# Approximately Ternary Homomorphisms and Derivations onC∗-Ternary Algebras

Authors: M. Eshaghi Gordji; A. Ebadian; N. Ghobadipour; J. M. Rassias; M. B. Savadkouhi
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We investigate the stability and superstability of ternary homomorphisms betweenC*-ternary algebras and derivations on C*-ternary algebras, associated with the following functional equation f((x2-x1)/3)+f((x1-3x3)/3)+f((3x1+3x3-x2)/3)=f(x1).


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## Body

## 1. Introduction

AC*-ternary algebra is a complex Banach space A, equipped with a ternary product (x,y,z)↣[x,y,z] of A3 into A, which is ℂ-linear in the outer variables, conjugate ℂ-linear in the middle variable, and associative in the sense that [x,y,[z,w,v]]=[x,[w,z,y],v]=[[x,y,z],w,v], and satisfies ∥[x,y,z]∥≤∥x∥·∥y∥·∥z∥ and ∥[x,x,x]∥=∥x∥3. If a C*-ternary algebra (A,[·,·,·]) has an identity, that is, an element e∈A such that x=[x,e,e]=[e,e,x] for all x∈A, then it is routine to verify that A, endowed with xoy:=[x,e,y] and x*:=[e,x,e], is a unital C*-algebra. Conversely, if (A,o) is a unital C*-algebra, then [x,y,z]:=xoy*oz makes A into a C*-ternary algebra. A ℂ-linear mapping H:A→B is called a C*-ternary algebra homomorphism if(1.1)H([x,y,z])=[H(x),H(y),H(z)],
for all x, y, z∈A. A ℂ-linear mapping δ:A→A is called a C*-ternary algebra derivation if(1.2)δ([x,y,z])=[δ(x),y,z]+[x,δ(y),z]+[x,y,δ(z)],
for all x, y, z∈A.Ternary structures and their generalization the so-calledn-ary structures raise certain hopes in view of their applications in physics (see [1–8]).We say a functional equationζ is stable if any function g satisfying the equation ζ approximately is near to true solution of ζ. Moreover, ζ is superstable if every approximately solution of ζ is an exact solution of it.The study of stability problems originated from a famous talk given by Ulam [9] in 1940: “Under what condition does there exist a homomorphism near an approximate homomorphism?” In the next year 1941, Hyers [10] answered affirmatively the question of Ulam for additive mappings between Banach spaces.A generalized version of the theorem of Hyers for approximately additive maps was given by Rassias [11] in 1978 as follows.Theorem 1.1.
Letf:E1→E2 be a mapping from a normed vector space E1 into a Banach space E2 subject to the inequality:
(1.3)‖f(x+y)-f(x)-f(y)‖≤ϵ(‖x‖p+‖y‖p),
for all x, y∈E1, where ϵ and p are constants with ϵ>0 and p<1. Then, there exists a unique additive mapping T:E1→E2 such that
(1.4)‖f(x)-T(x)‖≤2ϵ2-2p‖x‖p,
for all x∈E1.The stability phenomenon that was introduced and proved by Rassias is called Hyers-Ulam-Rassias stability. And then the stability problems of several functional equations have been extensively investigated by a number of authors, and there are many interesting results concerning this problem (see [12–27]).Throughout this paper, we assume thatA is a C*-ternary algebra with norm ∥·∥A and that B is a C*-ternary algebra with norm ∥·∥B. Moreover, we assume that n0∈ℕ is a positive integer and suppose that 𝕋1/no1:={eiθ;0≤θ≤2π/no}.


## 2. Superstability

In this section, first we investigate homomorphisms betweenC*-ternary algebras. We need the following Lemma in the main results of the paper.Lemma 2.1.
Letf:A→B be a mapping such that
(2.1)‖f(x2-x13)+f(x1-3x33)+f(3x1+3x3-x23)‖B≤‖f(x1)‖B,
for all x1, x2, x3∈A. Then f is additive.Proof.
Lettingx1=x2=x3=0 in (2.1), we get
(2.2)‖3f(0)‖B≤‖f(0)‖B.
So f(0)=0. Letting x1=x2=0 in (2.1), we get
(2.3)‖f(-x3)+f(x3)‖B≤‖f(0)‖B=0,
for all x3∈A. Hence f(-x3)=-f(x3) for all x3∈A. Letting x1=0 and x2=6x3 in (2.1), we get
(2.4)‖f(2x3)-2f(x3)‖B≤‖f(0)‖B=0,
for all x3∈A. Hence
(2.5)f(2x3)=2f(x3),
for all x3∈A. Letting x1=0 and x2=9x3 in (2.1), we get
(2.6)‖f(3x3)-f(x3)-2f(x3)‖B≤‖f(0)‖B=0,
for all x3∈A. Hence
(2.7)f(3x3)=3f(x3),
for all x3∈A. Letting x1=0 in (2.1), we get
(2.8)‖f(x23)+f(-x3)+f(x3-x23)‖B≤‖f(0)‖B=0,
for all x2, x3∈A. So
(2.9)f(x23)+f(-x3)+f(x3-x23)=0,
for all x2, x3∈A. Let t1=x3-(x2/3) and t2=x2/3 in (2.9). Then
(2.10)f(t2)-f(t1+t2)+f(t1)=0,
for all t1, t2∈A, this means that f is additive.Now, we prove the first result in superstability as follows.Theorem 2.2.
Letp≠1 and θ be nonnegative real numbers, and let f:A→B be a mapping such that
(2.11)‖f(x2-x13)+f(x1-3μx33)+μf(3x1+3x3-x23)‖B≤‖f(x1)‖B,(2.12)‖f([x1,x2,x3])-[f(x1),f(x2),f(x3)]‖B≤θ(‖x1‖A3p+‖x2‖A3p+‖x3‖A3p),
for all μ∈𝕋1/no1 and all x1, x2, x3∈A. Then, the mapping f:A→B is a C*-ternary algebra homomorphism.Proof.
Assumep>1.
Letμ=1 in (2.11). By Lemma 2.1, the mapping f:A→B is additive. Letting x1=x2=0 in (2.11), we get
(2.13)‖f(-μx3)+μf(x3)‖B≤‖f(0)‖B=0,
for all x3∈A and μ∈𝕋1. So
(2.14)-f(μx3)+μf(x3)=f(-μx3)+μf(x3)=0,
for all x3∈A and all μ∈𝕋1. Hence f(μx3)=μf(x3) for all x3∈A and all μ∈𝕋1/no1. By same reasoning as proof of Theorem 2.2 of [28], the mapping f:A→B is ℂ-linear. It follows from (2.12) that
(2.15)‖f([x1,x2,x3])-[f(x1),f(x2),f(x3)]‖B=lim⁡n→∞8n‖f([x1,x2,x3]2n⋅2n⋅2n)-[f(x12n),f(x22n),f(x32n)]‖B≤lim⁡n→∞8nθ8np(‖x1‖A3p+‖x2‖A3p+‖x3‖A3p)=0,
for all x1, x2, x3∈A. Thus,
(2.16)f([x1,x2,x3])=[f(x1),f(x2),f(x3)],
for all x1, x2, x3∈A. Hence, the mapping f:A→B is a C*-ternary algebra homomorphism. Similarly, one obtains the result for the case p<1.Now, we establish the superstability of derivations onC*-ternary algebras as follows.Theorem 2.3.
Letp≠1 and θ be nonnegative real numbers, and let f:A→A be a mapping satisfying (2.11) such that
(2.17)‖f([x1,x2,x3])-[f(x1),x2,x3]-[x1,f(x2),x3]-[x1,x2,f(x3)]‖A≤θ(‖x1‖A3p+‖x2‖A3p+‖x3‖A3p),
for all x1, x2, x3∈A. Then the mapping f:A→A is a C*-ternary derivation.Proof.
Assumep>1.
By the Theorem2.2, the mapping f:A→A is ℂ-linear. It follows from (2.17) that
(2.18)‖f([x1,x2,x3])-[f(x1),x2,x3]-[x1,f(x2),x3]-[x1,x2,f(x3)]‖A=lim⁡n→∞8n‖f([x1,x2,x3]8n)-[f(x12n),x22n,x32n]-[x12n,f(x22n),x32n]-[x12n,x22n,f(x32n)]([x1,x2,x3]8n)‖A≤lim⁡n→∞8nθ8np(‖x1‖A3p+‖x2‖A3p+‖x3‖A3p)=0,
for all x1, x2, x3∈A. So
(2.19)f([x1,x2,x3])=[f(x1),x2,x3]+[x1,f(x2),x3]+[x1,x2,f(x3)]
for all x1, x2, x3∈A. Thus, the mapping f:A→A is a C*-ternary derivation. Similarly, one obtains the result for the case p<1.


## 3. Stability

First we prove the generalized Hyers-Ulam-Rassias stability of homomorphisms inC*-ternary algebras.Theorem 3.1.
Letp>1 and θ be nonnegative real numbers, and let f:A→B be a mapping such that
(3.1)‖f(x2-x13)+f(x1-3μx33)+μf(3x1+3x3-x23)-f(x1)‖B≤θ(‖x1‖Ap+‖x2‖Ap+‖x3‖Ap),(3.2)‖f([x1,x2,x3])-[f(x1),f(x2),f(x3)]‖B≤θ(‖x1‖A3p+‖x2‖A3p+‖x3‖A3p),
for all μ∈𝕋1/no1, and all x1, x2, x3∈A. Then there exists a unique C*-ternary homomorphism H:A→B such that
(3.3)‖H(x1)-f(x1)‖B≤θ(1+2p)‖x1‖Ap1-31-p,
for all x1∈A.Proof.
Let us assumeμ=1, x2=2x1 and x3=0 in (3.1). Then we get
(3.4)‖3f(x13)-f(x1)‖B≤θ(1+2p)‖x1‖Ap,
for all x1∈A. So by induction, we have
(3.5)‖3nf(x13n)-f(x1)‖B≤θ(1+2p)‖x1‖Ap∑i=0n-13i(1-p),
for all x1∈A. Hence
(3.6)‖3n+mf(x13n+m)-3mf(x13m)‖B≤θ(1+2p)‖x1‖Ap∑i=0n-13(i+m)(1-p)≤θ(1+2p)‖x1‖Ap∑i=mn+m-13i(1-p),
for all nonnegative integers m and n with n≥m, and all x1∈A. It follows that the sequence {3nf(x1/3n)} is a Cauchy sequence for all x1∈A. Since B is complete, the sequence {3nf(x1/3n)} converges. Thus, one can define the mapping H:A→B by
(3.7)H(x1):=lim⁡n→∞3nf(x13n),
for all x1∈A. Moreover, letting m=0 and passing the limit n→∞ in (3.6), we get (3.3). It follows from (3.1) that
(3.8)‖H(x2-x13)+H(x1-3μx33)+μH(3x1+3x3-x23)-H(x1)‖B=lim⁡n→∞3n‖f(x2-x13n+1)+f(x1-3μx33n+1)+f(3x1+3x3-x23n+1)-f(x13n)‖B≤lim⁡n→∞3nθ3np(‖x1‖Ap+‖x2‖Ap+‖x3‖Ap)=0,
for all μ∈𝕋1/no1, and all x1,x2,x3∈A. So
(3.9)H(x2-x13)+H(x1-3μx33)+μH(3x1+3x3-x23)=H(x1),
for all μ∈𝕋1/no1, and all x1, x2, x3∈A. By the same reasoning as proof of Theorem 2.2 of [28], the mapping H:A→B is ℂ-linear.
Now, letH′:A→B be another additive mapping satisfying (3.3). Then, we have
(3.10)‖H(x1)-H′(x1)‖B=3n‖H(x13n)-H′(x13n)‖B≤3n(‖H(x13n)-f(x13n)‖B+‖H′(x13n)-f(x13n)‖B)≤2⋅3nθ(1+2p)3np(1-31-p)‖x‖Ap,
which tends to zero as n→∞ for all x1∈A. So we can conclude that H(x1)=H′(x1) for all x1∈A. This proves the uniqueness of H.
It follows from (3.2) that
(3.11)‖H([x1,x2,x3])-[H(x1),H(x2),H(x3)]‖B=lim⁡n→∞27n‖f([x1,x2,x3]3n⋅3n⋅3n)-[f(x13n),f(x23n),f(x33n)]‖B≤lim⁡n→∞27nθ27np(‖x1‖A3p+‖x2‖A3p+‖x3‖A3p)=0,
for all x1, x2, x3∈A.
Thus, the mappingH:A→B is a unique C*-ternary homomorphism satisfying (3.3).Theorem 3.2.
Letp<1 and θ be nonnegative real numbers, and let f:A→B be a mapping satisfying (3.1) and (3.2). Then, there exists a unique C*-ternary homomorphism H:A→B such that
(3.12)‖H(x1)-f(x1)‖B≤θ(1+2p)‖x1‖Ap31-p-1,
for all x1∈A.Proof.
The proof is similar to the proof of Theorem3.1.Now, we prove the generalized Hyers-Ulam-Rassias stability of derivations onC*-ternary algebras.Theorem 3.3.
Letp>1 and θ be nonnegative real numbers, and let f:A→A be a mapping such that
(3.13)‖f(x2-x13)+f(x1-3μx33)+μf(3x1+3x3-x23)-f(x1)‖A≤θ(‖x1‖Ap+‖x2‖Ap+‖x3‖Ap),(3.14)‖f([x1,x2,x3])-[f(x1),x2,x3]-[x1,f(x2),x3]-[x1,x2,f(x3)]‖A≤θ(‖x1‖A3p+‖x2‖A3p+‖x3‖A3p),
for all μ∈𝕋1/no1, and all x1, x2, x3∈A. Then, there exists a unique C*-ternary derivation D:A→A such that
(3.15)‖D(x1)-f(x1)‖A≤θ(1+2p)‖x1‖Ap1-31-p,
for all x1∈A.Proof.
By the same reasoning as in the proof of the Theorem3.1, there exists a unique ℂ-linear mapping D:A→A satisfying (3.15). The mapping D:A→A is defined by
(3.16)D(x1):=lim⁡n→∞3nf(x13n),
for all x1∈A. It follows from (3.14) that
(3.17)‖D([x1,x2,x3])-[D(x1),x2,x3]-[x1,D(x2),x3]-[x1,x2,D(x3)]‖A=lim⁡n→∞27n‖[x1,x2,x3]3n⋅3n⋅3n-[f(x13n),x23n,x33n]-[x13n,f(x23n),x33n]-[x13n,x23n,f(x33n)]‖A≤lim⁡n→∞27nθ27np(‖x1‖A3p+‖x2‖A3p+‖x3‖A3p)=0,
for all x1, x2, x3∈A. So
(3.18)D([x1,x2,x3])=[D(x1),x2,x3]+[x1,D(x2),x3]+[x1,x2,D(x3)]
for all x1, x2, x3∈A.
Thus, the mappingD:A→A is a unique C*-ternary derivation satisfying (3.15).Theorem 3.4.
Letp<1 and θ be nonnegative real numbers, and let f:A→A be a mapping satisfying (3.13)and (3.14). Then, there exists a unique C*-ternary derivation D:A→A such that
(3.19)‖D(x1)-f(x1)‖A≤θ(1+2p)‖x1‖Ap31-p-1,
for all x1∈A.Proof.
The proof is similar to the proof of Theorems3.1 and 3.3.


## 4. Conclusions

In this paper, we have analyzed some detailC*-ternary algebras and derivations on C*-ternary algebras, associated with the following functional equation:(4.1)f(x2-x13)+f(x1-3x33)+f(3x1+3x3-x23)=f(x1).
A detailed study of how we can have the generalized Hyers-Ulam-Rassias stability of homomorphisms and derivations on C*-ternary algebras is given.



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*Source: 984160-2012-03-25.xml*