D O I: 1 0 .1 5 0 1 / C o m m u a 1 _ 0 0 0 0 0 0 0 7 2 7 IS S N 1 3 0 3 –5 9 9 1
STABILITY AND SUPER STABILITY OF FUZZY APPROXIMATELY *-HOMOMORPHISMS
N. EGHBALI
Abstract. In this paper we introduce the concept of fuzzy Banach *-algebra. Then we study the stability and super stability of approximately *-homomorphisms in the fuzzy sense.
1. Introduction
It seems that the stability problem of functional equations had been …rst raised by Ulam [12]. In 1941, Hyers [3] showed that if > 0 and if f : E1 ! E2 is a
mapping between Banach spaces E1and E2with jjf(x+y) f(x) f(y)jj for all
x; y 2 E1, then there exists a unique T : E1! E2such that T (x + y) = T (x) + T (y)
with jjf(x) T (x)jj for all x; y 2 E1. In 1978, a generalized solution to Ulam’s
problem for approximately linear mappings was given by Th. M. Rassias [10]. Suppose E1 and E2 are two real Banach spaces and f : E1! E2 is a mapping. If
there exist 0 and 0 p < 1 such that jjf(x+y) f(x) f(y)jj (jjxjjp+jjyjjp)
for all x; y 2 E1, then there is a unique additive mapping T : E1! E2 such that
jjf(x) T (x)jj 2 jjxjjp=j2 2pj for every x 2 E
1. In 1991, Gajda [1] gave a
solution to this question for p > 1. For the case p = 1, Th. M. Rassias and Šemrl [11] showed that there exists a continuous real-valued function f : R ! R such that f can not be approximated with an additive map.
Gµavruta [2] generalized Rassias’s result: Let G be an abelian group and X a Banach space. Denote by ' : G G ! [0; 1) a function such that
~
'(x; y) =P1k=02 k'(2kx; 2ky) < 1
for all x; y 2 G. Suppose that f : G ! X is a mapping satisfying jjf(x + y) f (x) f (y)jj '(x; y)
for all x; y 2 G. Then there exists a unique additive mapping T : G ! X such that jjf(x) T (x)jj 1=2~'(x; x)
Received by the editors: October 10, 2013; Accepted: April 04, 2015.
2010 Mathematics Subject Classi…cation. Primary: 46S40; Secondary: 39B52, 39B82, 26E50, 46S50.
Key words and phrases. Fuzzy normed space; approximately *-homomorphism; stability.
c 2 0 1 5 A n ka ra U n ive rsity
for all x 2 G. Recently, Park [9] applied Gµavruta’s result to linear functional equations in Banach modules over a C*-algebra.
B. E. Johnson [4] also investigated almost algebra *-homomorphisms between Banach *-algebras.
Fuzzy notion introduced …rstly by Zadeh [13] that has been widely involved in di¤erent subjects of mathematics. Zadeh’s de…nition of a fuzzy set characterized by a function from a nonempty set X to [0; 1].
Later, in 1984 Katsaras [7] de…ned a fuzzy norm on a linear space to construct a fuzzy vector topological structure on the space. De…ning the class of approximately solutions of a given functional equation one can ask whether every mapping from this class can be somehow approximated by an exact solution of the considered equation in the fuzzy Banach *-algebra. To answer this question, we use here the de…nition of fuzzy normed spaces given in [7] to exhibit some reasonable notions of fuzzy approximately *-homomorphism in fuzzy normed algebras and we will prove that if A is a Banach *-algebra, then under some suitable conditions a fuzzy ap-proximately *-homomorphism f : A ! A can be approximated in a fuzzy sense by a *-homomorphism H : A ! A. This is applied to show that for a fuzzy approxi-mately map f : A ! A on a C*-algebra A, there exists a unique *-homomorphism H : A ! A such that f = H.
2. Preliminaries
In this section, we provide a collection of de…nitions and related results which are essential and used in the next discussions.
De…nition 2.1. Let X be a real linear space. A function N : X R ! [0; 1] is said to be a fuzzy norm on X if for all x; y 2 X and all t; s 2 R,
(N1) N (x; c) = 0 for c 0;
(N2) x = 0 if and only if N (x; c) = 1 for all c > 0; (N3) N (cx; t) = N (x;jcjt) if c 6= 0;
(N4) N (x + y; s + t) minfN(x; s); N(y; t)g;
(N5) N (x; :) is a non-decreasing function on R and limt!1N (x; t) = 1;
(N6) for x 6= 0, N(x; :) is (upper semi) continuous on R. The pair (X; N ) is called a fuzzy normed linear space. Example 2.2. Let (X; jj:jj) be a normed linear space. Then
N (x; t) = 8 < : 0; t 0; t jjxjj; 0 < t jjxjj; 1; t > jjxjj. is a fuzzy norm on X.
De…nition 2.3. Let (X; N ) be a fuzzy normed linear space and fxng be a
limn!1N (xn x; t) = 1 for all t > 0. In that case, x is called the limit of the
sequence fxng and we denote it by N limn!1xn= x.
De…nition 2.4. A sequence fxng in X is called Cauchy if for each " > 0 and each
t > 0 there exists n0such that for all n n0and all p > 0, we have N (xn+p xn; t) >
1 ".
It is known that every convergent sequence in a fuzzy normed space is Cauchy and if each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and furthermore the fuzzy normed space is called a complete fuzzy normed space.
Let X be an algebra and (X; N ) be complete fuzzy normed space. The pair (X; N ) is said to be a fuzzy Banach algebra if for every x; y 2 X and s; t 2 R we have N (xy; st) minfN(x; s); N(y; t)g.
De…nition 2.5. Let X be a linear space and ' : X X ! [0; 1). We say that ' is control function if we have
~
'(x; y) =P1n=02 n'(2nx; 2ny) < 1;
for all x; y 2 X.
We give the following results proved in [8].
Theorem 2.6. Let X be a linear space and (Y; N ) be a fuzzy Banach space. Suppose that ' : X X ! [0; 1) is a control function and f : X ! Y is a uniformly approximately additive function with respect to ' in the sense that
limt!1N (f (x + y) f (x) f (y); t'(x; y)) = 1
uniformly on X X. Then T (x) = N limn!1f (2 nx)
2n for all x 2 X exists and
de…nes an additive mapping T : X ! Y such that if for some > 0, > 0 N (f (x + y) f (x) f (y); '(x; y)) > ;
for all x; y 2 X, then
N (T (x) f (x); =2~'(x; x)) > ; for every x 2 X.
Corollary 2.7. Let X be a linear space and (Y; N ) be a fuzzy Banach space. Let ' : X X ! [0; 1) be a control function and f : X ! Y be a uniformly approximately additive function with respect to ' in the sense that
limt!1N (f (x + y) f (x) f (y); t'(x; y)) = 1
uniformly on X X. Then there is a unique additive mapping T : X ! Y such that
limt!1N (T (x) f (x); t~'(x; x)) = 1;
uniformly on X.
Theorem 2.8. Let X be a linear space and let (Z; N0) be a fuzzy normed space.
Let : X X ! Z be a function such that for some 0 < < 2, N0( (2x; 2y); t) N0( (x; y); t)
for all x; y 2 X and t > 0. Let (Y; N) be a fuzzy Banach space and let f : X ! Y be a mapping in the sense that
N (f (x + y) f (x) f (y); t) N0( (x; y); t)
for each t > 0 and x; y 2 X. Then there exists a unique additive mapping T : X ! Y such that
N (f (x) T (x); t) N0(2 (x;x) 2 ; t),
where x 2 X and t > 0.
3. Stability and super stability of fuzzy approximately *-homomorphisms on a fuzzy Banach *-algebra in uniform version We start our work with de…nition of fuzzy Banach *-algebra.
De…nition 3.1. A fuzzy Banach *-algebra A is a *-algebra A with a fuzzy complete N - norm N such that N (a; t) = N (a ; t) for all a 2 A.
Throughout this paper, let Asabe the set of self-adjoint elements of A and U (A)
the set of unitary elements in A.
Lemma 3.2. Let X be a fuzzy normed *-algebra and N limn!1xn = x. Then
N limn!1xn = x .
Proof. By De…nition 2.3 we have limt!1N (xn x; t) = 1. So limt!1N (xn
x ; t) = limt!1N ((xn x) ; t) = 1. It means that N limn!1xn = x .
Theorem 3.3. Let A be a fuzzy Banach *-algebra and let ' : A A ! [0; 1) be a control function and suppose that f : A ! A is a function such that
limt!1N (f ( x + y) f (x) f (y); t'(x; y)) = 1; (3.1)
uniformly on A A,
limt!1N (f (x ) f (x) ; t'(x; x)) = 1; (3.2)
uniformly on A, and
limt!1N (f (zw) f (z)f (w); t'(z; w)) = 1; (3.3)
uniformly on A A for all 2 T1 = f 2 C : j j = 1g, all z; w 2 A
sa, and all
x; y 2 A. Then there exists a unique algebra *-homomorphism H : A ! A such that
limt!1N (H(x) f (x); t~'(x; x)) = 1 (3.4)
uniformly on A.
Proof. Put = 1 2 T1. It follows from Theorem 2.6 and Corollary 2.7 that, there
exists a unique additive mapping H : A ! A such that the equality (3.4) holds. The additive mapping H : A ! A is given by H(x) = N limn!121nf (2nx) for
all x 2 A.
By the assumption we have,
for all 2 T1and all x 2 A. We have
N ( f (2nx) 2 f (2n 1x); t'(2n 1x; 2n 1x)) = N (f (2nx) 2f (2n 1x); j j 1t'(2n 1x; 2n 1x)) = N (f (2nx) 2f (2n 1x); t'(2n 1x; 2n 1x)); for all 2 T1and all x 2 A. On the other hand
N (f (2n x) f (2nx); t'(2n 1x; 2n 1x))
minfN(f(2n x) 2 f (2n 1x); t=2'(2n 1x; 2n 1x)); N (2 f (2n 1x) f (2nx); t=2'(2n 1x; 2n 1x))g; for all 2 T1and x 2 A. Thus
limt!1N (f (2n x) f (2nx); t'(2n 1x; 2n 1x)) = 1.
So
limt!1N (2 nf (2n x) 2 n f (2nx); 2 nt'(2n 1x; 2n 1x)) = 1.
Since limn!12 nt'(2n 1x; 2n 1x) = 0, there is some n0> 0 such that
2 nt'(2n 1x; 2n 1x) < t, for all n n0 and t > 0. Hence
N (2 nf (2n x) 2 n f (2nx); t) N (2 nf (2n x) 2 n f (2nx); 2 nt'(2n 1x; 2n 1x)).
Given " > 0 we can …nd some t0> 0 such that
N (2 nf (2n x) 2 n f (2nx); 2 nt'(2n 1x; 2n 1x)) 1 ",
for all x 2 A and all t t0. So N (2 nf (2n x) 2 n f (2nx); t) = 1 for all t > 0.
Hence by items (N5) and (N2) of de…nition 2.1 we have N limn!12 nf (2n x) = N limn!12 n f (2nx),
for all 2 T1and all x 2 A. Hence
H( x) = N limn!1f (2 n x)
2n = N limn!1 f (2
nx)
2n = H(x),
for all 2 T1and all x 2 A.
Now let 2 C ( 6= 0) and let M be an integer greater than 4j j. Then jMj < 1=4 < 1=3. By ([5], Theorem 1), there exist three elements 1; 2; 3 2 T1
such that 3M = 1+ 2+ 3. We have H(x) = H(3:1=3x) = 3H(1=3x) for all x 2 A. So H(1=3x) = 1=3H(x) for all x 2 A. Thus
H( x) = H(M33:Mx) = M H(1=3:3Mx) = M=3H( 1x + 2x + 3x)
= M=3(H( 1x) + H( 2x) + H( 3x)) = M=3( 1+ 2+ 3)H(x) = M33MH(x) =
H(x),
for all x 2 A. Hence
H( x + y) = H( x) + H( y) = H(x) + H(y),
for all ; 2 C ( ; 6= 0) and all x; y 2 A, and H(0x) = 0 = 0H(x) for all x 2 A. So the unique additive mapping H : A ! A is a C-linear mapping.
By using (3.2) we have
limt!1N (2 nf (2nx ) 2 nf (2nx) ; 2 nt'(x; x)) = 1.
Since limn!12 nt'(x; x) = 0, there is some n0> 0 such that 2 nt'(x; x) < t
for all n n0 and t > 0. Hence
Given " > 0 we can …nd some t0> 0 such that
N (2 nf (2nx ) 2 nf (2nx) ; 2 nt'(x; x)) 1 ",
for all x 2 A and all t t0. So N (2 nf (2nx ) 2 nf (2nx) ; t) = 1 for all t > 0.
Hence by items (N5) and (N2) of De…nition 2.1 we have
N limn!1(2 nf (2nx )) = N limn!12 nf (2nx) : (3.5)
By (3.5) and Lemma 3.2, we get H(x ) = N limn!1f (2 nx ) 2n = N limn!1(f (2 nx)) 2n = (N limn!1f (2 nx) 2n ) = H(x) , for all x 2 A.
Now it follows from (3.3) that
limt!1N (4 nf (2 nz2 nw) 4 nf (2 nz)f (2 nw); 4 nt'(2 nz; 2 nw)) = 1.
Since limn!14 nt'(2 nz; 2 nw) = 0, there is some n0> 0 such that
4 nt'(2 nz; 2 nw) < t,
for all n n0 and t > 0. Hence
N (4 nf (2 nz2 nw) 4 nf (2 nz)f (2 nw); t)
N (4 nf (2 nz2 nw) 4 nf (2 nz)f (2 nw); 4 nt'(2 nz; 2 nw)): Given " > 0 we can …nd some t0> 0 such that
N (4 nf (2 nz2 nw) 4 nf (2 nz)f (2 nw); 4 nt'(2 nz; 2 nw)) 1 ", for all x 2 A and all t t0. So N (4 nf (2 nz2 nw) 4 nf (2 nz)f (2 nw); t) = 1
for all t > 0. Hence by items (N5) and (N2) of de…nition 2.1 we have N limn!14 nf (2 nz2 nw) = N limn!14 nf (2 nz)f (2 nw),
for all z; w 2 Asa; but
P1 j=04 j'(2jz; 2jw) P1 j=02 j'(2jz; 2jw) for all z; w 2 Asa. So H(zw) = N limn!1f (4 nzw) 4n = N limn!1f (2 nz)f (2nw) 2n2n = N limn!1f (2 nz) 2n :N limn!1f (2 nw) 2n = H(z)H(w), for all z; w 2 Asa. For elements x; y 2 A, x = x+x2 +i x x 2i and y = y+y 2 +i y y 2i , where x1= x+x 2 ,
x2= x x2i , y1= y+y2 and y2= y2iy are self-adjoint. Since H is C-linear,
H(xy) = H(x1y1 x2y2+ i(x1y2+ x2y1)) = H(x1y1) H(x2y2) + iH(x1y2) +
iH(x2y1)
= H(x1)H(y1) H(x2)H(y2) + iH(x1)H(y2) + iH(x2)H(y1)
= (H(x1) + iH(x2))(H(y1) + iH(y2))
= H(x1+ ix2)H(y1+ iy2) = H(x)H(y),
for all x; y 2 A. Hence the additive mapping H is an algebra *-homomorphism satisfying the inequality (3.4), as desired.
The proof of the uniqueness property of H is similar to the proof of Corollary 2.7.
Corollary 3.4. Let A be a fuzzy Banach *-algebra, 0 and q > 0 ; q 6= 1. Suppose that f : A ! A is a function such that
limt!1N (f ( x + y) f (x) f (y); t (jjxjjq+ jjyjjq)) = 1; (3.6)
uniformly on A A,
limt!1N (f (x ) f (x) ; 2t jjxjjq) = 1; (3.7)
uniformly on A, and
limt!1N (f (zw) f (z)f (w); t (jjzjjq+ jjwjjq)) = 1; (3.8)
uniformly on A A for all 2 T1 = f 2 C : j j = 1g, all z; w 2 A
sa, and all
x; y 2 A. Then there exists a unique algebra *-homomorphism H : A ! A such that
limt!1N (H(x) f (x); 2 tjjxjj q
j1 2q 1j) = 1; (3.9)
uniformly on A.
Proof. Considering the control function '(x; y) = (jjxjjq+ jjyjjq) for some > 0, we obtain this corollary.
In the following example we will show that Corollary 3.4 does not necessarily hold for q = 1.
Example 3.5. Let X be a Banach *-algebra, x0 2 X and ; are real numbers
such that j j 1 (jjxjj + jjyjj) and j j jjxjj + jjyjj for every x; y 2 X. Put f (x) = x + x0jjxjj; (x 2 X).
Moreover for each fuzzy norm N on X, we have N (f (x + y) f (x) f (y); t(jjxjj + jjyjj)) = N ( x0(jjx + yjj jjxjj jjyjj); t(jjxjj + jjyjj))
= N ( x0;jjx+yjj jxjj jjyjjt(jjxjj+jjyjj) ) N ( x0; t) (x; y 2 X; t 2 R).
Therefore by the item (N5) of the De…nition 2.1, we get limt!1N (f (x + y) f (x) f (y); t(jjxjj + jjyjj)) = 1,
Also
N (f (xy) f (x)f (y); t(jjxjj + jjyjj))
= N ( xy + x0jjxyjj ( x + x0jjxjj)( y + x0jjyjj); t(jjxjj + jjyjj))
= N ( xy + x0jjxyjj 2xy xx0jjyjj x0yjjxjj 2x20jjxjjjjyjj; t(jjxjj + jjyjj))
minfN((1 ) xy;t(jjxjj + jjyjj)
5 ); N (jjxyjj x0; t(jjxjj + jjyjj) 5 ); N ( 2x20jjxjjjjyjj;t(jjxjj + jjyjj) 5 ); N ( xx0jjyjj; t(jjxjj + jjyjj) 5 ); N ( x0yjjxjj;t(jjxjj + jjyjj) 5 )g where x 2 X and t 2 R.
Taking into account the following inequalities
N ((1 ) xy;t(jjxjj + jjyjj) 5 ) = N ( xy; t(jjxjj + jjyjj) 5j1 j ) N ( xy; t=5); (3.10) N (jjxyjj x0;t(jjxjj + jjyjj) 5 ) = N (jjxyjjx0; t(jjxjj + jjyjj) 5j j ) N (jjxyjjx0; t=5); (3.11) N ( 2x20jjxjjjjyjj;t(jjxjj + jjyjj) 5 ) = N ( jjxjjjjyjjx 2 0; t 5j j) N ( jjxjjjjyjjx 2 0; t 5); (3.12) N ( xx0jjyjj;t(jjxjj + jjyjj) 5 ) = N ( xx0jjyjj; t(jjxjj + jjyjj) 5j j ) N ( xx0jjyjj; t=5); (3.13) N ( x0yjjxjj;t(jjxjj + jjyjj) 5 ) = N ( x0yjjxjj; t(jjxjj + jjyjj) 5j j ) N ( x0yjjxjj; t=5); (3.14) it can be easily seen that limt!1N (f (xy) f (x)f (y); t(jjxjj + jjyjj)) = 1 uniformly
on X X. Also we have N (f (x ) f (x) ; 2tjjxjj) = N ( x x + x0jjx jj x0jjxjj; 2tjjxjj) minfN( x0;2tjjxjj jjx jj); N ( x0; 2tjjxjj jjxjj )g:
So limt!1N (f (x ) f (x) ; 2tjjxjj) = 1 uniformly on X and therefore the conditions
of Corollary 3.4 are ful…lled.
Now we suppose that there exists a unique *-homomorphism H satisfying the conditions of Corollary 3.4. By the equation
limt!1N (f (x + y) f (x) f (y); t(jjxjj + jjyjj)) = 1; (3.15)
for given " > 0, we can …nd some t0> 0 such that
N (f (x + y) f (x) f (y); t(jjxjj + jjyjj)) 1 ",
for all x; y 2 X and all t t0. By using the simple induction on n, we shall show
that
N (f (2nx) 2nf (x); tn2njjxjj) 1 ": (3.16) Putting y = x in (3.15), we get (3.16) for n = 1. Let (3.16) holds for some positive integer n. Then
N (f (2n+1x) 2n+1f (x); t(n + 1)2n+1jjxjj) minfN(f(2n+1x) 2f (2nx); t(jj2nxjj + jj2nxjj)); N (2f (2nx) 2n+1f (x); 2tn(jj2n 1xjj + jj2n 1xjj)) 1 ":
This completes the induction argument. We observe that limn!1N (H(x) f (x); ntjjxjj) 1 ".
Hence
limn!1N (H(x) f (x); ntjjxjj) = 1: (3.17)
One may regard N (x; t) as the truth value of the statement ’the norm of x is less than or equal to the real number t. So (3.17) is a contradiction with the non-fuzzy sense. This means that there is no such the H.
Theorem 3.6. Let A be a C*-algebra and let f : A ! A be a bijective mapping satisfying f (xy) = f (x)f (y) and f (0) = 0 for which there exists function ' : A A ! [0:1) satisfying (3.1) and (3.3) such that
limt!1N (f (u ) f (u) ; t'(u; u)) = 1; (3.18)
for all u 2 U (A). Assume that N limn!1f (2 ne)
2n is invertible, where e is the
identity of A. Then the bijective mapping f is a bijective *-homomorphism. Proof. By the same reasoning as in the proof of Theorem 3.3 there exists a unique C-linear mapping H : A ! A such that
limt!1N (H(x) f (x); t~'(x; x)) = 1; (3.19)
for all x 2 A. The C-linear mapping H : A ! A is given by H(x) = N limn!1f (2
n x)
for all x 2 A.
By using (3.18) we have
limt!1N (2 nf (2nu ) 2 nf (2nu) ; 2 nt'(u; u)) = 1.
Since limn!12 nt'(u; u) = 0, there is some n0> 0 such that 2 nt'(u; u) < t
for all n n0 and t > 0. Hence
N (2 nf (2nu ) 2 nf (2nu) ; t) N (2 nf (2nu ) 2 nf (2nu) ; 2 nt'(u; u)).
Given " > 0 we can …nd some t0> 0 such that
N (2 nf (2nu ) 2 nf (2nu) ; 2 nt'(u; u)) 1 ",
for all x 2 A and all t t0. So N (2 nf (2nu ) 2 nf (2nu) ; t) = 1 for all t > 0.
Hence by items (N5) and (N2) of de…nition 2.1 we have
N limn!1(2 nf (2nu ) = N limn!12 nf (2nu) : (3.20)
By (3.20) and Lemma 3.2, we get H(u ) = N limn!1f (2 nu ) 2n = N limn!1(f (2 nu)) 2n = (N limn!1f (2 nu) 2n ) = H(u) ,
for all u 2 U (A).
Since H is C-linear and each x 2 A is a …nite linear combination of unitary elements [6], H(x ) = H(Pmj=1 juj) = Pm j=1 jH(uj) = Pm j=1 jH(uj) = ( Pm j=1 jH(uj)) = H(Pmj=1 juj) = H(x) , for all x 2 A.
Since f (xy) = f (x)f (y) for all x; y 2 A, H(xy) = N limn!1
f (2nxy)
2n = N limn!1
f (2nx)f (y)
2n = H(x)f (y) (3.21)
for all x; y 2 A. By the additivity of H and (3.21), 2nH(xy) = H(2nxy) = H(x(2ny)) = H(x)f (2ny), for all x; y 2 A. Hence
H(xy) = H(x)f (2
ny)
2n = H(x)
f (2ny)
2n ; (3.22)
for all x; y 2 A. Taking the N-limit in (3.22) as n ! 1, we obtain H(xy) = H(x)H(y),
for all x; y 2 A. By (3.21) we have,
H(x) = H(ex) = H(e)f (x); (3.23)
for all x 2 A. Since H(e) = N limn!12 n
e
2n is invertible and the mapping f is
bijective, the C-linear mapping H is a bijective *-homomorphism. Now we have,
H(e)H(x) = H(ex) = H(x) = H(e)f (x),
for all x 2 A. Since H(e) is invertible, H(x) = f(x) for all x 2 A. Hence the bijective mapping f is a bijective *-homomorphism.
4. Non-uniform type of Stability and super stability of fuzzy approximately *-homomorphisms
We are in a position to give non-uniform type of Theorems 3.3 and 3.6.
Theorem 4.1. Let (B; N0) be a fuzzy normed algebra, A a fuzzy Banach *-algebra
and let ' : A A ! B be a function such that for some 0 < < 2, N0('(2x; 2y); t) N0('(x; y); t)
for all x; y 2 A and t > 0. Let f : A ! A be a function such that N (f ( x + y) f (x) f (y); t) N0('(x; y); t),
for all x; y 2 A,
N (f (x ) f (x) ; t) N0('(x; x); t); (4.1) for all x 2 A and
N (f (zw) f (z)f (w); t) N0('(z; w); t); (4.2) for all t > 0, all 2 T1= f 2 C : j j = 1g, and all z; w 2 A
sa. Then there exists
a unique algebra *-homomorphism H : A ! A such that N (H(x) f (x); t) N0(2'(x;x)2 ; t)
for all x 2 A and all t > 0.
Proof. Theorem 2.8 shows that there exists an additive function H : A ! A such that
N (f (x) T (x); t) N0(2'(x;x) 2 ; t),
where x 2 A and t > 0.
Put = 1 2 T1. The additive mapping H : A ! A is given by H(x) =
N limn!121nf (2nx) for all x 2 A.
By assumption for each 2 T1,
N (f (2n x) 2 f (2n 1x); t) N0n 1x; 2n 1x); t), for all x 2 A. We have
N ( f (2nx) 2 f (2n 1x); t) = N (f (2nx) 2f (2n 1x); j j 1t) = N (f (2nx)
2f (2n 1x); t) N0n 1x; 2n 1x); t),
for all 2 T1and all x 2 A. So
N (f (2n x) f (2nx); t) minfN(f(2n x) 2 f (2n 1x); t=2); (4.3) N (2 f (2n 1x) f (2nx); t=2)g N0n 1x; 2n 1x); t=2);
for all 2 T1and all x 2 A. Taking n to in…nity in (4.3) and using the items (N2)
and (N5) of De…nition 2.1, we see that
N limn!12 nf (2n x) = N limn!12 n f (2nx),
for all 2 T1and all x 2 A.
Now by using the similar proof of the Theorem 3.3 the unique additive mapping H : A ! A is a C-linear mapping.
N (2 nf (2nx ) 2 nf (2nx) ; t) N0nx; 2nx); 2nt); (4.4) for all x 2 A. Taking n to in…nity in (4.4) and using the items (N2) and (N5) of De…nition 2.1, we see that
N limn!12 nf (2nx ) = N limn!12 nf (2nx) .
Again by using the similar proof of the Theorem 3.3 we have H(x ) = H(x) . Now it follows from (4.2) that
N (4 nf (2 nz2 nw) 4 nf (2 nz)f (2 nw); t) N0nz; 2nw); 4nt): (4.5) for all z; w 2 Asa. Taking n to in…nity in (4.5) and using the items (N2) and (N5)
of De…nition 2.1, we see that
N limn!14 nf (2 nz2 nw) = N limn!14 nf (2 nz)f (2 nw),
for all z; w 2 Asa. By the proof of Theorem 3.3, H is a *-homomorphism as desired.
To prove the uniqueness property of H, assume that H is another *-homomorphism satisfying N (f (x) H (x); t) N0(2'(x;x)
2 ; t). Since both H and H are additive
we deduce that
N (H(a) H (a); t) minfN(H(a) n 1f (na); t=2); N (n 1f (na) H (a); t=2)g
N0(2'(na;na)
2 ; nt=2)
for all a 2 A and all t > 0. Letting n tend to in…nity we get that H(a) = H (a) for all a 2 A.
Theorem 4.2. Let A be a C*-algebra, (B; N0) a fuzzy normed algebra and let ' : A A ! B be a function such that for some 0 < < 2,
N0('(2x; 2y); t) N0('(x; y); t)
for all x; y 2 A and t > 0. Let f : A ! A be a bijective mapping satisfying f (xy) = f (x)f (y) and f (0) = 0 such that
N (f ( x + y) f (x) f (y); t) N0('(x; y); t),
and
limt!1N (f (u ) f (u) ; t'(u; u)) = 1;
for all x; y 2 A and u 2 U (A). Assume that N limn!1f (2 n
e)
2n is invertible, where
e is the identity of A. Then the bijective mapping f is a bijective *-homomorphism. Proof. As same as the proof of the Theorems 3.6 and 4.1, we can prove this Theo-rem.
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Address : Department of Mathematics, Facualty of Mathematical Sciences, University of Mo-haghegh Ardabili, 56199-11367, Ardabil, Iran.
E-mail : nasrineghbali@gmail.com,eghbali@uma.ac.ir
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