# Nearly Jordan∗-Homomorphisms between Unital C∗-Algebras

Authors: A. Ebadian; S. Kaboli Gharetapeh; M. Eshaghi Gordji
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

LetA, B be two unital C∗-algebras. We prove that every almost unital almost
linear mapping h : A→B which satisfies h(3nuy+3nyu)=h(3nu)h(y)+h(y)h(3nu) for all u∈U(A), all y∈A, and all n=0,1,2,…, is a Jordan homomorphism. Also, for a unital
C∗-algebra A of real rank zero, every almost unital almost linear continuous mapping h:A→B is a Jordan homomorphism when h(3nuy+3nyu)=h(3nu)h(y)+h(y)h(3nu)
holds
for all u∈I1
(Asa), all y∈A, and all n=0,1,2,…. Furthermore, we investigate the Hyers-
Ulam-Aoki-Rassias stability of Jordan ∗-homomorphisms between unital C∗-algebras by using the fixed points methods.


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## Body

## 1. Introduction

The stability of functional equations was first introduced by Ulam [1] in 1940. More precisely, he proposed the following problem: given a group G1, a metric group (G2,d) and a positive number ϵ, does there exist a δ>0 such that if a function f:G1→G2 satisfies the inequality d(f(xy),f(x)f(y))<δ for all x,y∈G1, then there exists a homomorphism T:G1→G2 such that d(f(x),T(x))<ϵ for all x∈G1?. As mentioned above, when this problem has a solution, we say that the homomorphisms from G1 to G2 are stable. In 1941, Hyers [2] gave a partial solution of Ulam’s problem for the case of approximate additive mappings under the assumption that G1 and G2 are Banach spaces. Hyers theorem was generalized by Aoki [3] for additive mappings and by Th. M. Rassias [4] for linear mappings by considering an unbounded Cauchy difference. This phenomenon of stability is called the Hyers-Ulam-Aoki-Rassias stability.J. M. Rassias [5–7] established the stability of linear and nonlinear mappings with new control functions.During the last decades, several stability problems of functional equations have been investigated by many mathematicians. A large list of references concerning the stability of functional equations can be found in [8–10].Bourgin is the first mathematician dealing with the stability of ring homomorphisms. The topic of approximate ring homomorphisms was studied by a number of mathematicians, see [11–19] and references therein.Jun and Lee [20] proved the following: Let X and Y be Banach spaces. Denote by ϕ:X-{0}×Y-{0}→[0,∞) a function such that ϕ̃(x,y)=∑n=0∞3-nϕ(3nx,3ny)<∞ for all x,y∈X-{0}. Suppose that f:X→Y is a mapping satisfying (1.1)‖2f(x+y2)-f(x)-f(y)‖≤ϕ(x,y)
for all x,y∈X-{0}. Then there exists a unique additive mapping T:X→Y such that (1.2)‖f(x)-f(0)-T(x)‖≤13(ϕ̃(x,-x)+ϕ̃(-x,3x))
for all x∈X-{0}.Recently, C. Park and W. Park [21] applied the Jun and Lee's result to the Jensen's equation in Banach modules over a C*-algebra. Johnson (Theorem 7.2 of [22]) also investigated almost algebra *-homomorphisms between Banach *-algebras: Suppose that U and B are Banach *-algebras which satisfy the conditions of (Theorem 3.1 of [22]). Then for each positive ϵ and K, there is a positive δ such that if T∈L(U,B) with ∥T∥<K,∥T∨∥<δ and ∥T(x*)*-T(x)∥<δ∥x∥(x∈U), then there is a *-homomorphism T′:U→B with ∥T-T′∥<ϵ. Here L(U,B) is the space of bounded linear maps from U into B, and T∨(x,y)=T(xy)-T(x)T(y)(x,y∈U). See [22] for details. Throughout this paper, let A be a unital C*-algebra with unit e, and B a unital C*-algebra. Let U(A) be the set of unitary elements in A, Asa:={x∈Ax=x*}, and I1(Asa)={v∈Asa∣∥v∥=1,v∈Inv(A)}. In this paper, we prove that every almost unital almost linear mapping h:A→B is a Jordan homomorphism when h(3nuy+3nyu)=h(3nu)h(y)+h(y)h(3nu) holds for all u∈U(A), all y∈A, and all n=0,1,2,…, and that for a unital C*-algebra A of real rank zero (see [23]), every almost unital almost linear continuous mapping h:A→B is a Jordan homomorphism when h(3nuy+3nyu)=h(3nu)h(y)+h(y)h(3nu) holds for all u∈I1(Asa), all y∈A, and all n=0,1,2…. Furthermore, we investigate the Hyers-Ulam-Aoki-Rassias stability of Jordan *-homomorphisms between unital C*-algebras by using the fixed point methods.Note that a unitalC*-algebra is of real rank zero, if the set of invertible self-adjoint elements is dense in the set of self-adjoint elements (see [23]). We denote the algebric center of algebra A by Z(A).


## 2. Jordan*-Homomorphisms on Unital C*-Algebras

By a following similar way as in [24], we obtain the next theorem.Theorem 2.1.
Letf:A→B be a mapping such that f(0)=0 and that
(2.1)f(3nuy+3nyu)=f(3nu)f(y)+f(y)f(3nu)
for all u∈U(A), all y∈A, and all n=0,1,2,…. If there exists a function ϕ:(A-{0})2×A→[0,∞) such that ϕ̃(x,y,z)=∑n=0∞3-nϕ(3nx,3ny,3nz)<∞ for all x,y∈A-{0} and all z∈A and that
(2.2)‖2f(μx+μy2)-μf(x)-μf(y)+f(u*)-f(u)*‖≤ϕ(x,y,u),
for all μ∈𝕋 and all x,y∈A,u∈(U(A)∪{0}). Iflimn(f(3ne)/3n)∈U(B)∩Z(B), then the mapping f:A→B is a Jordan *-homomorphism.Proof.
Putu=0, μ=1 in (2.2), it follows from of [20, Theorem 1] that there exists a unique additive mapping T:A→B such that
(2.3)∥f(x)-T(x)∥≤13(ϕ̃(x,-x,0)+ϕ̃(-x,3x,0))
for all x∈A-{0}. This mapping is given by
(2.4)T(x)=lim⁡nf(3nx)3n
for all x∈A. By the same reasoning as the proof of [24, Theorem 1], T is ℂ-linear and *-preserving. It follows from (2.1) that
(2.5)T(uy+yu)=lim⁡nf(3nuy+3nyu)3n=lim⁡n[f(3nu)3nf(y)+f(y)f(3nu)3n]=T(u)f(y)+f(y)T(u)
for all u∈U(A), all y∈A. Since T is additive, then by (2.5), we have
(2.6)3nT(uy+yu)=T(u(3ny)+(3ny)u)=T(u)f(3ny)+f(3ny)T(u)
for all u∈U(A) and all y∈A. Hence,
(2.7)T(uy+yu)=lim⁡n[T(u)f(3ny)3n+f(3ny)3nT(u)]=T(u)T(y)+T(y)T(u)
for all u∈U(A) and all y∈A. By the assumption, we have
(2.8)T(e)=lim⁡nf(3ne)3n∈U(B)⋂Z(B)
hence, it follows by (2.5) and (2.7) that
(2.9)2T(e)T(y)=T(e)T(y)+T(y)T(e)=T(ye+ey)=T(e)f(y)+f(y)T(e)=2T(e)f(y)
for all y∈A. Since T(e) is invertible, then T(y)=f(y) for all y∈A. We have to show that f is Jordan homomorphism. To this end, let x∈A. By Theorem 4.1.7 of [25], x is a finite linear combination of unitary elements, that is, x=∑j=1ncjuj(cj∈ℂ,uj∈U(A)), and then it follows from (2.7) that
(2.10)f(xy+yx)=T(xy+yx)=T(∑j=1ncjujy+∑j=1ncjyuj)=∑j=1ncjT(ujy+yuj)=∑j=1ncj(T(ujy)+T(yuj))=∑j=1ncj(T(uj)T(y)+T(y)T(uj))=T(∑j=1ncjuj)T(y)+T(y)T(∑j=1ncjuj)=T(x)T(y)+T(y)T(x)=f(x)f(y)+f(y)f(x)
for all y∈A. And this completes the proof of theorem.Corollary 2.2.
Letp∈(0,1), θ∈[0,∞) be real numbers. Let f:A→B be a mapping such that f(0)=0 and that
(2.11)f(3nuy+3nyu)=f(3nu)f(y)+f(y)f(3nu)
for all u∈U(A), all y∈A, and all n=0,1,2,…. Suppose that
(2.12)‖2f(μx+μy2)-μf(x)-μf(y)+f(z*)-f(z)*‖≤θ(‖x‖p+‖y‖p+‖z‖p)
for all μ∈𝕋 and all x,y,z∈A. If limn(f(3ne)/3n)∈U(B)∩Z(B), then the mapping f:A→B is a Jordan *-homomorphism.Proof.
Settingϕ(x,y,z):=θ(∥x∥p+∥y∥p+∥z∥p) for all x,y,z∈A. Then by Theorem 2.1, we get the desired result.Theorem 2.3.
LetA be a C*-algebra of real rank zero. Let f:A→B be a continuous mapping such that f(0)=0 and that
(2.13)f(3nuy+3nyu)=f(3nu)f(y)+f(y)f(3nu)
for all u∈I1(Asa), all y∈A, and all n=0,1,2,…. Suppose that there exists a function ϕ:(A-{0})2×A→[0,∞) satisfying (2.2) and ϕ̃(x,y,z)<∞ for all x,y∈A-{0} and all z∈A. If limn(f(3ne)/3n)∈U(B)∩Z(B), then the mapping f:A→B is a Jordan *-homomorphism.Proof.
By the same reasoning as the proof of Theorem2.1, there exists a unique involutive ℂ-linear mapping T:A→B satisfying (2.3). It follows from (2.13) that
(2.14)T(uy+yu)=lim⁡nf(3nuy+3nyu)3n=lim⁡n[f(3nu)3nf(y)+f(y)f(3nu)3n]=T(u)f(y)+f(y)T(u)
for all u∈I1(Asa), and all y∈A. By additivity of T and (2.14), we obtain that
(2.15)3nT(uy+yu)=T(u(3ny)+(3ny)u)=T(u)f(3ny)+f(3ny)T(u)
for all u∈I1(Asa) and all y∈A. Hence,
(2.16)T(uy+yu)=lim⁡n[T(u)f(3ny)3n+f(3ny)3nT(u)]=T(u)T(y)+T(y)T(u)
for all u∈I1(Asa) and all y∈A. By the assumption, we have
(2.17)T(e)=lim⁡nf(3ne)3n∈U(B)∩Z(B).
Similar to the proof of Theorem 2.1, it follows from (2.14) and (2.16) that T=f on A. So T is continuous. On the other hand, A is real rank zero. One can easily show that I1(Asa) is dense in {x∈Asa:∥x∥=1}. Let v∈{x∈Asa:∥x∥=1}. Then there exists a sequence {zn} in I1(Asa) such that limnzn=v. Since T is continuous, it follows from (2.16) that
(2.18)T(vy+yv)=T(lim⁡n(zny+yzn))=lim⁡nT(zny+yzn)=lim⁡nT(zn)T(y)+lim⁡nT(y)T(zn)=T(lim⁡nzn)T(y)+T(y)T(lim⁡nzn)=T(v)T(y)+T(y)T(v)
for all y∈A. Now, let x∈A. Then we have x=x1+ix2, where x1:=(x+x*)/2 and x2=(x-x*)/2i are self adjoint.
First considerx1=0,x2≠0. Since T is ℂ-linear, it follows from (2.18) that
(2.19)f(xy+yx)=T(xy+yx)=T(ix2y+y(ix2))=T(i‖x2‖x2‖x2‖y+y(i‖x2‖x2‖x2‖))=T(i‖x2‖x2‖x2‖)T(y)+T(y)T(i‖x2‖x2‖x2‖)=T(ix2)T(y)+T(y)T(ix2)=T(x)T(y)+T(y)T(x)=f(x)f(y)+f(y)f(x)
for all y∈A.
Ifx2=0, x1≠0, then by (2.18), we have
(2.20)f(xy+yx)=T(xy+yx)=T(x1y+y(x1))=T(‖x1‖x1‖x1‖y+y(‖x1‖x1‖x1‖))=T(‖x1‖x1‖x1‖)T(y)+T(y)T(‖x1‖x1‖x1‖)=T(x1)T(y)+T(y)T(x1)=T(x)T(y)+T(y)T(x)=f(x)f(y)+f(y)f(x)
for all y∈A.
Finally, consider the case thatx1≠0, x2≠0. Then it follows from (2.18) that
(2.21)f(xy+yx)=T(xy+yx)=T(x1y+ix2y+yx1+y(ix2))=T(‖x1‖x1‖x1‖y+y(‖x1‖x1‖x1‖))+T(i‖x2‖x2‖x2‖y+y(i‖x2‖x2‖x1‖))=‖x1‖[T(x1‖x1‖)T(y)+T(y)T(x1‖x1‖)]+i‖x2‖[T(x2‖x2‖)T(y)+T(y)T(x2‖x1‖)]=[T(x1)+T(ix2)]T(y)+T(y)[T(x1)+T(ix2)]=T(x)T(y)+T(y)T(x)=f(x)f(y)+f(y)f(x)
for all y∈A. Hence, f(xy+yx)=f(x)f(y)+f(y)f(x) for all x,y∈A, and f is Jordan *-homomorphism.Corollary 2.4.
LetA be a C*-algebra of rank zero. Let p∈(0,1), θ∈[0,∞) be real numbers. Let f:A→B be a mapping such that f(0)=0 and that
(2.22)f(3nuy+3nyu)=f(3nu)f(y)+f(y)f(3nu)
for all u∈I1(Asa), all y∈A, and all n=0,1,2,…. Suppose that
(2.23)‖2f(μx+μy2)-μf(x)-μf(y)+f(z*)-f(z)*‖≤θ(‖x‖p+‖y‖p+‖z‖p)
for all μ∈𝕋 and all x,y,z∈A. If limnf(3ne)/3n∈U(B)∩Z(B), then the mapping f:A→B is a Jordan *-homomorphism.Proof.
Settingϕ(x,y,z):=θ(∥x∥p+∥y∥p+∥z∥p) for all x,y,z∈A. Then by Theorem 2.3, we get the desired result.


## 3. Stability of Jordan*-Homomorphisms: A Fixed Point Approach

We investigate the generalized Hyers-Ulam-Aoki-Rassias stability of Jordan*-homomorphisms on unital C*-algebras by using the alternative fixed point.Recently, Cădariu and Radu applied the fixed point method to the investigation of the functional equations. (See also [18, 26–43]).Theorem 3.1.
Letf:A→B be a mapping with f(0)=0 for which there exists a function ϕ:A5→[0,∞) satisfying
(3.1)‖f(μx+μy+μz3)+f(μx-2μy+μz3)+f(μx+μy-2μz3)-μf(x)+f(uv+uv)-f(v)f(u)-f(u)f(v)+f(w*)-f(w)*f(μx+μy+μz3)‖≤ϕ(x,y,z,u,v,w),
for all μ∈𝕋, and all x,y,z,u,v∈A,w∈U(A)∪{0}. If there exists an L<1 such that ϕ(x,y,z,u,v,w)≤3Lϕ(x/3,y/3,z/3,u/3,v/3,w/3) for all x,y,z,u,v,w∈A, then there exists a unique Jordan *-homomorphism h:A→B such that
(3.2)‖f(x)-h(x)‖≤L1-Lϕ(x,0,0,0,0,0)
for all x∈A.Proof.
It follows fromϕ(x,y,z,u,v,w)≤3Lϕ(x/3,y/3,z/3,u/3,v/3,w/3) that
(3.3)lim⁡j⁡3-jϕ(3jx,3jy,3jz,3ju,3jv,3jw)=0
for all x,y,z,u,v,w∈A.
Puty=z=w=u=0 and μ=1 in (3.1) to obtain
(3.4)‖3f(x3)-f(x)‖≤ϕ(x,0,0,0,0,0)
for all x∈A. Hence,
(3.5)‖13f(3x)-f(x)‖≤13ϕ(3x,0,0,0,0,0)≤Lϕ(x,0,0,0,0,0)
for all x∈A.
Consider the setX:={g∣g:A→B} and introduce the generalized metric on X:
(3.6)d(h,g):=inf⁡{C∈R+:‖g(x)-h(x)‖≤Cϕ(x,0,0,0,0,0)∀x∈A}.
It is easy to show that (X,d) is complete. Now we define the linear mapping J:X→X by
(3.7)J(h)(x)=13h(3x)
for all x∈A. By Theorem 3.1 of [44],
(3.8)d(J(g),J(h))≤Ld(g,h)
for all g,h∈X.
It follows from (3.5) that
(3.9)d(f,J(f))≤L.
Now, from the fixed point alternative [45], J has a unique fixed point in the set X1:={h∈X:d(f,h)<∞}. Let h be the fixed point of J. h is the unique mapping with
(3.10)h(3x)=3h(x)
for all x∈A satisfying there exists C∈(0,∞) such that
(3.11)‖h(x)-f(x)‖≤Cϕ(x,0,0,0,0,0)
for all x∈A. On the other hand, we have limnd(Jn(f),h)=0. It follows that
(3.12)lim⁡n⁡13nf(3nx)=h(x)
for all x∈A. It follows from d(f,h)≤(1/(1-L))d(f,J(f)), that
(3.13)d(f,h)≤L1-L.
This implies the inequality (3.2). It follows from (3.1), (3.3), and (3.12) that
(3.14)‖h(x+y+z3)+h(x-2y+z3)+h(x+y-2z3)-h(x)‖=lim⁡n⁡13n‖f(3n-1(x+y+z))+f(3n-1(x-2y+z))+f(3n-1(x+y-2z))-f(3nx)‖≤lim⁡n13nϕ(3nx,3ny,,3nz,0,0,0)=0
for all x,y,z∈A. So
(3.15)h(x+y+z3)+h(x-2y+z3)+h(x+y-2z3)=h(x)
for all x,y,z∈A. Put w=(x+y+z)/3, t=(x-2y+z)/3 and s=(x+y-2z)/3 in above equation, we get h(w+t+s)=h(w)+h(t)+h(s) for all w,t,s∈A. Hence, h is Cauchy additive. By putting y=z=x, v=w=0 in (3.1), we have
(3.16)‖f(μx)-μf(x)‖≤ϕ(x,x,,x,0,0,0)
for all μ∈𝕋 and all x∈A. It follows that
(3.17)‖h(μx)-μh(x)‖=lim⁡m⁡13m‖f(μ3mx)-μf(3mx)‖≤lim⁡m⁡13mϕ(3mx,3mx,3mx,0,0,0)=0
for all μ∈𝕋, and all x∈A. One can show that the mapping h:A→B is ℂ-linear. By putting x=y=z=u=v=0 in (3.1), it follows that
(3.18)‖h(w*)-(h(w))*‖=lim⁡m⁡‖13mf((3mw)*)-13m(f(3mw))*‖≤lim⁡m⁡13mϕ(0,0,0,0,0,3mw)=0
for all w∈U(A). By the same reasoning as the proof of Theorem 2.1, we can show that h:A→B is *-preserving.
Sinceh is ℂ-linear, by putting x=y=z=w=0 in (3.1), it follows that
(3.19)‖h(uv+vu)-h(u)h(v)-h(v)h(u)‖=lim⁡m⁡‖19mf(9m(uv+vu))-19m[f(3mu)f(3mv)+f(3mv)f(3mu)]‖≤lim⁡m⁡19mϕ(0,0,0,3mu,3mv,0)≤lim⁡m⁡13mϕ(0,0,0,3mu,3mv,0)=0
for all u,v∈A. Thus h:A→B is Jordan *-homomorphism satisfying (3.2), as desired.We prove the following Hyers-Ulam-Aoki-Rassias stability problem for Jordan*-homomorphisms on unital C*-algebras.Corollary 3.2.
Letp∈(0,1), θ∈[0,∞) be real numbers. Suppose f:A→A satisfies
(3.20)‖f(μx+μy+μz3)+f(μx-2μy+μz3)+f(μx+μy-2μz3)-μf(x)+f(uv+uv)-f(v)f(u)-f(u)f(v)+f(w*)-f(w)*(μx+μy+μz3)‖≤θ(‖x‖p+‖y‖p+‖z‖p+‖u‖p+‖v‖p+‖w‖p),
for all μ∈𝕋 and all x,y,z,u,v∈A,w∈U(A)∪{0}. Then there exists a unique Jordan *-homomorphism h:A→B such that such that
(3.21)‖f(x)-h(x)‖≤3pθ3-3p‖x‖p
for all x∈A.Proof.
Settingϕ(x,y,z,u,v,w):=θ(∥x∥p+∥y∥p+∥z∥p+∥u∥p+∥v∥p+∥w∥p) all x,y,z,u,v,w∈A. Then by L=3p-1 in Theorem 3.2, one can prove the result.



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