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# Approximation of linear mappings in Banach modules over $C ∗$-algebras

## Abstract

Let X, Y be Banach modules over a $C ∗$-algebra and let $r 1 ,…, r n ∈R$ be given. Using fixed-point methods, we prove the stability of the following functional equation in Banach modules over a unital $C ∗$-algebra:

$∑ j = 1 n f ( 1 2 ∑ 1 ≤ i ≤ n , i ≠ j r i x i − 1 2 r j x j ) + ∑ i = 1 n r i f( x i )=nf ( 1 2 ∑ i = 1 n r i x i ) .$

As an application, we investigate homomorphisms in unital $C ∗$-algebras.

MSC:39B72, 46L05, 47H10, 46B03, 47B48.

## 1 Introduction and preliminaries

We say a functional equation $(ζ)$ is stable if any function g satisfying the equation $(ζ)$ approximately is near to the true solution of $(ζ)$. We say that a functional equation is superstable if every approximate solution is an exact solution of it (see ). The stability problem of functional equations was originated from a question of Ulam  concerning the stability of group homomorphisms. Hyers  gave a first affirmative partial answer to the question of Ulam in Banach spaces. Hyers’ theorem was generalized by Aoki  for additive mappings and by T.M. Rassias  for linear mappings by considering an unbounded Cauchy difference. A generalization of the T.M. Rassias theorem was obtained by Găvruta  by replacing the unbounded Cauchy difference by a general control function in the spirit of T.M. Rassias’ approach.

The functional equation

$f(x+y)+f(x−y)=2f(x)+2f(y)$

is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. A Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof  for mappings $f:X→Y$, where X is a normed space and Y is a Banach space. Cholewa  noticed that the theorem of Skof is still true if the relevant domain X is replaced by an Abelian group. Czerwik  proved the Hyers-Ulam stability of the quadratic functional equation. J.M. Rassias [10, 11] introduced and investigated the stability problem of Ulam for the Euler-Lagrange quadratic functional equation

$f( a 1 x 1 + a 2 x 2 )+f( a 2 x 1 − a 1 x 2 )= ( a 1 2 + a 2 2 ) [ f ( x 1 ) + f ( x 2 ) ] .$
(1.1)

Grabiec  has generalized these results mentioned above.

The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see ).

Let X be a set. A function $d:X×X→[0,∞]$ is called a generalized metric on X if d satisfies the following conditions:

1. (1)

$d(x,y)=0$ if and only if $x=y$;

2. (2)

$d(x,y)=d(y,x)$ for all $x,y∈X$;

3. (3)

$d(x,z)≤d(x,y)+d(y,z)$ for all $x,y,z∈X$.

We recall a fundamental result in fixed-point theory.

Theorem 1.1 [44, 45]

Let $(X,d)$ be a complete generalized metric space and let $J:X→X$ be a strictly contractive mapping with Lipschitz constant $L<1$. Then, for each given element $x∈X$, either

$d ( J n x , J n + 1 x ) =∞$

for all nonnegative integers n or there exists a positive integer $n 0$ such that

1. (1)

$d( J n x, J n + 1 x)<∞$ for all $n≥ n 0$;

2. (2)

the sequence ${ J n x}$ converges to a fixed point $y ∗$ of J;

3. (3)

$y ∗$ is the unique fixed point of J in the set $Y={y∈X∣d( J n 0 x,y)<∞}$;

4. (4)

$d(y, y ∗ )≤ 1 1 − L d(y,Jy)$ for all $y∈Y$.

In 1996, Isac and T.M. Rassias  were the first to provide applications of stability theory of functional equations for the proof of new fixed-point theorems with applications. By using fixed-point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see ).

Recently, Park and Park  introduced and investigated the following additive functional equation of Euler-Lagrange type: (1.2)

whose solution is said to be a generalized additive mapping of Euler-Lagrange type.

In this paper, we introduce the following additive functional equation of Euler-Lagrange type which is somewhat different from (1.2):

$∑ j = 1 n f ( 1 2 ∑ 1 ≤ i ≤ n , i ≠ j r i x i − 1 2 r j x j ) + ∑ i = 1 n r i f( x i )=nf ( 1 2 ∑ i = 1 n r i x i ) ,$
(1.3)

where $r 1 ,…, r n ∈R$. Every solution of the functional equation (1.3) is said to be a generalized Euler-Lagrange type additive mapping.

Using fixed-point methods, we investigate the Hyers-Ulam stability of the functional equation (1.3) in Banach modules over a $C ∗$-algebra. These results are applied to investigate $C ∗$-algebra homomorphisms in unital $C ∗$-algebras. Also, ones can get the super-stability results after all theorems by putting the product of powers of norms as the control functions (see for more details [60, 61]).

Throughout this paper, assume that A is a unital $C ∗$-algebra with the norm $∥ ⋅ ∥ A$ and the unit e, B is a unital $C ∗$-algebra with the norm $∥ ⋅ ∥ B$, and X, Y are left Banach modules over a unital $C ∗$-algebra A with the norms $∥ ⋅ ∥ X$ and $∥ ⋅ ∥ Y$, respectively. Let $U(A)$ be the group of unitary elements in A and let $r 1 ,…, r n ∈R$.

## 2 Hyers-Ulam stability of the functional equation (1.3) in Banach modules over a $C ∗$-algebra

For any given mapping $f:X→Y$, $u∈U(A)$ and $μ∈C$, we define $D u , r 1 , … , r n f$ and $D μ , r 1 , … , r n f: X n →Y$ by

$D u , r 1 , … , r n f ( x 1 , … , x n ) : = ∑ j = 1 n f ( 1 2 ∑ 1 ≤ i ≤ n , i ≠ j r i u x i − 1 2 r j u x j ) + ∑ i = 1 n r i u f ( x i ) − n f ( 1 2 ∑ i = 1 n r i u x i )$

and

$D μ , r 1 , … , r n f ( x 1 , … , x n ) : = ∑ j = 1 n f ( 1 2 ∑ 1 ≤ i ≤ n , i ≠ j μ r i x i − 1 2 μ r j x j ) + ∑ i = 1 n μ r i f ( x i ) − n f ( 1 2 ∑ i = 1 n μ r i x i )$

for all $x 1 ,…, x n ∈X$.

Lemma 2.1 Let X and Y be linear spaces and let $r 1 ,…, r n$ be real numbers with $∑ k = 1 n r k ≠0$ and $r i ≠0$, $r j ≠0$ for some $1≤i. Assume that a mapping $L:X→Y$ satisfies the functional equation (1.3) for all $x 1 ,…, x n ∈X$. Then the mapping L is additive. Moreover, $L( r k x)= r k L(x)$ for all $x∈X$ and $1≤k≤n$.

Proof One can find a complete proof at . □

Lemma 2.2 Let X and Y be linear spaces and let $r 1 ,…, r n$ be real numbers with $r i ≠0$, $r j ≠0$ for some $1≤i. Assume that a mapping $L:X→Y$ with $L(0)=0$ satisfies the functional equation (1.3) for all $x 1 ,…, x n ∈X$. Then the mapping L is additive. Moreover, $L( r k x)= r k L(x)$ for all $x∈X$ and $1≤k≤n$.

Proof One can find a complete proof at . □

We investigate the Hyers-Ulam stability of a generalized Euler-Lagrange type additive mapping in Banach modules over a unital $C ∗$-algebra. Throughout this paper, let $r 1 ,…, r n$ be real numbers such that $r i ≠0$, $r j ≠0$ for fixed $1≤i.

Theorem 2.3 Let $f:X→Y$ be a mapping satisfying $f(0)=0$ for which there is a function $φ: X n →[0,∞)$ such that

$∥ D e , r 1 , … , r n f ( x 1 , … , x n ) ∥ Y ≤φ( x 1 ,…, x n )$
(2.1)

for all $x 1 ,…, x n ∈X$. Let

$φ i j (x,y):=φ(0,…,0, x ⏟ i th ,0,…,0, y ⏟ j th ,0,…,0)$

for all $x,y∈X$ and $1≤i. If there exists $0 such that

$φ(2 x 1 ,…,2 x n )≤2Cφ( x 1 ,…, x n )$

for all $x 1 ,…, x n ∈X$, then there exists a unique generalized Euler-Lagrange type additive mapping $L:X→Y$ such that

$∥ f ( x ) − L ( x ) ∥ Y ≤ 1 4 − 4 C { φ i j ( 2 x r i , 2 x r j ) + 2 φ i j ( x r i , − x r j ) + φ i j ( 2 x r i , 0 ) + 2 φ i j ( x r i , 0 ) + φ i j ( 0 , 2 x r j ) + 2 φ i j ( 0 , − x r j ) }$
(2.2)

for all $x∈X$. Moreover, $L( r k x)= r k L(x)$ for all $x∈X$ and $1≤k≤n$.

Proof For each $1≤k≤n$ with $k≠i,j$, let $x k =0$ in (2.1). Then we get the following inequality:

$∥ f ( − r i x i + r j x j 2 ) + f ( r i x i − r j x j 2 ) − 2 f ( r i x i + r j x j 2 ) + r i f ( x i ) + r j f ( x j ) ∥ Y ≤ φ ( 0 , … , 0 , x i ⏟ i th , 0 , … , 0 , x j ⏟ j th , 0 , … , 0 )$
(2.3)

for all $x i , x j ∈X$. Letting $x i =0$ in (2.3), we get

$∥ f ( − r j x j 2 ) − f ( r j x j 2 ) + r j f ( x j ) ∥ Y ≤ φ i j (0, x j )$
(2.4)

for all $x j ∈X$. Similarly, letting $x j =0$ in (2.3), we get

$∥ f ( − r i x i 2 ) − f ( r i x i 2 ) + r i f ( x i ) ∥ Y ≤ φ i j ( x i ,0)$
(2.5)

for all $x i ∈X$. It follows from (2.3), (2.4) and (2.5) that

$∥ f ( − r i x i + r j x j 2 ) + f ( r i x i − r j x j 2 ) − 2 f ( r i x i + r j x j 2 ) + f ( r i x i 2 ) + f ( r j x j 2 ) − f ( − r i x i 2 ) − f ( − r j x j 2 ) ∥ Y ≤ φ i j ( x i , x j ) + φ i j ( x i , 0 ) + φ i j ( 0 , x j )$
(2.6)

for all $x i , x j ∈X$. Replacing $x i$ and $x j$ by $2 x r i$ and $2 y r j$ in (2.6), we get

$∥ f ( − x + y ) + f ( x − y ) − 2 f ( x + y ) + f ( x ) + f ( y ) − f ( − x ) − f ( − y ) ∥ Y ≤ φ i j ( 2 x r i , 2 y r j ) + φ i j ( 2 x r i , 0 ) + φ i j ( 0 , 2 y r j )$
(2.7)

for all $x,y∈X$. Putting $y=x$ in (2.7), we get

$∥ 2 f ( x ) − 2 f ( − x ) − 2 f ( 2 x ) ∥ Y ≤ φ i j ( 2 x r i , 2 x r j ) + φ i j ( 2 x r i , 0 ) + φ i j ( 0 , 2 x r j )$
(2.8)

for all $x∈X$. Replacing x and y by $x 2$ and $− x 2$ in (2.7), respectively, we get

$∥ f ( x ) + f ( − x ) ∥ Y ≤ φ i j ( x r i , − x r j ) + φ i j ( x r i , 0 ) + φ i j ( 0 , − x r j )$
(2.9)

for all $x∈X$. It follows from (2.8) and (2.9) that

$∥ 1 2 f ( 2 x ) − f ( x ) ∥ Y ≤ 1 4 ψ(x)$
(2.10)

for all $x∈X$, where

$ψ ( x ) : = φ i j ( 2 x r i , 2 x r j ) + 2 φ i j ( x r i , − x r j ) + φ i j ( 2 x r i , 0 ) + 2 φ i j ( x r i , 0 ) + φ i j ( 0 , 2 x r j ) + 2 φ i j ( 0 , − x r j ) .$

Consider the set $W:={g:X→Y}$ and introduce the generalized metric on $W$:

$d(g,h)=inf { C ∈ R + : ∥ g ( x ) − h ( x ) ∥ Y ≤ C ψ ( x ) , ∀ x ∈ X } .$

It is easy to show that $(W,d)$ is complete.

Now, we consider the linear mapping $J:W→W$ such that

$Jg(x):= 1 2 g(2x)$
(2.11)

for all $x∈X$. By Theorem 3.1 of , $d(Jg,Jh)≤Cd(g,h)$ for all $g,h∈W$. Hence, $d(f,Jf)≤ 1 4$.

By Theorem 1.1, there exists a mapping $L:X→Y$ such that

1. (1)

L is a fixed point of J, i.e.,

$L(2x)=2L(x)$
(2.12)

for all $x∈X$. The mapping L is a unique fixed point of J in the set

$Z= { g ∈ W : d ( f , g ) < ∞ } .$

This implies that L is a unique mapping satisfying (2.12) such that there exists $C∈(0,∞)$ satisfying

$∥ L ( x ) − f ( x ) ∥ Y ≤Cψ(x)$

for all $x∈X$.

1. (2)

$d( J n f,L)→0$ as $n→∞$. This implies the equality

$lim n → ∞ f ( 2 n x ) 2 n =L(x)$

for all $x∈X$.

1. (3)

$d(f,L)≤ 1 1 − C d(f,Jf)$, which implies the inequality $d(f,L)≤ 1 4 − 4 C$. This implies that the inequality (2.2) holds.

Since $φ(2 x 1 ,…,2 x n )≤2Cφ( x 1 ,…, x n )$, it follows that

$∥ D e , r 1 , … , r n L ( x 1 , … , x n ) ∥ Y = lim k → ∞ 1 2 k ∥ D e , r 1 , … , r n f ( 2 k x 1 , … , 2 k x n ) ∥ Y ≤ lim k → ∞ 1 2 k φ ( 2 k x 1 , … , 2 k x n ) ≤ lim k → ∞ C k φ ( x 1 , … , x n ) = 0$

for all $x 1 ,…, x n ∈X$. Therefore, the mapping $L:X→Y$ satisfies the equation (1.3) and $L(0)=0$. Hence, by Lemma 2.2, L is a generalized Euler-Lagrange type additive mapping and $L( r k x)= r k L(x)$ for all $x∈X$ and $1≤k≤n$. This completes the proof. □

Theorem 2.4 Let $f:X→Y$ be a mapping satisfying $f(0)=0$ for which there is a function $φ: X n →[0,∞)$ satisfying

$∥ D u , r 1 , … , r n f ( x 1 , … , x n ) ∥ ≤φ( x 1 ,…, x n )$
(2.13)

for all $x 1 ,…, x n ∈X$ and $u∈U(A)$. If there exists $0 such that

$φ(2 x 1 ,…,2 x n )≤2Cφ( x 1 ,…, x n )$

for all $x 1 ,…, x n ∈X$, then there exists a unique A-linear generalized Euler-Lagrange type additive mapping $L:X→Y$ satisfying (2.2) for all $x∈X$. Moreover, $L( r k x)= r k L(x)$ for all $x∈X$ and $1≤k≤n$.

Proof By Theorem 2.3, there exists a unique generalized Euler-Lagrange type additive mapping $L:X→Y$ satisfying (2.2), and moreover $L( r k x)= r k L(x)$ for all $x∈X$ and $1≤k≤n$. By the assumption, for each $u∈U(A)$, we get

$∥ D u , r 1 , … , r n L ( 0 , … , 0 , x ⏟ i th , 0 , … , 0 ) ∥ Y = lim k → ∞ 1 2 k ∥ D u , r 1 , … , r n f ( 0 , … , 0 , 2 k x ⏟ i th , 0 , … , 0 ) ∥ Y ≤ lim k → ∞ 1 2 k φ ( 0 , … , 0 , 2 k x ⏟ i th , 0 , … , 0 ) ≤ lim k → ∞ C k φ ( 0 , … , 0 , x ⏟ i th , 0 , … , 0 ) = 0$

for all $x∈X$. So, we have

$r i uL(x)=L( r i ux)$

for all $u∈U(A)$ and $x∈X$. Since $L( r i x)= r i L(x)$ for all $x∈X$ and $r i ≠0$,

$L(ux)=uL(x)$

for all $u∈U(A)$ and $x∈X$. By the same reasoning as in the proofs of  and ,

$L(ax+by)=L(ax)+L(by)=aL(x)+bL(y)$

for all $a,b∈A$ ($a,b≠0$) and $x,y∈X$. Since $L(0x)=0=0L(x)$ for all $x∈X$, the unique generalized Euler-Lagrange type additive mapping $L:X→Y$ is an A-linear mapping. This completes the proof. □

Theorem 2.5 Let $f:X→Y$ be a mapping satisfying $f(0)=0$ for which there is a function $φ: X n →[0,∞)$ such that

$∥ D e , r 1 , … , r n f ( x 1 , … , x n ) ∥ Y ≤φ( x 1 ,…, x n )$
(2.14)

for all $x 1 ,…, x n ∈X$. If there exists $0 such that

$φ( x 1 ,…, 2 n )≤ C 2 φ(2 x 1 ,…,2 x n )$

for all $x 1 ,…, x n ∈X$, then there exists a unique generalized Euler-Lagrange type additive mapping $L:X→Y$ such that

$∥ f ( x ) − L ( x ) ∥ Y ≤ C 4 − 4 C { φ i j ( 2 x r i , 2 x r j ) + 2 φ i j ( x r i , − x r j ) + φ i j ( 2 x r i , 0 ) + 2 φ i j ( x r i , 0 ) + φ i j ( 0 , 2 x r j ) + 2 φ i j ( 0 , − x r j ) }$
(2.15)

for all $x∈X$, where $φ i j$ is defined in the statement of Theorem  2.3. Moreover, $L( r k x)= r k L(x)$ for all $x∈X$ and $1≤k≤n$.

Proof It follows from (2.10) that

$∥ f ( x ) − f ( x 2 ) ∥ Y ≤ 1 2 ψ ( x 2 ) ≤ C 4 ψ(x)$

for all $x∈X$, where ψ is defined in the proof of Theorem 2.3. The rest of the proof is similar to the proof of Theorem 2.3. □

Theorem 2.6 Let $f:X→Y$ be a mapping with $f(0)=0$ for which there is a function $φ: X n →[0,∞)$ satisfying

$∥ D u , r 1 , … , r n f ( x 1 , … , x n ) ∥ ≤φ( x 1 ,…, x n )$
(2.16)

for all $x 1 ,…, x n ∈X$ and $u∈U(A)$. If there exists $0 such that

$φ( x 1 ,…, 2 n )≤ C 2 φ(2 x 1 ,…,2 x n )$

for all $x 1 ,…, x n ∈X$, then there exists a unique A-linear generalized Euler-Lagrange type additive mapping $L:X→Y$ satisfying (2.15) for all $x∈X$. Moreover, $L( r k x)= r k L(x)$ for all $x∈X$ and all $1≤k≤n$.

Proof The proof is similar to the proof of Theorem 2.4. □

Remark 2.7 In Theorems 2.5 and 2.6, one can assume that $∑ k = 1 n r k ≠0$ instead of $f(0)=0$.

## 3 Homomorphisms in unital $C ∗$-algebras

In this section, we investigate $C ∗$-algebra homomorphisms in unital $C ∗$-algebras. We use the following lemma in the proof of the next theorem.

Lemma 3.1 

Let $f:A→B$ be an additive mapping such that $f(μx)=μf(x)$ for all $x∈A$ and $μ∈ S 1 n o 1 :={ e i θ ;0≤θ≤2π n o }$. Then the mapping $f:A→B$ is -linear.

Note that a -linear mapping $H:A→B$ is called a homomorphism in $C ∗$-algebras if H satisfies $H(xy)=H(x)H(y)$ and $H( x ∗ )=H ( x ) ∗$ for all $x,y∈A$.

Theorem 3.2 Let $f:A→B$ be a mapping with $f(0)=0$ for which there is a function $φ: A n →[0,∞)$ satisfying (3.1) (3.2) (3.3)

for all $x, x 1 ,…, x n ∈A$, $u∈U(A)$, $k∈N$ and $μ∈ S 1$. If there exists $0 such that

$φ(2 x 1 ,…,2 x n )≤2Cφ( x 1 ,…, x n )$

for all $x 1 ,…, x n ∈A$, then the mapping $f:A→B$ is a $C ∗$-algebra homomorphism.

Proof Since $|J|≥3$, letting $μ=1$ and $x k =0$ for all $1≤k≤n$ ($k≠i,j$) in (3.1), we get

$f ( − r i x i + r j x j 2 ) +f ( r i x i − r j x j 2 ) + r i f( x i )+ r j f( x j )=2f ( r i x i + r j x j 2 )$

for all $x i , x j ∈A$. By the same reasoning as in the proof of Lemma 2.1, the mapping f is additive and $f( r k x)= r k f(x)$ for all $x∈A$ and $k=i,j$. So, by letting $x i =x$ and $x k =0$ for all $1≤k≤n$, $k≠i$, in (3.1), we get $f(μx)=μf(x)$ for all $x∈A$ and $μ∈ S 1$. Therefore, by Lemma 3.1, the mapping f is -linear. Hence, it follows from (3.2) and (3.3) that

$∥ f ( u ∗ ) − f ( u ) ∗ ∥ B = lim k → ∞ 1 2 k ∥ f ( 2 k u ∗ ) − f ( 2 k u ) ∗ ∥ B ≤ lim k → ∞ 1 2 k φ ( 2 k u , … , 2 k u ⏟ n times ) ≤ lim k → ∞ C k φ ( u , … , u ⏟ n times ) = 0 , ∥ f ( u x ) − f ( u ) f ( x ) ∥ B = lim k → ∞ 1 2 k ∥ f ( 2 k u x ) − f ( 2 k u ) f ( x ) ∥ B ≤ lim k → ∞ 1 2 k φ ( 2 k u x , … , 2 k u x ⏟ n times ) ≤ lim k → ∞ C k φ ( u x , … , u x ⏟ n times ) = 0$

for all $x∈A$ and $u∈U(A)$. So, we have $f( u ∗ )=f ( u ) ∗$ and $f(ux)=f(u)f(x)$ for all $x∈A$ and $u∈U(A)$. Since f is -linear and each $x∈A$ is a finite linear combination of unitary elements (see ), i.e., $x= ∑ k = 1 m λ k u k$, where $λ k ∈C$ and $u k ∈U(A)$ for all $1≤k≤n$, we have

$f ( x ∗ ) = f ( ∑ k = 1 m λ ¯ k u k ∗ ) = ∑ k = 1 m λ ¯ k f ( u k ∗ ) = ∑ k = 1 m λ ¯ k f ( u k ) ∗ = ( ∑ k = 1 m λ k f ( u k ) ) ∗ = f ( ∑ k = 1 m λ k u k ) ∗ = f ( x ) ∗ , f ( x y ) = f ( ∑ k = 1 m λ k u k y ) = ∑ k = 1 m λ k f ( u k y ) = ∑ k = 1 m λ k f ( u k ) f ( y ) = f ( ∑ k = 1 m λ k u k ) f ( y ) = f ( x ) f ( y )$

for all $x,y∈A$. Therefore, the mapping $f:A→B$ is a $C ∗$-algebra homomorphism. This completes the proof. □

The following theorem is an alternative result of Theorem 3.2.

Theorem 3.3 Let $f:A→B$ be a mapping with $f(0)=0$ for which there is a function $φ: A n →[0,∞)$ satisfying (3.4) (3.5)

for all $x, x 1 ,…, x n ∈A$, $u∈U(A)$, $k∈N$ and $μ∈ S 1$. If there exists $0 such that

$φ( x 1 ,…, 2 n )≤ C 2 φ(2 x 1 ,…,2 x n )$

for all $x 1 ,…, x n ∈A$, then the mapping $f:A→B$ is a $C ∗$-algebra homomorphism.

Remark 3.4 In Theorems 3.2 and 3.3, one can assume that $∑ k = 1 n r k ≠0$ instead of $f(0)=0$.

Theorem 3.5 Let $f:A→B$ be a mapping with $f(0)=0$ for which there is a function $φ: A n →[0,∞)$ satisfying (3.2), (3.3) and

$∥ D μ , r 1 , … , r n f ( x 1 , … , x n ) ∥ B ≤φ( x 1 ,…, x n )$
(3.6)

for all $x 1 ,…, x n ∈A$ and $μ∈ S 1$. Assume that $lim k → ∞ 1 2 k f( 2 k e)$ is invertible. If there exists $0 such that

$φ(2 x 1 ,…,2 x n )≤2Cφ( x 1 ,…, x n )$

for all $x 1 ,…, x n ∈A$, then the mapping $f:A→B$ is a $C ∗$-algebra homomorphism.

Proof Consider the $C ∗$-algebras A and B as left Banach modules over the unital $C ∗$-algebra . By Theorem 2.4, there exists a unique -linear generalized Euler-Lagrange type additive mapping $H:A→B$ defined by

$H(x)= lim k → ∞ 1 2 k f ( 2 k x )$

for all $x∈A$. By (3.2) and (3.3), we get

$∥ H ( u ∗ ) − H ( u ) ∗ ∥ B = lim k → ∞ 1 2 k ∥ f ( 2 k u ∗ ) − f ( 2 k u ) ∗ ∥ B ≤ lim k → ∞ 1 2 k φ ( 2 k u , … , 2 k u ⏟ n times ) = 0 , ∥ H ( u x ) − H ( u ) f ( x ) ∥ B = lim k → ∞ 1 2 k ∥ f ( 2 k u x ) − f ( 2 k u ) f ( x ) ∥ B ≤ lim k → ∞ 1 2 k φ ( 2 k u x , … , 2 k u x ⏟ n times ) = 0$

for all $u∈U(A)$ and $x∈A$. So, we have $H( u ∗ )=H ( u ) ∗$ and $H(ux)=H(u)f(x)$ for all $u∈U(A)$ and $x∈A$. Therefore, by the additivity of H, we have

$H(ux)= lim k → ∞ 1 2 k H ( 2 k u x ) =H(u) lim k → ∞ 1 2 k f ( 2 k x ) =H(u)H(x)$
(3.7)

for all $u∈U(A)$ and all $x∈A$. Since H is -linear and each $x∈A$ is a finite linear combination of unitary elements, i.e., $x= ∑ k = 1 m λ k u k$, where $λ k ∈C$ and $u k ∈U(A)$ for all $1≤k≤n$, it follows from (3.7) that

$H ( x y ) = H ( ∑ k = 1 m λ k u k y ) = ∑ k = 1 m λ k H ( u k y ) = ∑ k = 1 m λ k H ( u k ) H ( y ) = H ( ∑ k = 1 m λ k u k ) H ( y ) = H ( x ) H ( y ) , H ( x ∗ ) = H ( ∑ k = 1 m λ ¯ k u k ∗ ) = ∑ k = 1 m λ ¯ k H ( u k ∗ ) = ∑ k = 1 m λ ¯ k H ( u k ) ∗ = ( ∑ k = 1 m λ k H ( u k ) ) ∗ = H ( ∑ k = 1 m λ k u k ) ∗ = H ( x ) ∗$

for all $x,y∈A$. Since $H(e)= lim k → ∞ 1 2 k f( 2 k e)$ is invertible and

$H(e)H(y)=H(ey)=H(e)f(y)$

for all $y∈A$, it follows that $H(y)=f(y)$ for all $y∈A$. Therefore, the mapping $f:A→B$ is a $C ∗$-algebra homomorphism. This completes the proof. □

The following theorem is an alternative result of Theorem 3.5.

Theorem 3.6 Let $f:A→B$ be a mapping with $f(0)=0$ for which there is a function $φ: A n →[0,∞)$ satisfying (3.4), (3.5) and

$∥ D μ , r 1 , … , r n f ( x 1 , … , x n ) ∥ B ≤φ( x 1 ,…, x n )$

for all $x 1 ,…, x n ∈A$ and $μ∈ S 1$. Assume that $lim k → ∞ 2 k f( e 2 k )$ is invertible. If there exists $0 such that

$φ( x 1 ,…, 2 n )≤ C 2 φ(2 x 1 ,…,2 x n )$

for all $x 1 ,…, x n ∈A$, then the mapping $f:A→B$ is a $C ∗$-algebra homomorphism.

Remark 3.7 In Theorem 3.6, one can assume that $∑ k = 1 n r k ≠0$ instead of $f(0)=0$.

Theorem 3.8 Let $f:A→B$ be a mapping with $f(0)=0$ for which there is a function $φ: A n →[0,∞)$ satisfying (3.2), (3.3) and

$∥ D μ , r 1 , … , r n f ( x 1 , … , x n ) ∥ B ≤φ( x 1 ,…, x n )$
(3.8)

for all $x 1 ,…, x n ∈A$ and $μ=i,1$. Assume that $lim k → ∞ 1 2 k f( 2 k e)$ is invertible and for each fixed $x∈A$ the mapping $t↦f(tx)$ is continuous in $t∈R$. If there exists $0 such that

$φ(2 x 1 ,…,2 x n )≤2Cφ( x 1 ,…, x n )$

for all $x 1 ,…, x n ∈A$, then the mapping $f:A→B$ is a $C ∗$-algebra homomorphism.

Proof Put $μ=1$ in (3.8). By the same reasoning as in the proof of Theorem 2.3, there exists a unique generalized Euler-Lagrange type additive mapping $H:A→B$ defined by

$H(x)= lim k → ∞ f ( 2 k x ) 2 k$

for all $x∈A$. By the same reasoning as in the proof of , the generalized Euler-Lagrange type additive mapping $H:A→B$ is -linear. By the same method as in the proof of Theorem 2.4, we have

$∥ D μ , r 1 , … , r n H ( 0 , … , 0 , x ⏟ j th , 0 , … , 0 ) ∥ Y = lim k → ∞ 1 2 k ∥ D μ , r 1 , … , r n f ( 0 , … , 0 , 2 k x ⏟ j th , 0 , … , 0 ) ∥ Y ≤ lim k → ∞ 1 2 k φ ( 0 , … , 0 , 2 k x ⏟ j th , 0 , … , 0 ) = 0$

for all $x∈A$ and so

$r j μH(x)=H( r j μx)$

for all $x∈A$. Since $H( r j x)= r j H(x)$ for all $x∈X$ and $r j ≠0$,

$H(μx)=μH(x)$

for all $x∈A$ and $μ=i,1$. For each $λ∈C$, we have $λ=s+it$, where $s,t∈R$. Thus, it follows that

$H ( λ x ) = H ( s x + i t x ) = s H ( x ) + t H ( i x ) = s H ( x ) + i t H ( x ) = ( s + i t ) H ( x ) = λ H ( x )$

for all $λ∈C$ and $x∈A$ and so

$H(ζx+ηy)=H(ζx)+H(ηy)=ζH(x)+ηH(y)$

for all $ζ,η∈C$ and $x,y∈A$. Hence, the generalized Euler-Lagrange type additive mapping $H:A→B$ is -linear.

The rest of the proof is the same as in the proof of Theorem 3.5. This completes the proof. □

The following theorem is an alternative result of Theorem 3.8.

Theorem 3.9 Let $f:A→B$ be a mapping with $f(0)=0$ for which there is a function $φ: A n →[0,∞)$ satisfying (3.4), (3.5) and

$∥ D μ , r 1 , … , r n f ( x 1 , … , x n ) ∥ B ≤φ( x 1 ,…, x n ),$

for all $x, x 1 ,…, x n ∈A$ and $μ=i,1$. Assume that $lim k → ∞ 2 k f( e 2 k )$ is invertible and for each fixed $x∈A$ the mapping $t↦f(tx)$ is continuous in $t∈R$. If there exists $0 such that

$φ( x 1 ,…, 2 n )≤ C 2 φ(2 x 1 ,…,2 x n )$

for all $x 1 ,…, x n ∈A$, then the mapping $f:A→B$ is a $C ∗$-algebra homomorphism.

Proof We omit the proof because it is very similar to last theorem. □

Remark 3.10 In Theorem 3.9, one can assume that $∑ k = 1 n r k ≠0$ instead of $f(0)=0$.

## References

1. 1.

Eshaghi Gordji M, Najati A, Ebadian A:Stability and superstability of Jordan homomorphisms and Jordan derivations on Banach algebras and $C ∗$-algebras: a fixed point approach. Acta Math. Sci. 2011, 31: 1911–1922.

2. 2.

Ulam SM: A Collection of the Mathematical Problems. Interscience, New York; 1960.

3. 3.

Hyers DH: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 1941, 27: 222–224. 10.1073/pnas.27.4.222

4. 4.

Aoki T: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Jpn. 1950, 2: 64–66. 10.2969/jmsj/00210064

5. 5.

Rassias TM: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 1978, 72: 297–300. 10.1090/S0002-9939-1978-0507327-1

6. 6.

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.1994.1211

7. 7.

Skof F: Proprietà locali e approssimazione di operatori. Rend. Semin. Mat. Fis. Milano 1983, 53: 113–129. 10.1007/BF02924890

8. 8.

Cholewa PW: Remarks on the stability of functional equations. Aequ. Math. 1984, 27: 76–86. 10.1007/BF02192660

9. 9.

Czerwik S: On the stability of the quadratic mapping in normed spaces. Abh. Math. Semin. Univ. Hamb. 1992, 62: 59–64. 10.1007/BF02941618

10. 10.

Rassias JM: On the stability of the Euler-Lagrange functional equation. Chin. J. Math 1992, 20: 185–190.

11. 11.

Rassias JM: On the stability of the non-linear Euler-Lagrange functional equation in real normed linear spaces. J. Math. Phys. Sci. 1994, 28: 231–235.

12. 12.

Grabiec A: The generalized Hyers-Ulam stability of a class of functional equations. Publ. Math. (Debr.) 1996, 48: 217–235.

13. 13.

Agarwal RP, Cho Y, Saadati R, Wang S: Nonlinear L -fuzzy stability of cubic functional equations. J. Inequal. Appl. 2012., 2012: Article ID 77

14. 14.

Baktash E, Cho Y, Jalili M, Saadati R, Vaezpour SM: On the stability of cubic mappings and quadratic mappings in random normed spaces. J. Inequal. Appl. 2008., 2008: Article ID 902187

15. 15.

Cho Y, Eshaghi Gordji M, Zolfaghari S: Solutions and stability of generalized mixed type QC functional equations in random normed spaces. J. Inequal. Appl. 2010., 2010: Article ID 403101

16. 16.

Cho Y, Park C, Rassias TM, Saadati R: Inner product spaces and functional equations. J. Comput. Anal. Appl. 2011, 13: 296–304.

17. 17.

Cho Y, Park C, Saadati R: Functional inequalities in non-Archimedean Banach spaces. Appl. Math. Lett. 2010, 60: 1994–2002.

18. 18.

Cho Y, Saadati R: Lattice non-Archimedean random stability of ACQ-functional equations. Adv. Differ. Equ. 2011., 2011: Article ID 31

19. 19.

Cho Y, Saadati R, Shabanian S, Vaezpour SM: On solution and stability of a two-variable functional equations. Discrete Dyn. Nat. Soc. 2011., 2011: Article ID 527574

20. 20.

Czerwik P: Functional Equations and Inequalities in Several Variables. World Scientific, Singapore; 2002.

21. 21.

Forti GL: Comments on the core of the direct method for proving Hyers-Ulam stability of functional equations. J. Math. Anal. Appl. 2004, 295: 127–133. 10.1016/j.jmaa.2004.03.011

22. 22.

Forti GL: Elementary remarks on Ulam-Hyers stability of linear functional equations. J. Math. Anal. Appl. 2007, 328: 109–118. 10.1016/j.jmaa.2006.04.079

23. 23.

Gajda Z: On stability of additive mappings. Int. J. Math. Math. Sci. 1991, 14: 431–434. 10.1155/S016117129100056X

24. 24.

Gao ZX, Cao HX, Zheng WT, Xu L: Generalized Hyers-Ulam-Rassias stability of functional inequalities and functional equations. J. Math. Inequal. 2009, 3: 63–77.

25. 25.

Gǎvruta P: On the stability of some functional equations. In Stability of Mappings of Hyers-Ulam Type. Hadronic Press, Palm Harbor; 1994:93–98.

26. 26.

Gǎvruta P: On a problem of G. Isac and T.M. Rassias concerning the stability of mappings. J. Math. Anal. Appl. 2001, 261: 543–553. 10.1006/jmaa.2001.7539

27. 27.

Gǎvruta P: On the Hyers-Ulam-Rassias stability of the quadratic mappings. Nonlinear Funct. Anal. Appl. 2004, 9: 415–428.

28. 28.

Hyers DH, Isac G, Rassias TM: Stability of Functional Equations in Several Variables. Birkhäuser, Basel; 1998.

29. 29.

Hyers DH, Isac G, Rassias TM: On the asymptoticity aspect of Hyers-Ulam stability of mappings. Proc. Am. Math. Soc. 1998, 126: 425–430. 10.1090/S0002-9939-98-04060-X

30. 30.

Jung S: Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor; 2001.

31. 31.

Khodaei H, Eshaghi Gordji M, Kim S, Cho Y: Approximation of radical functional equations related to quadratic and quartic mappings. J. Math. Anal. Appl. 2012, 397: 284–297.

32. 32.

Miheţ D, Radu V: On the stability of the additive Cauchy functional equation in random normed spaces. J. Math. Anal. Appl. 2008, 343: 567–572. 10.1016/j.jmaa.2008.01.100

33. 33.

Mohammadi M, Cho Y, Park C, Vetro P, Saadati R: Random stability of an additive-quadratic-quartic functional equation. J. Inequal. Appl. 2010., 2010: Article ID 754210

34. 34.

Najati A, Cho Y: Generalized Hyers-Ulam stability of the pexiderized Cauchy functional equation in non-Archimedean spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 309026

35. 35.

Najati A, Kang J, Cho Y: Local stability of the pexiderized Cauchy and Jensen’s equations in fuzzy spaces. J. Inequal. Appl. 2011., 2011: Article ID 78

36. 36.

Park C: On the stability of the linear mapping in Banach modules. J. Math. Anal. Appl. 2002, 275: 711–720. 10.1016/S0022-247X(02)00386-4

37. 37.

Park C:Linear functional equations in Banach modules over a $C ∗$-algebra. Acta Appl. Math. 2003, 77: 125–161. 10.1023/A:1024014026789

38. 38.

Rassias JM: Solution of the Ulam stability problem for Euler-Lagrange quadratic mappings. J. Math. Anal. Appl. 1998, 220: 613–639. 10.1006/jmaa.1997.5856

39. 39.

Rassias TM: On the stability of the quadratic functional equation and its applications. Stud. Univ. Babeş-Bolyai, Math. 1998, XLIII: 89–124.

40. 40.

Rassias TM: The problem of S.M. Ulam for approximately multiplicative mappings. J. Math. Anal. Appl. 2000, 246: 352–378. 10.1006/jmaa.2000.6788

41. 41.

Rassias TM, Šemrl P: On the Hyers-Ulam stability of linear mappings. J. Math. Anal. Appl. 1993, 173: 325–338. 10.1006/jmaa.1993.1070

42. 42.

Rassias TM, Shibata K: Variational problem of some quadratic functionals in complex analysis. J. Math. Anal. Appl. 1998, 228: 234–253. 10.1006/jmaa.1998.6129

43. 43.

Saadati R, Cho Y, Vahidi J: The stability of the quartic functional equation in various spaces. Comput. Math. Appl. 2010, 60: 1994–2002. 10.1016/j.camwa.2010.07.034

44. 44.

Cădariu L, Radu V: Fixed points and the stability of Jensen’s functional equation. J. Inequal. Pure Appl. Math. 2003., 4: Article ID 4

45. 45.

Diaz J, Margolis B: A fixed point theorem of the alternative for contractions on a generalized complete metric space. Bull. Am. Math. Soc. 1968, 74: 305–309. 10.1090/S0002-9904-1968-11933-0

46. 46.

Isac G, Rassias TM: Stability of ψ -additive mappings: applications to nonlinear analysis. Int. J. Math. Math. Sci. 1996, 19: 219–228. 10.1155/S0161171296000324

47. 47.

Cădariu L, Radu V: On the stability of the Cauchy functional equation: a fixed point approach. Grazer Math. Ber. 2004, 346: 43–52.

48. 48.

Cădariu L, Radu V: Fixed point methods for the generalized stability of functional equations in a single variable. Fixed Point Theory Appl. 2008., 2008: Article ID 749392

49. 49.

Cho Y, Kang J, Saadati R: Fixed points and stability of additive functional equations on the Banach algebras. J. Comput. Anal. Appl. 2012, 14: 1103–1111.

50. 50.

Cho Y, Kang S, Sadaati R: Nonlinear random stability via fixed-point method. J. Appl. Math. 2012., 2012: Article ID 902931

51. 51.

Cho Y, Saadati R, Vahidi J:Approximation of homomorphisms and derivations on non-Archimedean Lie $C ∗$-algebras via fixed point method. Discrete Dyn. Nat. Soc. 2012., 2012: Article ID 373904

52. 52.

Eshaghi Gordji M, Cho Y, Ghaemi MB, Majani H: Approximately quintic and sextic mappings form r -divisible groups into Šerstnev probabilistic Banach spaces: fixed point method. Discrete Dyn. Nat. Soc. 2011., 2011: Article ID 572062

53. 53.

Eshaghi Gordji M, Ramezani M, Cho Y, Baghani H: Approximate Lie brackets: a fixed point approach. J. Inequal. Appl. 2012., 2012: Article ID 125

54. 54.

Miheţ D: The fixed point method for fuzzy stability of the Jensen functional equation. Fuzzy Sets Syst. 2009, 160: 1663–1667. 10.1016/j.fss.2008.06.014

55. 55.

Mirzavaziri M, Moslehian MS: A fixed point approach to stability of a quadratic equation. Bull. Braz. Math. Soc. 2006, 37: 361–376. 10.1007/s00574-006-0016-z

56. 56.

Park C: Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras. Fixed Point Theory Appl. 2007., 2007: Article ID 50175

57. 57.

Park C: Generalized Hyers-Ulam-Rassias stability of quadratic functional equations: a fixed point approach. Fixed Point Theory Appl. 2008., 2008: Article ID 493751

58. 58.

Radu V: The fixed point alternative and the stability of functional equations. Fixed Point Theory 2003, 4: 91–96.

59. 59.

Park C, Park J: Generalized Hyers-Ulam stability of an Euler-Lagrange type additive mapping. J. Differ. Equ. Appl. 2006, 12: 1277–1288. 10.1080/10236190600986925

60. 60.

Eshaghi Gordji M:Jordan -homomorphisms between unital $C ∗$-algebras: a fixed point approach. Fixed Point Theory 2011, 12: 341–348.

61. 61.

Eshaghi Gordji M, Fazeli A:Stability and superstability of homomorphisms on $C ∗$-ternary algebras. An. Univ. “Ovidius” Constanţa, Ser. Mat. 2012, 20: 173–187.

62. 62.

Najati A, Park C:Stability of a generalized Euler-Lagrange type additive mapping and homomorphisms in $C ∗$-algebras II. J. Nonlinear Sci. Appl. 2010, 3: 123–143.

63. 63.

Park C:Homomorphisms between Lie $J C ∗$-algebras and Cauchy-Rassias stability of Lie $J C ∗$-algebra derivations. J. Lie Theory 2005, 15: 393–414.

64. 64.

Park C:Homomorphisms between Poisson $J C ∗$-algebras. Bull. Braz. Math. Soc. 2005, 36: 79–97. 10.1007/s00574-005-0029-z

65. 65.

Kadison RV, Ringrose JR: Fundamentals of the Theory of Operator Algebras. Academic Press, New York; 1983.

## Acknowledgements

The authors are grateful to the reviewers for their valuable comments and suggestions.

## Author information

Authors

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Correspondence to Yeol Je Cho or Reza Saadati.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.

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Park, C., Cho, Y.J. & Saadati, R. Approximation of linear mappings in Banach modules over $C ∗$-algebras. J Inequal Appl 2013, 185 (2013). https://doi.org/10.1186/1029-242X-2013-185

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### Keywords

• fixed point
• Hyers-Ulam stability
• super-stability
• generalized Euler-Lagrange type additive mapping
• homomorphism
• $C ∗$-algebra 