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# Approximation of homomorphisms and derivations on Lie ${C}^{\ast }$-algebras via fixed point method

## Abstract

In this paper, using fixed point methods, we prove the generalized Hyers-Ulam stability of homomorphisms in ${C}^{\ast }$-algebras and Lie ${C}^{\ast }$-algebras and of derivations on C-algebras and Lie C-algebras for an m-variable additive functional equation.

MSC:39A10, 39B52, 39B72, 46L05, 47H10, 46B03.

## 1 Introduction and preliminaries

The stability problem of functional equations originated from a question of Ulam [1] concerning the stability of group homomorphisms:

Let $\left({G}_{1},\ast \right)$ be a group and let $\left({G}_{2},\diamond ,d\right)$ be a metric group with the metric $d\left(\cdot ,\cdot \right)$ . Given $ϵ>0$ , does there exist a $\delta \left(ϵ\right)>0$ such that if a mapping $h:{G}_{1}\to {G}_{2}$ satisfies the inequality $d\left(h\left(x\ast y\right),h\left(x\right)\diamond h\left(y\right)\right)<\delta$ for all $x,y\in {G}_{1}$ , then there is a homomorphism $H:{G}_{1}\to {G}_{2}$ with $d\left(h\left(x\right),H\left(x\right)\right)<ϵ$ for all $x\in {G}_{1}$ ?

If the answer is affirmative, we say that the equation of homomorphism $H\left(x\ast y\right)=H\left(x\right)\diamond H\left(y\right)$ is stable.

Since Ulam’s question, recently, many authors have given many answers and proved many kinds of functional equations in various spaces, for example, Banach algebras [2], random normed spaces [38], fuzzy normed spaces [9, 10], non-Archimedean Banach spaces [11], non-Archimedean lattice random spaces [12], inner product spaces [1315] and others [1621].

In this paper, using the fixed point method, we prove the generalized Hyers-Ulam stability of homomorphisms and derivations in Lie ${C}^{\ast }$-algebras for the following additive functional equation (see [22]):

$\sum _{i=1}^{m}f\left(m{x}_{i}+\sum _{j=1,j\ne i}^{m}{x}_{j}\right)+f\left(\sum _{i=1}^{m}{x}_{i}\right)=2f\left(\sum _{i=1}^{m}m{x}_{i}\right)$
(1.1)

for all $m\in \mathbb{N}$ with $m\ge 2$.

## 2 Stability of homomorphisms and derivations in ${C}^{\ast }$-algebras

Throughout this section, assume that A is a ${C}^{\ast }$-algebra with a norm ${\parallel \cdot \parallel }_{A}$ and B is a ${C}^{\ast }$-algebra with a norm ${\parallel \cdot \parallel }_{B}$.

For any mapping $f:A\to B$, we define

${D}_{\mu }f\left({x}_{1},\dots ,{x}_{m}\right):=\sum _{i=1}^{m}\mu f\left(m{x}_{i}+\sum _{j=1,j\ne i}^{m}{x}_{j}\right)+f\left(\mu \sum _{i=1}^{m}{x}_{i}\right)-2f\left(\mu \sum _{i=1}^{m}m{x}_{i}\right)$

for all $\mu \in {\mathbb{T}}^{1}:=\left\{\nu \in \mathbb{C}:|\nu |=1\right\}$ and ${x}_{1},\dots ,{x}_{m}\in A$.

Recall that a -linear mapping $H:A\to B$ is called a homomorphism in ${C}^{\ast }$-algebras if H satisfies $H\left(xy\right)=H\left(x\right)H\left(y\right)$ and $H\left({x}^{\ast }\right)=H{\left(x\right)}^{\ast }$ for all $x,y\in A$.

Recently, in [23], O’Regan et al. proved the generalized Hyers-Ulam stability of homomorphisms in ${C}^{\ast }$-algebras for the functional equation ${D}_{\mu }f\left({x}_{1},\dots ,{x}_{m}\right)=0$.

Theorem 2.1 [23]

Let $f:A\to B$ be a mapping for which there are functions $\phi :{A}^{m}\to \left[0,\mathrm{\infty }\right)$, $\psi :{A}^{2}\to \left[0,\mathrm{\infty }\right)$ and $\eta :A\to \left[0,\mathrm{\infty }\right)$ such that

$\begin{array}{c}\underset{j\to \mathrm{\infty }}{lim}{m}^{-j}\phi \left({m}^{j}{x}_{1},\dots ,{m}^{j}{x}_{m}\right)=0,\hfill \\ {\parallel {D}_{\mu }f\left({x}_{1},\dots ,{x}_{m}\right)\parallel }_{B}\le \phi \left({x}_{1},\dots ,{x}_{m}\right),\hfill \\ {\parallel f\left(xy\right)-f\left(x\right)f\left(y\right)\parallel }_{B}\le \psi \left(x,y\right),\hfill \\ \underset{j\to \mathrm{\infty }}{lim}{m}^{-2j}\psi \left({m}^{j}x,{m}^{j}y\right)=0,\hfill \\ {\parallel f\left({x}^{\ast }\right)-f{\left(x\right)}^{\ast }\parallel }_{B}\le \eta \left(x\right),\hfill \\ \underset{j\to \mathrm{\infty }}{lim}{m}^{-j}\eta \left({m}^{j}x\right)=0\hfill \end{array}$

for all $\mu \in {\mathbb{T}}^{1}$ and ${x}_{1},\dots ,{x}_{m},x,y\in A$. If there exists $0 such that

$\phi \left(mx,0,\dots ,0\right)\le mL\phi \left(x,0,\dots ,0\right)$

for all $x\in A$, then there exists a unique homomorphism $H:A\to B$ such that

${\parallel f\left(x\right)-H\left(x\right)\parallel }_{B}\le \frac{1}{m-mL}\phi \left(x,0,\dots ,0\right)$

for all $x\in A$.

Theorem 2.2 [23]

Let $f:A\to B$ be a mapping for which there are functions $\phi :{A}^{m}\to \left[0,\mathrm{\infty }\right)$, $\psi :{A}^{2}\to \left[0,\mathrm{\infty }\right)$ and $\eta :A\to \left[0,\mathrm{\infty }\right)$ such that

$\begin{array}{c}\underset{j\to \mathrm{\infty }}{lim}{m}^{j}\phi \left({m}^{-j}{x}_{1},\dots ,{m}^{-j}{x}_{m}\right)=0,\hfill \\ {\parallel {D}_{\mu }f\left({x}_{1},\dots ,{x}_{m}\right)\parallel }_{B}\le \phi \left({x}_{1},\dots ,{x}_{m}\right),\hfill \\ {\parallel f\left(xy\right)-f\left(x\right)f\left(y\right)\parallel }_{B}\le \psi \left(x,y\right),\hfill \\ \underset{j\to \mathrm{\infty }}{lim}{m}^{2j}\psi \left({m}^{-j}x,{m}^{-j}y\right)=0,\hfill \\ {\parallel f\left({x}^{\ast }\right)-f{\left(x\right)}^{\ast }\parallel }_{B}\le \eta \left(x\right),\hfill \\ \underset{j\to \mathrm{\infty }}{lim}{m}^{j}\eta \left({m}^{-j}x\right)=0\hfill \end{array}$

for all $\mu \in {\mathbb{T}}^{1}$ and ${x}_{1},\dots ,{x}_{m},x,y\in A$. If there exists $0 such that

$\phi \left(x,0,\dots ,0\right)\le \frac{L}{m}\phi \left(mx,0,\dots ,0\right)$

for all $x\in A$, then there exists a unique homomorphism $H:A\to B$ such that

${\parallel f\left(x\right)-H\left(x\right)\parallel }_{B}\le \frac{L}{m-mL}\phi \left(x,0,\dots ,0\right)$

for all $x\in A$.

Recall that a -linear mapping $\delta :A\to A$ is called a derivation on A if δ satisfies $\delta \left(xy\right)=\delta \left(x\right)y+x\delta \left(y\right)$ for all $x,y\in A$.

In [23], also, O’Regan et al. proved the generalized Hyers-Ulam stability of derivations on ${C}^{\ast }$-algebras for the functional equation ${D}_{\mu }f\left({x}_{1},\dots ,{x}_{m}\right)=0$.

Theorem 2.3 [23]

Let $f:A\to B$ be a mapping for which there are functions $\phi :{A}^{m}\to \left[0,\mathrm{\infty }\right)$ and $\psi :{A}^{2}\to \left[0,\mathrm{\infty }\right)$ such that

$\begin{array}{c}\underset{j\to \mathrm{\infty }}{lim}{m}^{-j}\phi \left({m}^{j}{x}_{1},\dots ,{m}^{j}{x}_{m}\right)=0,\hfill \\ {\parallel {D}_{\mu }f\left({x}_{1},\dots ,{x}_{m}\right)\parallel }_{B}\le \phi \left({x}_{1},\dots ,{x}_{m}\right),\hfill \\ {\parallel f\left(xy\right)-f\left(x\right)y-xf\left(y\right)\parallel }_{B}\le \psi \left(x,y\right),\hfill \\ \underset{j\to \mathrm{\infty }}{lim}{m}^{-2j}\psi \left({m}^{j}x,{m}^{j}y\right)=0\hfill \end{array}$

for all $\mu \in {\mathbb{T}}^{1}$ and ${x}_{1},\dots ,{x}_{m},x,y\in A$. If there exists $0 such that

$\phi \left(mx,0,\dots ,0\right)\le mL\phi \left(x,0,\dots ,0\right)$

for all $x\in A$, then there exists a unique derivation $\delta :A\to A$ such that

${\parallel f\left(x\right)-\delta \left(x\right)\parallel }_{B}\le \frac{1}{m-mL}\phi \left(x,0,\dots ,0\right)$

for all $x\in A$.

Theorem 2.4 [23]

Let $f:A\to B$ be a mapping for which there are functions $\phi :{A}^{m}\to \left[0,\mathrm{\infty }\right)$ and $\psi :{A}^{2}\to \left[0,\mathrm{\infty }\right)$ such that

$\begin{array}{r}\underset{j\to \mathrm{\infty }}{lim}{m}^{j}\phi \left({m}^{-j}{x}_{1},\dots ,{m}^{-j}{x}_{m}\right)=0,\\ {\parallel {D}_{\mu }f\left({x}_{1},\dots ,{x}_{m}\right)\parallel }_{B}\le \phi \left({x}_{1},\dots ,{x}_{m}\right),\\ {\parallel f\left(xy\right)-f\left(x\right)y-xf\left(y\right)\parallel }_{B}\le \psi \left(x,y\right),\\ \underset{j\to \mathrm{\infty }}{lim}{m}^{2j}\psi \left({m}^{-j}x,{m}^{-j}y\right)=0\end{array}$

for all $\mu \in {\mathbb{T}}^{1}$ and ${x}_{1},\dots ,{x}_{m},x,y\in A$. If there exists $0 such that

$\phi \left(mx,0,\dots ,0\right)\le \frac{L}{m}\phi \left(x,0,\dots ,0\right)$

for all $x\in A$, then there exists a unique derivation $\delta :A\to A$ such that

${\parallel f\left(x\right)-\delta \left(x\right)\parallel }_{B}\le \frac{L}{m-mL}\phi \left(x,0,\dots ,0\right)$

for all $x\in A$.

## 3 Stability of homomorphisms in Lie ${C}^{\ast }$-algebras

A ${C}^{\ast }$-algebra , endowed with the Lie product

$\left[x,y\right]:=\frac{xy-yx}{2}$

on , is called a Lie ${C}^{\ast }$-algebra (see [16, 2426]).

Definition 3.1 Let A and B be Lie ${C}^{\ast }$-algebras. A -linear mapping $H:A\to B$ is called a Lie ${C}^{\ast }$-algebra homomorphism if $H\left(\left[x,y\right]\right)=\left[H\left(x\right),H\left(y\right)\right]$ for all $x,y\in A$.

Throughout this section, assume that A is a Lie ${C}^{\ast }$-algebra with a norm ${\parallel \cdot \parallel }_{A}$ and B is a Lie ${C}^{\ast }$-algebra with a norm ${\parallel \cdot \parallel }_{B}$.

Now, we prove the generalized Hyers-Ulam stability of homomorphisms in Lie ${C}^{\ast }$-algebras for the functional equation ${D}_{\mu }f\left({x}_{1},\dots ,{x}_{m}\right)=0$.

Theorem 3.2 Let $f:A\to B$ be a mapping for which there are functions $\phi :{A}^{m}\to \left[0,\mathrm{\infty }\right)$ and $\psi :{A}^{2}\to \left[0,\mathrm{\infty }\right)$ such that

$\underset{j\to \mathrm{\infty }}{lim}{m}^{-j}\phi \left({m}^{j}{x}_{1},\dots ,{m}^{j}{x}_{m}\right)=0,$
(3.1)
${\parallel {D}_{\mu }f\left({x}_{1},\dots ,{x}_{m}\right)\parallel }_{B}\le \phi \left({x}_{1},\dots ,{x}_{m}\right),$
(3.2)
${\parallel f\left(\left[x,y\right]\right)-\left[f\left(x\right),f\left(y\right)\right]\parallel }_{B}\le \psi \left(x,y\right),$
(3.3)
$\underset{j\to \mathrm{\infty }}{lim}{m}^{-2j}\psi \left({m}^{j}x,{m}^{j}y\right)=0$
(3.4)

for all $\mu \in {\mathbb{T}}^{1}$ and ${x}_{1},\dots ,{x}_{m},x,y\in A$. If there exists $0 such that

$\phi \left(mx,0,\dots ,0\right)\le mL\phi \left(x,0,\dots ,0\right)$

for all $x\in A$, then there exists a unique Lie ${C}^{\ast }$-algebra homomorphism $H:A\to B$ such that

${\parallel f\left(x\right)-H\left(x\right)\parallel }_{B}\le \frac{1}{m-mL}\phi \left(x,0,\dots ,0\right)$
(3.5)

for all $x\in A$.

Proof By the same method as in the proof of Theorem 2.1, we can find the mapping $H:A\to B$ given by

$H\left(x\right)=\underset{n\to \mathrm{\infty }}{lim}\frac{f\left({m}^{n}x\right)}{{m}^{n}}$

for all $x\in A$. Thus it follows from (3.3) that

$\begin{array}{rcl}{\parallel H\left(\left[x,y\right]\right)-\left[H\left(x\right),H\left(y\right)\right]\parallel }_{B}& =& \underset{n\to \mathrm{\infty }}{lim}\frac{1}{{m}^{2n}}{\parallel f\left({m}^{2n}\left[x,y\right]\right)-\left[f\left({m}^{n}x\right),f\left({m}^{n}y\right)\right]\parallel }_{B}\\ \le & \underset{n\to \mathrm{\infty }}{lim}\frac{1}{{m}^{2n}}\psi \left({m}^{n}x,{m}^{n}y\right)=0\end{array}$

for all $x,y\in A$, and so

$H\left(\left[x,y\right]\right)=\left[H\left(x\right),H\left(y\right)\right]$

for all $x,y\in A$. Therefore, $H:A\to B$ is a Lie ${C}^{\ast }$-algebra homomorphism satisfying (3.5). This completes the proof. □

Corollary 3.3 Let $0 and θ be nonnegative real numbers. If $f:A\to B$ is a mapping such that

$\begin{array}{c}{\parallel {D}_{\mu }f\left({x}_{1},\dots ,{x}_{m}\right)\parallel }_{B}\le \theta \left({\parallel {x}_{1}\parallel }_{A}^{r}+{\parallel {x}_{2}\parallel }_{A}^{r}+\cdots +{\parallel {x}_{m}\parallel }_{A}^{r}\right),\hfill \\ {\parallel f\left(\left[x,y\right]\right)-\left[f\left(x\right),f\left(y\right)\right]\parallel }_{B}\le \theta \cdot {\parallel x\parallel }_{A}^{r}\cdot {\parallel y\parallel }_{A}^{r}\hfill \end{array}$

for all $\mu \in {\mathbb{T}}^{1}$ and ${x}_{1},\dots ,{x}_{m},x,y\in A$, then there exists a unique Lie ${C}^{\ast }$-algebra homomorphism $H:A\to B$ such that

${\parallel f\left(x\right)-H\left(x\right)\parallel }_{B}\le \frac{\theta }{m-{m}^{r}}{\parallel x\parallel }_{A}^{r}$

for all $x\in A$.

Proof The proof follows from Theorem 3.2 by taking

$\phi \left({x}_{1},\dots ,{x}_{m}\right)=\theta \left({\parallel {x}_{1}\parallel }_{A}^{r}+{\parallel {x}_{2}\parallel }_{A}^{r}+\cdots +{\parallel {x}_{m}\parallel }_{A}^{r}\right),\phantom{\rule{2em}{0ex}}\psi \left(x,y\right):=\theta \cdot {\parallel x\parallel }_{A}^{r}\cdot {\parallel y\parallel }_{A}^{r}$

for all ${x}_{1},\dots ,{x}_{m},x,y\in A$ and putting $L={m}^{r-1}$. □

Theorem 3.4 Let $f:A\to B$ be a mapping for which there are functions $\phi :{A}^{m}\to \left[0,\mathrm{\infty }\right)$ and $\psi :{A}^{2}\to \left[0,\mathrm{\infty }\right)$ satisfying (3.1)-(3.4) for all $\mu \in {\mathbb{T}}^{1}$ and ${x}_{1},\dots ,{x}_{m},x,y\in A$. If there exists $0 such that

$\phi \left(x,0,\dots ,0\right)\le \frac{L}{m}\phi \left(x,0,\dots ,0\right)$

for all $x\in A$, then there exists a unique Lie ${C}^{\ast }$-algebra homomorphism $H:A\to B$ such that

${\parallel f\left(x\right)-H\left(x\right)\parallel }_{B}\le \frac{L}{m-mL}\phi \left(x,0,\dots ,0\right)$

for all $x\in A$.

Corollary 3.5 Let $r>1$ and θ be nonnegative real numbers. If $f:A\to B$ is a mapping such that

$\begin{array}{c}{\parallel {D}_{\mu }f\left({x}_{1},\dots ,{x}_{m}\right)\parallel }_{B}\le \theta \cdot \left({\parallel {x}_{1}\parallel }_{A}^{r}+{\parallel {x}_{2}\parallel }_{A}^{r}+\cdots +{\parallel {x}_{m}\parallel }_{A}^{r}\right),\hfill \\ {\parallel f\left(\left[x,y\right]\right)-\left[f\left(x\right),f\left(y\right)\right]\parallel }_{B}\le \theta \cdot {\parallel x\parallel }_{A}^{r}\cdot {\parallel y\parallel }_{A}^{r}\hfill \end{array}$

for all $\mu \in {\mathbb{T}}^{1}$ and ${x}_{1},\dots ,{x}_{m},x,y\in A$, then there exists a unique Lie ${C}^{\ast }$-algebra homomorphism $H:A\to B$ such that

${\parallel f\left(x\right)-H\left(x\right)\parallel }_{B}\le \frac{\theta }{{m}^{r}-m}{\parallel x\parallel }_{A}^{r}$

for all $x\in A$.

Proof The proof follows from Theorem 3.4 by taking

$\begin{array}{c}\phi \left({x}_{1},\dots ,{x}_{m}\right)=\theta \cdot \left({\parallel {x}_{1}\parallel }_{A}^{r}+{\parallel {x}_{2}\parallel }_{A}^{r}+\cdots +{\parallel {x}_{m}\parallel }_{A}^{r}\right),\hfill \\ \psi \left(x,y\right):=\theta \cdot {\parallel x\parallel }_{A}^{r}\cdot {\parallel y\parallel }_{A}^{r}\hfill \end{array}$

for all ${x}_{1},\dots ,{x}_{m},x,y\in A$ and putting $L={m}^{1-r}$. □

## 4 Stability of derivations in Lie ${C}^{\ast }$-algebras

Definition 4.1 Let A be a Lie ${C}^{\ast }$-algebra. A -linear mapping $\delta :A\to A$ is called a Lie derivation if $\delta \left(\left[x,y\right]\right)=\left[\delta \left(x\right),y\right]+\left[x,\delta \left(y\right)\right]$ for all $x,y\in A$.

Throughout this section, assume that A is a Lie ${C}^{\ast }$-algebra with a norm ${\parallel \cdot \parallel }_{A}$.

Finally, we prove the generalized Hyers-Ulam stability of derivations on Lie ${C}^{\ast }$-algebras for the functional equation ${D}_{\mu }f\left({x}_{1},\dots ,{x}_{m}\right)=0$.

Theorem 4.2 Let $f:A\to A$ be a mapping for which there are functions $\phi :{A}^{m}\to \left[0,\mathrm{\infty }\right)$ and $\psi :{A}^{2}\to \left[0,\mathrm{\infty }\right)$ such that

$\underset{j\to \mathrm{\infty }}{lim}{m}^{-j}\phi \left({m}^{j}{x}_{1},\dots ,{m}^{j}{x}_{m}\right)=0,$
(4.1)
${\parallel {D}_{\mu }f\left({x}_{1},\dots ,{x}_{m}\right)\parallel }_{B}\le \phi \left({x}_{1},\dots ,{x}_{m}\right),$
(4.2)
${\parallel f\left(\left[x,y\right]\right)-\left[f\left(x\right),y\right]-\left[x,f\left(y\right)\right]\parallel }_{B}\le \psi \left(x,y\right),$
(4.3)
$\underset{j\to \mathrm{\infty }}{lim}{m}^{-2j}\psi \left({m}^{j}x,{m}^{j}y\right)=0$
(4.4)

for all $\mu \in {\mathbb{T}}^{1}$ and ${x}_{1},\dots ,{x}_{m},x,y\in A$. If there exists $0 such that

$\phi \left(mx,0,\dots ,0\right)\le mL\phi \left(x,0,\dots ,0\right)$

for all $x\in A$, then there exists a unique Lie derivation $\delta :A\to A$ such that

${\parallel f\left(x\right)-\delta \left(x\right)\parallel }_{B}\le \frac{1}{m-mL}\phi \left(x,0,\dots ,0\right)$
(4.5)

for all $x\in A$.

Proof By the same method as in the proof of Theorem 2.3, there exists a unique -linear mapping $\delta :A\to A$ satisfying (3.5). Also, we can find the mapping $\delta :A\to A$ given by

$\delta \left(x\right)=\underset{n\to \mathrm{\infty }}{lim}\frac{f\left({m}^{n}x\right)}{{m}^{n}}$
(4.6)

for all $x\in A$. Thus it follows from (4.3), (4.4) and (4.6) that

$\begin{array}{r}{\parallel \delta \left(\left[x,y\right]\right)-\left[\delta \left(x\right),y\right]-\left[x,\delta \left(y\right)\right]\parallel }_{A}\\ \phantom{\rule{1em}{0ex}}=\underset{n\to \mathrm{\infty }}{lim}\frac{1}{{m}^{2n}}{\parallel f\left({m}^{2n}\left[x,y\right]\right)-\left[f\left({m}^{n}x\right),\cdot {m}^{n}y\right]-\left[{m}^{n}x,f\left({m}^{n}y\right)\right]\parallel }_{A}\\ \phantom{\rule{1em}{0ex}}\le \underset{n\to \mathrm{\infty }}{lim}\frac{1}{{m}^{2n}}\psi \left({m}^{n}x,{m}^{n}y\right)=0\end{array}$

for all $x,y\in A$, and so

$\delta \left(\left[x,y\right]\right)=\left[\delta \left(x\right),y\right]+\left[x,\delta \left(y\right)\right]$

for all $x,y\in A$. Thus $\delta :A\to A$ is a Lie derivation satisfying (4.5). □

Corollary 4.3 Let $0 and θ be nonnegative real numbers. If $f:A\to A$ is a mapping such that

$\begin{array}{c}{\parallel {D}_{\mu }f\left({x}_{1},\dots ,{x}_{m}\right)\parallel }_{B}\le \theta \cdot \left({\parallel {x}_{1}\parallel }_{A}^{r}+\cdots {\parallel {x}_{m}\parallel }_{A}^{r}\right),\hfill \\ {\parallel f\left(\left[x,y\right]\right)-\left[f\left(x\right),y\right]-\left[x,f\left(y\right)\right]\parallel }_{A}\le \theta \cdot {\parallel x\parallel }_{A}^{r}\cdot {\parallel y\parallel }_{A}^{r}\hfill \end{array}$

for all $\mu \in {\mathbb{T}}^{1}$ and ${x}_{1},\dots ,{x}_{m},x,y\in A$, then there exists a unique derivation $\delta :A\to A$ such that

${\parallel f\left(x\right)-\delta \left(x\right)\parallel }_{A}\le \frac{\theta }{m-{m}^{r}}{\parallel x\parallel }_{A}^{r}$

for all $x\in A$.

Proof The proof follows from Theorem 4.2 by taking

$\phi \left({x}_{1},\dots ,{x}_{m}\right):=\theta \cdot \left({\parallel {x}_{1}\parallel }_{A}^{r}+\cdots +{\parallel {x}_{m}\parallel }_{A}^{r}\right)$

and

$\psi \left(x,y\right):=\theta \cdot {\parallel x\parallel }_{A}^{r}\cdot {\parallel y\parallel }_{A}^{r}$

for all ${x}_{1},\dots ,{x}_{m},x,y\in A$ and putting $L={m}^{r-1}$. □

Theorem 4.4 Let $f:A\to A$ be a mapping for which there are functions $\phi :{A}^{m}\to \left[0,\mathrm{\infty }\right)$ and $\psi :{A}^{2}\to \left[0,\mathrm{\infty }\right)$ such that

$\begin{array}{c}\begin{array}{r}\underset{j\to \mathrm{\infty }}{lim}{m}^{j}\phi \left({m}^{-j}{x}_{1},\dots ,{m}^{-j}{x}_{m}\right)=0,\\ {\parallel {D}_{\mu }f\left({x}_{1},\dots ,{x}_{m}\right)\parallel }_{B}\le \phi \left({x}_{1},\dots ,{x}_{m}\right),\end{array}\hfill \\ \begin{array}{r}{\parallel f\left(\left[x,y\right]\right)-\left[f\left(x\right),y\right]-\left[x,f\left(y\right)\right]\parallel }_{B}\le \psi \left(x,y\right),\\ \underset{j\to \mathrm{\infty }}{lim}{m}^{2j}\psi \left({m}^{-j}x,{m}^{-j}y\right)=0\end{array}\hfill \end{array}$

for all $\mu \in {\mathbb{T}}^{1}$ and ${x}_{1},\dots ,{x}_{m},x,y\in A$. If there exists $0 such that

$\phi \left(mx,0,\dots ,0\right)\le \frac{L}{m}\phi \left(x,0,\dots ,0\right)$

for all $x\in A$, then there exists a unique Lie derivation $\delta :A\to A$ such that

${\parallel f\left(x\right)-\delta \left(x\right)\parallel }_{B}\le \frac{L}{m-mL}\phi \left(x,0,\dots ,0\right)$

for all $x\in A$.

Proof The proof is similar to the proof of Theorem 4.2. □

Corollary 4.5 Let $r>1$ and θ be nonnegative real numbers. If $f:A\to A$ is a mapping such that

$\begin{array}{c}{\parallel {D}_{\mu }f\left({x}_{1},\dots ,{x}_{m}\right)\parallel }_{B}\le \theta \cdot \left({\parallel {x}_{1}\parallel }_{A}^{r}+\cdots {\parallel {x}_{m}\parallel }_{A}^{r}\right),\hfill \\ {\parallel f\left(\left[x,y\right]\right)-\left[f\left(x\right),y\right]-\left[x,f\left(y\right)\right]\parallel }_{A}\le \theta \cdot {\parallel x\parallel }_{A}^{r}\cdot {\parallel y\parallel }_{A}^{r}\hfill \end{array}$

for all μ ${\mathbb{T}}^{1}$ and ${x}_{1},\dots ,{x}_{m},x,y\in A$, then there exists a unique Lie derivation $\delta :A\to A$ such that

${\parallel f\left(x\right)-\delta \left(x\right)\parallel }_{A}\le \frac{\theta }{{m}^{r}-m}{\parallel x\parallel }_{A}^{r}$

for all $x\in A$.

Proof The proof follows from Theorem 4.4 by taking

$\phi \left({x}_{1},\dots ,{x}_{m}\right):=\theta \cdot \left({\parallel {x}_{1}\parallel }_{A}^{r}+\cdots {\parallel {x}_{m}\parallel }_{A}^{r}\right)$

and

$\psi \left(x,y\right):=\theta \cdot {\parallel x\parallel }_{A}^{r}\cdot {\parallel y\parallel }_{A}^{r}$

for all ${x}_{1},\dots ,{x}_{m},x,y\in A$ and putting $L={m}^{1-r}$. □

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25. 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.

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

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## Acknowledgements

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant Number: 2012-0008170).

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Cho, Y.J., Saadati, R. & Yang, YO. Approximation of homomorphisms and derivations on Lie ${C}^{\ast }$-algebras via fixed point method. J Inequal Appl 2013, 415 (2013). https://doi.org/10.1186/1029-242X-2013-415

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