 2.2 shows that D is -linear. The rest of the proof is similar to the proof of Theorem 2.3. □
The authors would like to thank the referee for careful reading of the paper and giving some useful suggestions.
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