# Stability of homomorphisms on fuzzy Lie $C ∗$-algebras via fixed point method

## Abstract

In this paper, first, we define fuzzy $C ∗$-algebras and fuzzy Lie $C ∗$-algebras; then, using fixed point methods, we prove the generalized Hyers-Ulam stability of homomorphisms in fuzzy $C ∗$-algebras and fuzzy 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 with a question of Ulam  concerning the stability of group homomorphisms: let $( G 1 ,∗)$ be a group and let $( G 2 ,⋄,d)$ be a metric group with the metric $d(⋅,⋅)$. Given $ϵ>0$, does there exist a $δ(ϵ)>0$ such that if a mapping $h: G 1 → G 2$ satisfies the inequality $d(h(x∗y),h(x)⋄h(y))<δ$ for all $x,y∈ G 1$, then there is a homomorphism $H: G 1 → G 2$ with $d(h(x),H(x))<ϵ$ for all $x∈ G 1$? If the answer is affirmative, we would say that the equation of homomorphism $H(x∗y)=H(x)⋄H(y)$ is stable. We recall a fundamental result in fixed-point theory. Let Ω be a set. A function $d:Ω×Ω→[0,∞]$ is called a generalized metric on Ω if d satisfies

1. (1)

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

2. (2)

$d(x,y)=d(y,x)$ for all $x,y∈Ω$;

3. (3)

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

Theorem 1.1 

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

$d( J n x, J n + 1 x)<∞$, $∀n≥ n 0$;

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 $Γ={y∈Ω∣d( J n 0 x,y)<∞}$;

4. (4)

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

In this paper, using the fixed point method, we prove the generalized Hyers-Ulam stability of homomorphisms and derivations in fuzzy Lie $C ∗$-algebras for the following additive functional equation :

$∑ i = 1 m f ( m x i + ∑ j = 1 , j ≠ i m x j ) +f ( ∑ i = 1 m x i ) =2f ( ∑ i = 1 m m x i ) (m∈N,m≥2).$
(1.1)

We use the definition of fuzzy normed spaces given in  to investigate a fuzzy version of the Hyers-Ulam stability for the Cauchy-Jensen functional equation in the fuzzy normed algebra setting (see also ).

Definition 1.2 

Let X be a real vector space. A function $N:X×R→[0,1]$ is called a fuzzy norm on X if for all $x,y∈X$ and all $s,t∈R$,

($N 1$) $N(x,t)=0$ for $t≤0$;

($N 2$) $x=0$ if and only if $N(x,t)=1$ for all $t>0$;

($N 3$) $N(cx,t)=N(x, t | c | )$ if $c≠0$;

($N 4$) $N(x+y,s+t)≥min{N(x,s),N(y,t)}$;

($N 5$) $N(x,⋅)$ is a non-decreasing function of and $lim t → ∞ N(x,t)=1$;

($N 6$) for $x≠0$, $N(x,⋅)$ is continuous on .

The pair $(X,N)$ is called a fuzzy normed vector space.

Definition 1.3 

1. (1)

Let $(X,N)$ be a fuzzy normed vector space. A sequence ${ x n }$ in X is said to be convergent or converge if there exists an $x∈X$ such that $lim n → ∞ N( x n −x,t)=1$ for all $t>0$. In this case, x is called the limit of the sequence ${ x n }$ and we denote it by $N- lim n → ∞ x n =x$.

2. (2)

Let $(X,N)$ be a fuzzy normed vector space. A sequence ${ x n }$ in X is called Cauchy if for each $ε>0$ and each $t>0$ there exists an $n 0 ∈N$ such that for all $n≥ n 0$ and all $p>0$, we have $N( x n + p − x n ,t)>1−ε$.

It is well-known that every convergent sequence in a fuzzy normed vector space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed vector space is called a fuzzy Banach space.

We say that a mapping $f:X→Y$ between fuzzy normed vector spaces X and Y is continuous at a point $x 0 ∈X$ if for each sequence ${ x n }$ converging to $x 0$ in X, then the sequence ${f( x n )}$ converges to $f( x 0 )$. If $f:X→Y$ is continuous at each $x∈X$, then $f:X→Y$ is said to be continuous on X (see [4, 10]).

Definition 1.4 

A fuzzy normed algebra $(X,μ,∗, ∗ ′ )$ is a fuzzy normed space $(X,N,∗)$ with algebraic structure such that

($N 7$) $N(xy,ts)≥N(x,t)∗N(y,s)$ for all $x,y∈X$ and all $t,s>0$, in which is a continuous t-norm.

Every normed algebra $(X,∥⋅∥)$ defines a fuzzy normed algebra $(X,N,min)$, where

$N(x,t)= t t + ∥ x ∥$

for all $t>0$ iff

$∥xy∥≤∥x∥∥y∥+s∥y∥+t∥x∥(x,y∈X;t,s>0).$

This space is called the induced fuzzy normed algebra.

Definition 1.5 (1) Let $(X,N,∗)$ and $(Y,N,∗)$ be fuzzy normed algebras. An -linear mapping $f:X→Y$ is called a homomorphism if $f(xy)=f(x)f(y)$ for all $x,y∈X$.

1. (2)

An -linear mapping $f:X→X$ is called a derivation if $f(xy)=f(x)y+xf(y)$ for all $x,y∈X$.

Definition 1.6 Let $(U,N,∗, ∗ ′ )$ be a fuzzy Banach algebra, then an involution on is a mapping $u→ u ∗$ from into which satisfies

1. (i)

$u ∗ ∗ =u$ for $u∈U$;

2. (ii)

$( α u + β v ) ∗ = α ¯ u ∗ + β ¯ v ∗$;

3. (iii)

$( u v ) ∗ = v ∗ u ∗$ for $u,v∈U$.

If, in addition $N( u ∗ u,ts)=N(u,t)∗N(u,s)$ for $u∈U$ and $t>0$, then is a fuzzy $C ∗$-algebra.

## 2 Stability of homomorphisms in fuzzy $C ∗$-algebras

Throughout this section, assume that A is a fuzzy $C ∗$-algebra with norm $N A$ and that B is a fuzzy $C ∗$-algebra with norm $N B$.

For a given mapping $f:A→B$, we define

$D μ f( x 1 ,…, x m ):= ∑ i = 1 m μf ( m x i + ∑ j = 1 , j ≠ i m x j ) +f ( μ ∑ i = 1 m x i ) −2f ( μ ∑ i = 1 m m x i )$

for all $μ∈ T 1 :={ν∈C:|ν|=1}$ and all $x 1 ,…, x m ∈A$.

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

We prove the generalized Hyers-Ulam stability of homomorphisms in fuzzy $C ∗$-algebras for the functional equation $D μ f( x 1 ,…, x m )=0$.

Theorem 2.1 Let $f:A→B$ be a mapping for which there are functions $φ: A m ×(0,∞)→[0,1]$, $ψ: A 2 ×(0,∞)→[0,1]$ and $η:A×(0,∞)→[0,1]$ such that

$N B ( D μ f ( x 1 , … , x m ) , t ) ≥φ( x 1 ,…, x m ,t),$
(2.1)
$lim j → ∞ φ ( m j x 1 , … , m j x m , m j t ) =1,$
(2.2)
$N B ( f ( x y ) − f ( x ) f ( y ) , t ) ≥ψ(x,y,t),$
(2.3)
$lim j → ∞ ψ ( m j x , m j y , m 2 j t ) =1,$
(2.4)
$N B ( f ( x ∗ ) − f ( x ) ∗ , t ) ≥η(x,t),$
(2.5)
$lim j → ∞ η ( m j x , m j t ) =1$
(2.6)

for all $μ∈ T 1$, all $x 1 ,…, x m ,x,y∈A$ and $t>0$. If there exists an $L<1$ such that

$φ(mx,0,…,0,mLt)≥φ(x,0,…,0,t)$
(2.7)

for all $x∈A$ and $t>0$, then there exists a unique homomorphism $H:A→B$ such that

$N B ( f ( x ) − H ( x ) , t ) ≥φ ( x , 0 , … , 0 , ( m − m L ) t )$
(2.8)

for all $x∈A$ and $t>0$.

Proof Consider the set $X:={g:A→B}$ and introduce the generalized metric on X:

$d(g,h)=inf { C ∈ R + : N B ( g ( x ) − h ( x ) , C t ) ≥ φ ( x , 0 , … , 0 , t ) , ∀ x ∈ A , t > 0 } .$

It is easy to show that $(X,d)$ is complete. Now, we consider the linear mapping $J:X→X$ such that $Jg(x):= 1 m g(mx)$ for all $x∈A$. By Theorem 3.1 of , $d(Jg,Jh)≤Ld(g,h)$ for all $g,h∈X$. Letting $μ=1$, $x= x 1$ and $x 2 =⋯= x m =0$ in equation (2.1), we get

$N B ( f ( m x ) − m f ( x ) , t ) ≥φ(x,0,…,0,t)$
(2.9)

for all $x∈A$ and $t>0$. Therefore

$N B ( f ( x ) − 1 m f ( m x ) , t ) ≥φ(x,0,…,0,mt)$

for all $x∈A$ and $t>0$. Hence $d(f,Jf)≤ 1 m$. By Theorem 1.1, there exists a mapping $H:A→B$ such that

1. (1)

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

$H(mx)=mH(x)$
(2.10)

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

$Y= { g ∈ X : d ( f , g ) < ∞ } .$

This implies that H is a unique mapping satisfying equation (2.10) such that there exists $C∈(0,∞)$ satisfying

$N B ( H ( x ) − f ( x ) , C t ) ≥φ(x,0,…,0,t)$

for all $x∈A$ and $t>0$.

1. (2)

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

$lim n → ∞ f ( m n x ) m n =H(x)$
(2.11)

for all $x∈A$.

1. (3)

$d(f,H)≤ 1 1 − L d(f,Jf)$, which implies the inequality $d(f,H)≤ 1 m − m L$. This implies that the inequality (2.8) holds.

It follows from equations (2.1), (2.2), and (2.11) that

$N B ( ∑ i = 1 m H ( m x i + ∑ j = 1 , j ≠ i m x j ) + H ( ∑ i = 1 m x i ) − 2 H ( ∑ i = 1 m m x i ) , t ) = lim n → ∞ N B ( ∑ i = 1 m f ( m n + 1 x i + ∑ j = 1 , j ≠ i m m n x j ) + f ( ∑ i = 1 m m n x i ) − 2 f ( ∑ i = 1 m m n + 1 x i ) , m n t ) ≤ lim n → ∞ φ ( m n x 1 , … , m n x m , m n t ) = 1$

for all $x 1 ,…, x m ∈A$ and $t>0$. So

$∑ i = 1 m H ( m x i + ∑ j = 1 , j ≠ i m x j ) +H ( ∑ i = 1 m x i ) =2H ( ∑ i = 1 m m x i )$

for all $x 1 ,…, x m ∈A$.

By a similar method to above, we get $μH(mx)=H(mμx)$ for all $μ∈ T 1$ and all $x∈A$. Thus one can show that the mapping $H:A→B$ is -linear.

It follows from equations (2.3), (2.4), and (2.11) that

$N B ( H ( x y ) − H ( x ) H ( y ) , t ) = lim n → ∞ N B ( f ( m n x y ) − f ( m n x ) f ( m n y ) , m n t ) ≤ lim n → ∞ ψ ( m n x , m n y , m 2 n t ) = 1$

for all $x,y∈A$. So $H(xy)=H(x)H(y)$ for all $x,y∈A$. Thus $H:A→B$ is a homomorphism satisfying equation (2.7), as desired.

Also by equations (2.5), (2.6), (2.11), and by a similar method we have $H( x ∗ )=H ( x ) ∗$. □

## 3 Stability of homomorphisms in fuzzy Lie $C ∗$-algebras

A fuzzy $C ∗$-algebra , endowed with the Lie product

$[x,y]:= x y − y x 2$

on , is called a fuzzy Lie $C ∗$-algebra (see ).

Definition 3.1 Let A and B be fuzzy Lie $C ∗$-algebras. A -linear mapping $H:A→B$ is called a fuzzy Lie $C ∗$-algebra homomorphism if $H([x,y])=[H(x),H(y)]$ for all $x,y∈A$.

Throughout this section, assume that A is a fuzzy Lie $C ∗$-algebra with norm $N A$ and that B is a fuzzy Lie $C ∗$-algebra with norm $N B$.

We prove the generalized Hyers-Ulam stability of homomorphisms in fuzzy Lie $C ∗$-algebras for the functional equation $D μ f( x 1 ,…, x m )=0$.

Theorem 3.2 Let $f:A→B$ be a mapping for which there are functions $φ: A m ×(0,∞)→[0,1]$ and $ψ: A 2 ×(0,∞)→[0,1]$ such that

$lim j → ∞ φ ( m j x 1 , … , m j x m , m j t ) =1,$
(3.1)
$N B ( D μ f ( x 1 , … , x m ) , t ) ≥φ( x 1 ,…, x m ,t),$
(3.2)
$N B ( f ( [ x , y ] ) − [ f ( x ) , f ( y ) ] , t ) ≥ψ(x,y,t),$
(3.3)
$lim j → ∞ ψ ( m j x , m j y , m 2 j t ) =1$
(3.4)

for all $μ∈ T 1$, all $x 1 ,…, x m ,x,y∈A$ and $t>0$. If there exists an $L<1$ such that

$φ(mx,0,…,0,mlt)≥φ(x,0,…,0,t)$

for all $x∈A$ and $t>0$, then there exists a unique homomorphism $H:A→B$ such that

$N B ( f ( x ) − H ( x ) , t ) ≥φ ( x , 0 , … , 0 , ( m − m L ) t )$
(3.5)

for all $x∈A$ and $t>0$.

Proof By the same reasoning as the proof of Theorem 2.1, we can find that the mapping $H:A→B$ is given by

$H(x)= lim n → ∞ f ( m n x ) m n$

for all $x∈A$.

It follows from equation (3.3) that

$N B ( H ( [ x , y ] ) − [ H ( x ) , H ( y ) ] , t ) = lim n → ∞ N B ( f ( m 2 n [ x , y ] ) − [ f ( m n x ) , f ( m n y ) ] , m 2 n t ) ≥ lim n → ∞ ψ ( m n x , m n y , m 2 n t ) = 1$

for all $x,y∈A$ and $t>0$. So

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

for all $x,y∈A$.

Thus $H:A→B$ is a fuzzy Lie $C ∗$-algebra homomorphism satisfying equation (3.5), as desired. □

Corollary 3.3 Let $0 and θ be nonnegative real numbers, and let $f:A→B$ be a mapping such that

$N B ( D μ f ( x 1 , … , x m ) , t ) ≥ t t + θ ( ∥ x 1 ∥ A r + ∥ x 2 ∥ A r + ⋯ + ∥ x m ∥ A r ) ,$
(3.6)
$N B ( f ( [ x , y ] ) − [ f ( x ) , f ( y ) ] , t ) ≥ t t + θ ⋅ ∥ x ∥ A r ⋅ ∥ y ∥ A r$
(3.7)

for all $μ∈ T 1$, all $x 1 ,…, x m ,x,y∈A$ and $t>0$. Then there exists a unique homomorphism $H:A→B$ such that

$N B ( f ( x ) − H ( x ) , t ) ≤ t t + θ m − m r ∥ x ∥ A r$

for all $x∈A$ and $t>0$.

Proof The proof follows from Theorem 3.2 by taking

$φ ( x 1 , … , x m , t ) = t t + θ ( ∥ x 1 ∥ A r + ∥ x 2 ∥ A r + ⋯ + ∥ x m ∥ A r ) , ψ ( x , y , t ) : = t t + θ ⋅ ∥ x ∥ A r ⋅ ∥ y ∥ A r$

for all $x 1 ,…, x m ,x,y∈A$ and $t>0$. Putting $L= m r − 1$, we get the desired result. □

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Correspondence to Sung Jin Lee.

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Vahidi, J., Lee, S.J. Stability of homomorphisms on fuzzy Lie $C ∗$-algebras via fixed point method. J Inequal Appl 2014, 33 (2014). https://doi.org/10.1186/1029-242X-2014-33

• homomorphism in $C ∗$-algebras and Lie $C ∗$-algebras 