# On the stability of ∗-derivations on Banach ∗-algebras

- Choonkil Park
^{1}and - Abasalt Bodaghi
^{2}Email author

**2012**:138

https://doi.org/10.1186/1687-1847-2012-138

© Park and Bodaghi; licensee Springer 2012

**Received: **12 June 2012

**Accepted: **30 July 2012

**Published: **7 August 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.

### Keywords

∗-derivation Banach ∗-algebra Jordan ∗-derivation stability superstability## 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 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 [10] and for the quadratic functional equation [9]. The Hyers-Ulam 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 Hyers-Ulam 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 fixed-point 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 constant*$L<1$.

*Then*,

*for each element*$\alpha \in \mathrm{\Omega}$,

*either*$d({\mathcal{T}}^{n}\alpha ,{\mathcal{T}}^{n+1}\alpha )=\mathrm{\infty}$

*for all*$n\ge 0$,

*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 fixed*-*point*${\beta}^{\ast}$*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)
$d(\beta ,{\beta}^{\ast})\le \frac{1}{1-L}d(\beta ,\mathcal{T}\beta )$

*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 that*$f:X\to Y$*is an additive mapping*. *Then* *f* *is*$\mathbb{C}$-*linear if and only if*$f(\mu x)=\mu f(x)$*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 Hyers-Ulam stability of ∗-derivations as follows:

**Theorem 2.3**

*Let*${n}_{0}\in \mathbb{N}$

*be fixed*, $f:A\to B$

*a mapping with*$f(0)=0$

*and 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 all*$a,b,x,y,z\in A$.

*If there exists a constant*$k\in (0,1)$,

*such that*

*for all*$a,b,x,y,z\in A$,

*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

*d*on $\mathrm{\Omega}\times \mathrm{\Omega}$ as follows:

*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

*a*by 2

*a*in the inequality (5) and using (2) and (4), we get

*D*is a fixed point of

*T*and that ${\mathrm{\Psi}}^{n}f\to D$ as $n\to \mathrm{\infty}$. Thus,

*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

*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

*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 let*$f:A\to B$

*be mappings with*$f(0)=0$

*such that*

*for all*$\mu \in {\mathbb{T}}_{\frac{1}{{n}_{0}}}^{1}$

*and all*$a,b,x,y,z\in A$.

*Then there exists a unique*∗-

*derivation*

*D*

*on*

*A*

*satisfying*

*for all*$a\in A$.

*Proof*The proof follows from Theorem 2.3 by taking

for all $a,b,x,y,z\in A$ and $k={2}^{r-1}$. □

In the following corollaries, we show that under some conditions the superstability for the inequality (1) is valid.

**Corollary 2.5** *Let*$f:A\to B$*be 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 with*$0<{\sum}_{j=1}^{5}{r}_{j}\ne 1$

*and 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 all*$a,b,x,y,z\in A$. *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 $x\mapsto ax-xa$, 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*, $f:A\to A$

*a mapping with*$f(0)=0$

*and 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 all*$a,b,c\in A$.

*If there exists a constant*$k\in (0,1)$,

*such that*

*for all*$a,b,c\in A$,

*then there exists a unique Jordan*∗-

*derivation*

*D*

*on*

*A*

*satisfying*

*for all*$a\in A$.

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 let*$f:A\to A$

*be mappings with*$f(0)=0$

*such that*

*for all*$\mu \in {\mathbb{T}}_{\frac{1}{{n}_{0}}}^{1}$

*and all*$a,b,c,d\in A$.

*Then there exists a unique Jordan*∗-

*derivation*

*D*

*on*

*A*

*satisfying*

*for all*$a\in A$.

**Corollary 2.9** *Let*$f:A\to A$*be 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 with*$0<{\sum}_{j=1}^{3}{r}_{j}\ne 1$

*and 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 for all*$a,b,c\in A$. *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*, $f:A\to B$

*a mapping with*$f(0)=0$

*and 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 all*$a,b,x,y,z\in A$.

*If there exists a constant*$k\in (0,1)$,

*such that*

*for all*$a,b,x,y,z\in A$,

*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

*d*on $\mathrm{\Omega}\times \mathrm{\Omega}$ as follows:

*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

*D*is a fixed point of Φ such that

*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 let*$f:A\to B$

*be mappings with*$f(0)=0$

*such that*

*for all*$\mu \in {\mathbb{T}}_{\frac{1}{{n}_{0}}}^{1}$

*and all*$a,b,x,y,z\in A$.

*Then there exists a unique*∗-

*derivation*

*D*

*on*

*A*

*satisfying*

*for all*$a\in A$.

*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 $k={2}^{r-1}$. □

In the following corollary, we show that when *f* is an additive mapping, the superstability for the inequality (13) holds.

**Corollary 3.3** *Let*$f:A\to B$*be 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. □

*for all*$\mu \in {\mathbb{T}}_{\frac{1}{{n}_{0}}}^{1}$*and all*$a,b,x,y,z\in A$. *If*${r}_{j}$ ($1\le j\le 5$), *δ* *are nonnegative real numbers such that*$0<{\sum}_{j=1}^{5}{r}_{j}\ne 1$, *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 2*a* 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*, $f:A\to B$

*a mapping with*$f(0)=0$

*and 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 all*$a,b,x,y,z\in A$.

*If there exists a constant*$k\in (0,1)$

*such that*

*for all*$a,b,x,y,z\in A$,

*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:

*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*

*a*by 3

*a*in the inequality (26) and use from (23) and (25), we have

*a*,

*b*by 3

*a*, −

*a*in (22), respectively, we get

*D*is a fixed point of Φ such that

*a*by ${3}^{n}a$ in (29) and divide both sides of the resulting inequality by ${3}^{n}$, we have

*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. □

## Declarations

### Acknowledgement

The authors would like to thank the referee for careful reading of the paper and giving some useful suggestions.

## Authors’ Affiliations

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