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 $(G_1,\ast )$ be a group and let $(G_2,\diamond ,d)$ be a metric group with the metric $d(\cdot ,\cdot )$ . Given $ϵ>0$ , does there exist a $\delta (ϵ)>0$ such that if a mapping $h:G_1\to G_2$ satisfies the inequality $d(h(x\ast y),h(x)\diamond h(y))<\delta$ for all $x,y\in G_1$ , then there is a homomorphism $H:G_1\to G_2$ with $d(h(x),H(x))<ϵ$ for all $x\in G_1$ ?

If the answer is affirmative, we say that the equation of homomorphism $H(x\ast y)=H(x)\diamond H(y)$ 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 [3–8], fuzzy normed spaces [9, 10], non-Archimedean Banach spaces [11], non-Archimedean lattice random spaces [12], inner product spaces [13–15] and others [16–21].

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]):

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

(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

$$D_{\mu }f(x_1,\dots ,x_m):=\sum _{i=1}^m\mu f(m{x}_i+\sum _{j=1,j\ne i}^m{x}_j)+f\left(\mu \sum _{i=1}^m{x}_i\right)-2f\left(\mu \sum _{i=1}^{m}m{x}_i\right)$$

for all $\mu \in {\mathbb{T}}^1:=\{\nu \in \mathbb{C}:|\nu |=1\}$ and $x_1,\dots ,x_m\in A$.

Recall that a ℂ-linear mapping $H:A\to B$ is called a homomorphism in $C^{\ast }$-algebras if H satisfies $H(xy)=H(x)H(y)$ and $H(x^{\ast })=H{(x)}^{\ast }$ 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 $D_{\mu }f(x_1,\dots ,x_m)=0$.

Theorem 2.1 [23]

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

$$\begin{array}{c}\underset{j\to \mathrm{\infty }}{lim}{m}^{-j}\phi (m^j{x}_1,\dots ,m^j{x}_m)=0,\hfill \\ {\parallel D_{\mu }f(x_1,\dots ,x_m)\parallel }_B\le \phi (x_1,\dots ,x_m),\hfill \\ {\parallel f(xy)-f(x)f(y)\parallel }_B\le \psi (x,y),\hfill \\ \underset{j\to \mathrm{\infty }}{lim}{m}^{-2j}\psi (m^{j}x,m^{j}y)=0,\hfill \\ {\parallel f\left(x^{\ast }\right)-f{(x)}^{\ast }\parallel }_B\le \eta (x),\hfill \\ \underset{j\to \mathrm{\infty }}{lim}{m}^{-j}\eta \left(m^{j}x\right)=0\hfill \end{array}$$

for all $\mu \in {\mathbb{T}}^1$ and $x_1,\dots ,x_m,x,y\in A$. If there exists $0<L<1$ such that

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

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

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

for all $x\in A$.

Theorem 2.2 [23]

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

$$\begin{array}{c}\underset{j\to \mathrm{\infty }}{lim}{m}^j\phi (m^{-j}{x}_1,\dots ,m^{-j}{x}_m)=0,\hfill \\ {\parallel D_{\mu }f(x_1,\dots ,x_m)\parallel }_B\le \phi (x_1,\dots ,x_m),\hfill \\ {\parallel f(xy)-f(x)f(y)\parallel }_B\le \psi (x,y),\hfill \\ \underset{j\to \mathrm{\infty }}{lim}{m}^{2j}\psi (m^{-j}x,m^{-j}y)=0,\hfill \\ {\parallel f\left(x^{\ast }\right)-f{(x)}^{\ast }\parallel }_B\le \eta (x),\hfill \\ \underset{j\to \mathrm{\infty }}{lim}{m}^j\eta \left(m^{-j}x\right)=0\hfill \end{array}$$

for all $\mu \in {\mathbb{T}}^1$ and $x_1,\dots ,x_m,x,y\in A$. If there exists $0<L<1$ such that

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

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

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

for all $x\in A$.

Recall that a ℂ-linear mapping $\delta :A\to A$ is called a derivation on A if δ satisfies $\delta (xy)=\delta (x)y+x\delta (y)$ 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 $D_{\mu }f(x_1,\dots ,x_m)=0$.

Theorem 2.3 [23]

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

$$\begin{array}{c}\underset{j\to \mathrm{\infty }}{lim}{m}^{-j}\phi (m^j{x}_1,\dots ,m^j{x}_m)=0,\hfill \\ {\parallel D_{\mu }f(x_1,\dots ,x_m)\parallel }_B\le \phi (x_1,\dots ,x_m),\hfill \\ {\parallel f(xy)-f(x)y-xf(y)\parallel }_B\le \psi (x,y),\hfill \\ \underset{j\to \mathrm{\infty }}{lim}{m}^{-2j}\psi (m^{j}x,m^{j}y)=0\hfill \end{array}$$

for all $\mu \in {\mathbb{T}}^1$ and $x_1,\dots ,x_m,x,y\in A$. If there exists $0<L<1$ such that

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

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

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

for all $x\in A$.

Theorem 2.4 [23]

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

$$\begin{array}{r}\underset{j\to \mathrm{\infty }}{lim}{m}^j\phi (m^{-j}{x}_1,\dots ,m^{-j}{x}_m)=0,\\ {\parallel D_{\mu }f(x_1,\dots ,x_m)\parallel }_B\le \phi (x_1,\dots ,x_m),\\ {\parallel f(xy)-f(x)y-xf(y)\parallel }_B\le \psi (x,y),\\ \underset{j\to \mathrm{\infty }}{lim}{m}^{2j}\psi (m^{-j}x,m^{-j}y)=0\end{array}$$

for all $\mu \in {\mathbb{T}}^1$ and $x_1,\dots ,x_m,x,y\in A$. If there exists $0<L<1$ such that

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

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

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

for all $x\in A$.

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

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

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

on , is called a Lie $C^{\ast }$-algebra (see [16, 24–26]).

Definition 3.1 Let A and B be Lie $C^{\ast }$-algebras. A ℂ-linear mapping $H:A\to B$ is called a Lie $C^{\ast }$-algebra homomorphism if $H([x,y])=[H(x),H(y)]$ 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 $D_{\mu }f(x_1,\dots ,x_m)=0$.

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

$$\underset{j\to \mathrm{\infty }}{lim}{m}^{-j}\phi (m^j{x}_1,\dots ,m^j{x}_m)=0,$$

(3.1)

$${\parallel D_{\mu }f(x_1,\dots ,x_m)\parallel }_B\le \phi (x_1,\dots ,x_m),$$

(3.2)

$${\parallel f([x,y])-[f(x),f(y)]\parallel }_B\le \psi (x,y),$$

(3.3)

$$\underset{j\to \mathrm{\infty }}{lim}{m}^{-2j}\psi (m^{j}x,m^{j}y)=0$$

(3.4)

for all $\mu \in {\mathbb{T}}^1$ and $x_1,\dots ,x_m,x,y\in A$. If there exists $0<L<1$ such that

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

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

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

(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

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

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

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

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

$$H([x,y])=[H(x),H(y)]$$

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 $0<r<1$ and θ be nonnegative real numbers. If $f:A\to B$ is a mapping such that

$$\begin{array}{c}{\parallel D_{\mu }f(x_1,\dots ,x_m)\parallel }_B\le \theta ({\parallel x_1\parallel }_A^r+{\parallel x_2\parallel }_A^r+\cdots +{\parallel x_m\parallel }_A^r),\hfill \\ {\parallel f([x,y])-[f(x),f(y)]\parallel }_B\le \theta \cdot {\parallel x\parallel }_A^r\cdot {\parallel y\parallel }_A^r\hfill \end{array}$$

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

$${\parallel f(x)-H(x)\parallel }_B\le \frac{\theta }{m-m^r}{\parallel x\parallel }_A^r$$

for all $x\in A$.

Proof The proof follows from Theorem 3.2 by taking

$$\phi (x_1,\dots ,x_m)=\theta ({\parallel x_1\parallel }_A^r+{\parallel x_2\parallel }_A^r+\cdots +{\parallel x_m\parallel }_A^r),\psi (x,y):=\theta \cdot {\parallel x\parallel }_A^r\cdot {\parallel y\parallel }_A^r$$

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 $\phi :A^m\to [0,\mathrm{\infty })$ and $\psi :A^2\to [0,\mathrm{\infty })$ 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 $0<L<1$ such that

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

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

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

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

$$\begin{array}{c}{\parallel D_{\mu }f(x_1,\dots ,x_m)\parallel }_B\le \theta \cdot ({\parallel x_1\parallel }_A^r+{\parallel x_2\parallel }_A^r+\cdots +{\parallel x_m\parallel }_A^r),\hfill \\ {\parallel f([x,y])-[f(x),f(y)]\parallel }_B\le \theta \cdot {\parallel x\parallel }_A^r\cdot {\parallel y\parallel }_A^r\hfill \end{array}$$

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

$${\parallel f(x)-H(x)\parallel }_B\le \frac{\theta }{m^r-m}{\parallel x\parallel }_A^r$$

for all $x\in A$.

Proof The proof follows from Theorem 3.4 by taking

$$\begin{array}{c}\phi (x_1,\dots ,x_m)=\theta \cdot ({\parallel x_1\parallel }_A^r+{\parallel x_2\parallel }_A^r+\cdots +{\parallel x_m\parallel }_A^r),\hfill \\ \psi (x,y):=\theta \cdot {\parallel x\parallel }_A^r\cdot {\parallel y\parallel }_A^r\hfill \end{array}$$

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 ℂ-linear mapping $\delta :A\to A$ is called a Lie derivation if $\delta ([x,y])=[\delta (x),y]+[x,\delta (y)]$ 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 $D_{\mu }f(x_1,\dots ,x_m)=0$.

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

$$\underset{j\to \mathrm{\infty }}{lim}{m}^{-j}\phi (m^j{x}_1,\dots ,m^j{x}_m)=0,$$

(4.1)

$${\parallel D_{\mu }f(x_1,\dots ,x_m)\parallel }_B\le \phi (x_1,\dots ,x_m),$$

(4.2)

$${\parallel f([x,y])-[f(x),y]-[x,f(y)]\parallel }_B\le \psi (x,y),$$

(4.3)

$$\underset{j\to \mathrm{\infty }}{lim}{m}^{-2j}\psi (m^{j}x,m^{j}y)=0$$

(4.4)

for all $\mu \in {\mathbb{T}}^1$ and $x_1,\dots ,x_m,x,y\in A$. If there exists $0<L<1$ such that

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

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

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

(4.5)

for all $x\in A$.

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

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

(4.6)

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

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

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

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

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

Corollary 4.3 Let $0<r<1$ and θ be nonnegative real numbers. If $f:A\to A$ is a mapping such that

$$\begin{array}{c}{\parallel D_{\mu }f(x_1,\dots ,x_m)\parallel }_B\le \theta \cdot ({\parallel x_1\parallel }_A^r+\cdots {\parallel x_m\parallel }_A^r),\hfill \\ {\parallel f([x,y])-[f(x),y]-[x,f(y)]\parallel }_A\le \theta \cdot {\parallel x\parallel }_A^r\cdot {\parallel y\parallel }_A^r\hfill \end{array}$$

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

$${\parallel f(x)-\delta (x)\parallel }_A\le \frac{\theta }{m-m^r}{\parallel x\parallel }_A^r$$

for all $x\in A$.

Proof The proof follows from Theorem 4.2 by taking

$$\phi (x_1,\dots ,x_m):=\theta \cdot ({\parallel x_1\parallel }_A^r+\cdots +{\parallel x_m\parallel }_A^r)$$

and

$$\psi (x,y):=\theta \cdot {\parallel x\parallel }_A^r\cdot {\parallel y\parallel }_A^r$$

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 $\phi :A^m\to [0,\mathrm{\infty })$ and $\psi :A^2\to [0,\mathrm{\infty })$ such that

$$\begin{array}{c}\begin{array}{r}\underset{j\to \mathrm{\infty }}{lim}{m}^j\phi (m^{-j}{x}_1,\dots ,m^{-j}{x}_m)=0,\\ {\parallel D_{\mu }f(x_1,\dots ,x_m)\parallel }_B\le \phi (x_1,\dots ,x_m),\end{array}\hfill \\ \begin{array}{r}{\parallel f([x,y])-[f(x),y]-[x,f(y)]\parallel }_B\le \psi (x,y),\\ \underset{j\to \mathrm{\infty }}{lim}{m}^{2j}\psi (m^{-j}x,m^{-j}y)=0\end{array}\hfill \end{array}$$

for all $\mu \in {\mathbb{T}}^1$ and $x_1,\dots ,x_m,x,y\in A$. If there exists $0<L<1$ such that

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

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

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

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

$$\begin{array}{c}{\parallel D_{\mu }f(x_1,\dots ,x_m)\parallel }_B\le \theta \cdot ({\parallel x_1\parallel }_A^r+\cdots {\parallel x_m\parallel }_A^r),\hfill \\ {\parallel f([x,y])-[f(x),y]-[x,f(y)]\parallel }_A\le \theta \cdot {\parallel x\parallel }_A^r\cdot {\parallel y\parallel }_A^r\hfill \end{array}$$

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

$${\parallel f(x)-\delta (x)\parallel }_A\le \frac{\theta }{m^r-m}{\parallel x\parallel }_A^r$$

for all $x\in A$.

Proof The proof follows from Theorem 4.4 by taking

$$\phi (x_1,\dots ,x_m):=\theta \cdot ({\parallel x_1\parallel }_A^r+\cdots {\parallel x_m\parallel }_A^r)$$

and

$$\psi (x,y):=\theta \cdot {\parallel x\parallel }_A^r\cdot {\parallel y\parallel }_A^r$$

for all $x_1,\dots ,x_m,x,y\in A$ and putting $L=m^{1-r}$. □