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On the stability of ∗derivations on Banach ∗algebras
Advances in Difference Equations volume 2012, Article number: 138 (2012)
Abstract
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.
1 Introduction
The basic problem of the stability of functional equations asks whether an approximate solution of the Cauchy functional equation f(x+y)=f(x)+f(y) can be approximated by a solution of this equation [21]. 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 [26] concerning the stability of group homomorphisms, affirmatively answered for Banach spaces by Hyers [16]. The Hyers’ theorem was generalized by Aoki [2] and Bourgin [8] for additive mappings by considering an unbounded Cauchy difference. In [22], Th. M. Rassias succeeded in extending the result of the Hyers’ theorem by weakening the condition for the Cauchy difference controlled by {\parallel a\parallel}^{r}+{\parallel b\parallel}^{r}, r\in [0,1) to be unbounded. Gǎvruta generalized the Rassias’ result in [15] 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, [12] and [17]). The stability of ∗derivations and of quadratic ∗derivations with the Cauchy functional equation and the Jensen functional equation on Banach ∗algebras was investigated in [18]. Jang and Park [18] proved the superstability of ∗derivations and of quadratic ∗derivations on {C}^{\ast}algebras. In [1], An, Cui, and Park investigated Jordan ∗derivations on {C}^{\ast}algebras and Jordan ∗derivations on J{C}^{\ast}algebras associated with a special functional inequality.
In 2003, Cǎdariu and Radu employed the fixedpoint 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 [10] and for the quadratic functional equation [9]. The HyersUlam stability of the Jensen functional equation was studied by this method in [19]. 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 {C}^{\ast}ternary rings, see [4]).
The stability and the superstability of homomorphisms on {C}^{\ast}algebras by using the fixed point alternative (Theorem 2.1) were proved in [14]. The HyersUlam stability of ∗homomorphisms in unital {C}^{\ast}algebras associated with the Trif functional equation, and of linear ∗derivations on unital {C}^{\ast}algebras have earlier been established by Park and Hou in [20].
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 [18]. In fact, we obtain an extension and refinement of their results on Banach ∗algebras. So the condition of being {C}^{\ast}algebra for A in [18] 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 \mathbb{C}linear mapping D:A\to B is said to be derivation on A if D(ab)=D(a)\cdot b+a\cdot D(b) for all a,b\in A. In addition, if D satisfies the additional condition D({a}^{\ast})=D{(a)}^{\ast} for all a\in A, 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 [25]).
Theorem 2.1 (The fixedpoint alternative [11])
Let(\mathrm{\Omega},d)be a complete generalized metric space and\mathcal{T}:\mathrm{\Omega}\to \mathrm{\Omega}be a mapping with Lipschitz constantL<1. Then, for each element\alpha \in \mathrm{\Omega}, eitherd({\mathcal{T}}^{n}\alpha ,{\mathcal{T}}^{n+1}\alpha )=\mathrm{\infty}for alln\ge 0, or there exists a natural number{n}_{0}such that:

(i)
d({\mathcal{T}}^{n}\alpha ,{\mathcal{T}}^{n+1}\alpha )<\mathrm{\infty} for all n\ge {n}_{0};

(ii)
the sequence \{{\mathcal{T}}^{n}\alpha \} is convergent to a fixedpoint {\beta}^{\ast} of \mathcal{T};

(iii)
{\beta}^{\ast} is the unique fixed point of \mathcal{T} in the set \mathrm{\Lambda}=\{\beta \in \mathrm{\Omega}:d({\mathcal{T}}^{{n}_{0}}\alpha ,\beta )<\mathrm{\infty}\};

(iv)
d(\beta ,{\beta}^{\ast})\le \frac{1}{1L}d(\beta ,\mathcal{T}\beta ) for all \beta \in \mathrm{\Lambda}.
To achieve our maim in this section, we shall use the following lemma which is proved in [13].
Lemma 2.2 Let{n}_{0}\in \mathbb{N}be a positive integer and let X, Y be complex vector spaces. Suppose thatf:X\to Yis an additive mapping. Then f is\mathbb{C}linear if and only iff(\mu x)=\mu f(x)for all x in X and μ in{\mathbb{T}}_{\frac{1}{{n}_{0}}}^{1}:=\{{e}^{i\theta}:0\le \theta \le \frac{2\pi}{{n}_{0}}\}.
We establish the HyersUlam stability of ∗derivations as follows:
Theorem 2.3 Let{n}_{0}\in \mathbb{N}be fixed, f:A\to Ba mapping withf(0)=0and let\psi :{A}^{5}\to [0,\mathrm{\infty})be a function such that
for all\mu \in {\mathbb{T}}_{\frac{1}{{n}_{0}}}^{1}and alla,b,x,y,z\in A. If there exists a constantk\in (0,1), such that
for alla,b,x,y,z\in A, then there exists a unique ∗derivation D on A satisfying
where\tilde{\psi}(a)=\psi (a,a,0,0,0).
Proof First, we provide the conditions of Theorem 2.1. We consider the set
and define the mapping d on \mathrm{\Omega}\times \mathrm{\Omega} as follows:
if there exist such constant C, and d({g}_{1},{g}_{2})=\mathrm{\infty}, otherwise. Similar to the proof of [7], Theorem 2.2], we can show that d is a generalized metric on Ω and the metric space (\mathrm{\Omega},d) is complete. We define a mapping \mathrm{\Psi}:\mathrm{\Omega}\to \mathrm{\Omega} by
for all a\in A. We show that Ψ is strictly contractive on Ω. Given {g}_{1},{g}_{2}\in \mathrm{\Omega}, let c\in (0,\mathrm{\infty}) be an arbitrary constant with d({g}_{1},{g}_{2})\le c. It means that
for all a\in A. Substituting a by 2a in the inequality (5) and using (2) and (4), we get
for all a\in A. Then d(\mathrm{\Psi}{g}_{1},\mathrm{\Psi}{g}_{2}(a))\le ck. This shows that
for all {g}_{1},{g}_{2}\in \mathrm{\Omega}. To achieve inequality (3), we prove that d(\mathrm{\Psi}f,f)<\mathrm{\infty}. Putting a=b, x=y=z=0, and \mu =1 (in fact 1\in {\mathbb{T}}_{\frac{1}{n}}^{1} for all n\in \mathbb{N}) in (1), we obtain
for all a\in A. Hence,
for all a\in A. We conclude from (6) that d(\mathrm{\Psi}f,f)\le \frac{1}{2}. It follows from Theorem 2.1 that d({\mathcal{J}}^{n}g,{\mathcal{J}}^{n+1}g)<\mathrm{\infty} for all n\ge 0, and thus in this theorem we have {n}_{0}=0. Therefore, the parts (iii) and (iv) of Theorem 2.1 hold on the whole Ω. Hence, there exists a unique mapping D:A\to B such that D is a fixed point of T and that {\mathrm{\Psi}}^{n}f\to D as n\to \mathrm{\infty}. Thus,
for all a\in A, and so
The above inequality shows that (3) is true for all a\in A. It follows from (2) that
Now, we replace a by {2}^{n}a and put b=x=y=z=0 in (1). We divide both sides of the resulting inequality by {2}^{n}, and let n tend to infinity. It follows from (1), (7), and (8) that
for all a\in A and all \mu \in {\mathbb{T}}_{\frac{1}{{n}_{0}}}^{1}. Now, by Lemma 2.2, D is \mathbb{C}linear. Replacing x, y by {2}^{n}x, {2}^{n}y, respectively, and putting a=b=z=0 in (1), we have
Taking the limit as n tend to infinity, we get D(xy)=D(x)\cdot y+x\cdot D(y) for all x,y\in A. If we put a=b=x=y=0 and substitute z by {2}^{n}z in (1) and we divide the both sides of the obtained inequality by {2}^{n}, then we get
for all z\in A. Passing to the limit as n\to \mathrm{\infty} in (9), we conclude that D({z}^{\ast})=D{(z)}^{\ast} for all z\in A. Thus, D is a ∗derivation. □
Corollary 2.4 Let{n}_{0}\in \mathbb{N}be fixed, r\in (0,1), and letf:A\to Bbe mappings withf(0)=0such that
for all\mu \in {\mathbb{T}}_{\frac{1}{{n}_{0}}}^{1}and alla,b,x,y,z\in A. Then there exists a unique ∗derivation D on A satisfying
for alla\in A.
Proof The proof follows from Theorem 2.3 by taking
for all a,b,x,y,z\in A and k={2}^{r1}. □
In the following corollaries, we show that under some conditions the superstability for the inequality (1) is valid.
Corollary 2.5 Letf:A\to Bbe an additive mapping satisfying (1) and\psi :{A}^{5}\to [0,\mathrm{\infty})be a function satisfying (2). Then f is a ∗derivation.
Proof It follows immediately from additivity of f that f(0)=0. Thus, f({2}^{n}a)={2}^{n}f(a) for all a\in A. Now, by the proof of Theorem 2.3, f is a ∗derivation. □
Corollary 2.6 Let{r}_{j} (1\le j\le 5), δ be nonnegative real numbers with0<{\sum}_{j=1}^{5}{r}_{j}\ne 1and letf:A\to Bbe a mapping withf(0)=0such that
for all\mu \in {\mathbb{T}}_{\frac{1}{{n}_{0}}}^{1}and alla,b,x,y,z\in A. Then f is a ∗derivation on A.
Proof If we put a=b=x=y=z=0 and \mu =1 in (10), we get f(0)=0. Again, putting a=b, x=y=z=0, and \mu =1 in (10), we conclude that f(2a)=2f(a), and by induction we have f(a)=\frac{f({2}^{n}a)}{{2}^{n}} for all a\in A and n\in \mathbb{N}. Now, we can obtain the desired result by Theorem 2.3. □
A bounded \mathbb{C}linear mapping D:A\to A is said to be Jordan derivation on A if D({a}^{2})=D(a)\cdot a+a\cdot D(a) for all a,b\in A. Note that the mapping x\mapsto axxa, 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 [3]. 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.
Theorem 2.7 Let{n}_{0}\in \mathbb{N}be fixed, f:A\to Aa mapping withf(0)=0and let\psi :{A}^{3}\to [0,\mathrm{\infty})be a function such that
for all\mu \in {\mathbb{T}}_{\frac{1}{{n}_{0}}}^{1}and alla,b,c\in A. If there exists a constantk\in (0,1), such that
for alla,b,c\in A, then there exists a unique Jordan ∗derivation D on A satisfying
for alla\in A.
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 2.8 Let{n}_{0}\in \mathbb{N}be fixed, r\in (0,1), and letf:A\to Abe mappings withf(0)=0such that
for all\mu \in {\mathbb{T}}_{\frac{1}{{n}_{0}}}^{1}and alla,b,c,d\in A. Then there exists a unique Jordan ∗derivation D on A satisfying
for alla\in A.
Corollary 2.9 Letf:A\to Abe an additive mapping satisfying (11) and let\psi :{A}^{3}\to [0,\mathrm{\infty})be a function satisfying (12). Then f is a Jordan ∗derivation.
Corollary 2.10 Let{r}_{j} (1\le j\le 3), δ be nonnegative real numbers with0<{\sum}_{j=1}^{3}{r}_{j}\ne 1and letf:A\to Bbe a mapping withf(0)=0such that
for all\mu \in {\mathbb{T}}_{\frac{1}{{n}_{0}}}^{1}and for alla,b,c\in A. 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.
Theorem 3.1 Let{n}_{0}\in \mathbb{N}be fixed, f:A\to Ba mapping withf(0)=0and let\varphi :{A}^{5}\to [0,\mathrm{\infty})be a function such that
for all\mu \in {\mathbb{T}}_{\frac{1}{{n}_{0}}}^{1}and alla,b,x,y,z\in A. If there exists a constantk\in (0,1), such that
for alla,b,x,y,z\in A, then there exists a unique ∗derivation D on A satisfying
where\tilde{\varphi}(a)=\varphi (a,0,0,0,0).
Proof Similar to the proof of Theorem 2.3, we consider the set
and define the mapping d on \mathrm{\Omega}\times \mathrm{\Omega} as follows:
if there exist such constant C, and d({g}_{1},{g}_{2})=\mathrm{\infty}, otherwise. The metric space (\mathrm{\Omega},d) is complete and also the mapping \mathrm{\Phi}:\mathrm{\Omega}\to \mathrm{\Omega} defined by
is strictly contractive on Ω. Putting \mu =1 and b=x=y=z=0 in (13), we obtain
for all a\in A. By (14), we get
for all a\in A. So by (17), we have d(\mathrm{\Phi}f,f)\le k. Since the parts (iii) and (iv) of Theorem 2.1 hold on the whole Ω, there exists a unique mapping D:A\to B such that D is a fixed point of Φ such that
for all a\in A, and so
So (15) holds for all a\in A. It follows from (14) that
Letting a=b and putting x=y=z=0 in (13), we have
Now, replacing a by {2}^{n}a in (20) and dividing both sides of the resulting inequality by {2}^{n}, and letting n\to \mathrm{\infty}, by (13), (18), and (19), we have
for all a\in A and all \mu \in {\mathbb{T}}_{\frac{1}{{n}_{0}}}^{1}. By Lemma 2.2, D is \mathbb{C}linear.
The rest of the proof is similar to the proof of Theorem 2.3. □
Corollary 3.2 Let{n}_{0}\in \mathbb{N}be fixed, r\in (0,1), and letf:A\to Bbe mappings withf(0)=0such that
for all\mu \in {\mathbb{T}}_{\frac{1}{{n}_{0}}}^{1}and alla,b,x,y,z\in A. Then there exists a unique ∗derivation D on A satisfying
for alla\in A.
Proof The proof follows from Theorem 3.1 by taking \varphi (a,b,x,y,z)=\delta ({\parallel a\parallel}^{r}+{\parallel b\parallel}^{r}+{\parallel x\parallel}^{r}+{\parallel y\parallel}^{r}+{\parallel z\parallel}^{r}) for all a,b,x,y,z\in A and k={2}^{r1}. □
In the following corollary, we show that when f is an additive mapping, the superstability for the inequality (13) holds.
Corollary 3.3 Letf:A\to Bbe an additive mapping satisfying (13) and let\psi :{A}^{5}\to [0,\mathrm{\infty})be a function satisfying (14). Then f is a ∗derivation.
Proof The proof is similar to the proof of Corollary 2.5. □
Corollary 3.4 Let f:A\to B be a mapping with f(0)=0 such that
for all\mu \in {\mathbb{T}}_{\frac{1}{{n}_{0}}}^{1}and alla,b,x,y,z\in A. If{r}_{j} (1\le j\le 5), δ are nonnegative real numbers such that0<{\sum}_{j=1}^{5}{r}_{j}\ne 1, then f is a ∗derivation.
Proof Putting a=b=x=y=z=0 and \mu =1 in (21), we obtain f(0)=0. Replacing a by 2a and setting b=x=y=z=0 and \mu =1 in (21), we have f(2a)=2f(a), and thus f(a)=\frac{f({2}^{n}a)}{{2}^{n}} for all a\in A and n\in \mathbb{N}. Now, Theorem 3.1 shows that f is a ∗derivation on A. □
Theorem 3.5 Let{n}_{0}be a fixed natural number, f:A\to Ba mapping withf(0)=0and let\varphi :{A}^{5}\to [0,\mathrm{\infty})be a function such that
for all\mu \in {\mathbb{T}}_{\frac{1}{{n}_{0}}}^{1}and alla,b,x,y,z\in A. If there exists a constantk\in (0,1)such that
for alla,b,x,y,z\in A, then there exists a unique ∗derivation D on A satisfying
where{\tilde{\varphi}}_{1}(a)=\varphi (a,a,0,0,0)and{\tilde{\varphi}}_{2}(a)=\varphi (a,3a,0,0,0).
Proof Suppose that the set Ω as in the proof of Theorem 3.1. We introduce the generalized metric on Ω as follows:
if there exists such constant c, and d(g,h)=\mathrm{\infty}, otherwise. One can prove that the metric space (\mathrm{\Omega},d) is complete. Define the mapping \mathrm{\Phi}:\mathrm{\Omega}\to \mathrm{\Omega}via
Given g,h\in \mathrm{\Omega}, let c\in (0,\mathrm{\infty}) be an arbitrary constant with d(g,h)\le c, that is,
for all a\in A. If we substitute a by 3a in the inequality (26) and use from (23) and (25), we have
for all a\in A. Thus, d(\mathrm{\Psi}{g}_{1},\mathrm{\Psi}{g}_{2}(a))\le ck. This shows that Ψ is strictly contractive on Ω. Letting \mu =1, b=a and x=y=z=0 in (22), we have
for all a\in A. Putting \mu =1, x=y=z=0 and replacing a, b by 3a, −a in (22), respectively, we get
for all a\in A. Hence,
for all a\in A. The above inequality shows that d(\mathrm{\Phi}f,f)\le \frac{1}{3}. Now, by Theorem 2.1, there exists a unique mapping D:A\to B such that D is a fixed point of Φ such that
for all a\in A, and thus d(f,D)\le \frac{1}{1k}d(\mathrm{\Phi}f,f)\le \frac{1}{3(1k)}. So we proved the inequality (24). By the inequality (23), we get
Letting a=b and putting x=y=z=0 in (22), we have
where {\tilde{\varphi}}_{3}(a)=\varphi (a,a,0,0,0). If we replace a by {3}^{n}a in (29) and divide both sides of the resulting inequality by {3}^{n}, we have
Let n tend to infinity. It follows from (22), (27), and (28) that D(\mu a)=\mu D(a) for all a\in A and all \mu \in {\mathbb{T}}_{\frac{1}{{n}_{0}}}^{1}. Similar to the above and again from (22) by applying (27) and (28), we can prove that 2D(\frac{a+b}{2})=D(a)+D(b) for all a,b\in A. Since f(0)=0, we have D(0)=0. Therefore, 2D(\frac{a}{2})=D(a) for every a\in A, and thus
for all a,b\in A. Now, Lemma 2.2 shows that D is \mathbb{C}linear. The rest of the proof is similar to the proof of Theorem 2.3. □
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The study presented here was carried out in collaboration between all authors. AB suggested to write the current article. All authors read and approved the final manuscript.
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Park, C., Bodaghi, A. On the stability of ∗derivations on Banach ∗algebras. Adv Differ Equ 2012, 138 (2012). https://doi.org/10.1186/168718472012138
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DOI: https://doi.org/10.1186/168718472012138