# Approximately Ternary Homomorphisms onC*-Ternary Algebras

Authors: Eon Wha Shim; Su Min Kwon; Yun Tark Hyen; Yong Hun Choi; Abasalt Bodaghi
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

Gordji et al. established the Hyers-Ulam stability and the superstability ofC*-ternary homomorphisms and C*-ternary derivations on C*-ternary algebras, associated with the following
functional equation: fx2-x1/3+fx1-3x3/3+f3x1+3x3-x2/3=fx1, by the direct method. Under the conditions in the main theorems, we can show that the related mappings must be zero. In this paper, we correct the conditions and prove the corrected theorems. Furthermore, we prove the Hyers-Ulam stability and the superstability of C*-ternary homomorphisms and C*-ternary derivations on C*-ternary algebras by using a fixed point approach.


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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 x∘y:=[x,e,y] and x*:=[e,x,e], is a unital C*-algebra. Conversely, if (A,∘) is a unital C*-algebra, then [x,y,z]:=x∘y*∘z makes A into a C*-ternary algebra. A ℂ-linear mapping H:A→B between C*-ternary algebras is called a C*-ternary homomorphism if
(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 derivation if
(2)δ([x,y,z])=[δ(x),y,z]+[x,δ(y),z]+[x,y,δ(z)](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 [1–4]).The stability problem of functional equations is originated from the following question of Ulam [5]: under what condition does there exist an additive mapping near an approximately additive mapping? In 1941, Hyers [6] gave a partial affirmative answer to the question of Ulam in the context of Banach spaces. In 1978, Rassias [7] extended the theorem of Hyers by considering the unbounded Cauchy difference ∥f(x+y)-f(x)-f(y)∥≤ε(∥x∥p+∥y∥p),(ε>0,p∈[0,1)). 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 [8–12]).Gordji et al. [13] proved the Hyers-Ulam stability and the superstability of C*-ternary homomorphisms and C*-ternary derivations on C*-ternary algebras, associated with the functional equation
(3)f(x2-x13)+f(x1-3x33)+f(3x1+3x3-x23)=f(x1)
by applying the direct method. Under the conditions in the main theorems of [13], we can show that the related mappings must be zero.In this paper, we change the conditions of [13] and establish the corrected theorems. Moreover, we prove the Hyers-Ulam stability and the superstability of C*-ternary homomorphisms and C*-ternary derivations on C*-ternary algebras by employing a fixed point method. In fact, we show that some results of [13] are the special cases of our results.


## 2. Superstability: Direct Method

Throughout this paper, we assume thatA is a C*-ternary algebra with norm ∥·∥ and that B is a C*-ternary algebra with norm ∥·∥. Moreover, we assume that n0∈ℕ is a positive integer and suppose that 𝕋1/n01:={eiθ;0≤θ≤2π/n0}.In this section, we modify some results of [13]. Recall that a functional equation is called superstable if every approximate solution is an exact solution of it.Lemma 1 (see [13]).
Letf:A→B be a mapping such that
(4)∥f(x2-x13)+f(x1-3x33)+f(3x1+3x3-x23)∥≤∥f(x1)∥
for all x1,x2,x3∈A. Then, f is additive.We correct the statements of  [13, Theorem 2.2] as follows.Theorem 2.
Letp≠1 and θ be nonnegative real numbers, and let f:A→B be a mapping such that
(5)∥f(μx2-x13)+f(x1-3μx33)+μf(3x1+3x3-x23)∥≤∥f(x1)∥,(6)∥f([x1,x2,x3])-[f(x1),f(x2),f(x3)]∥≤θ(∥x1∥3p+∥x2∥3p+∥x3∥3p)
for all μ∈𝕋1/n01 and all x1,x2,x3∈A. Then, the mapping f:A→B is a C*-ternary homomorphism.Proof.
The proof is the same as in the proof of  [13, Theorem 2.2].In the following result, we correct Theorem3 from [13]. Since the proof is similar, it is omitted.Theorem 3.
Letp≠1 and θ be nonnegative real numbers, and let f:A→A be a mapping satisfying (5) and
(7)∥f([x1,x2,x3])-[f(x1),x2,x3]a-[x1,f(x2),x3]-[x1,x2,f(x3)]∥≤θ(∥x1∥3p+∥x2∥3p+∥x3∥3p)
for all x1,x2,x3∈A. Then, the mapping f:A→A is a C*-ternary derivation.


## 3. Hyers-Ulam Stability: Direct Method

In this section, we prove the Hyers-Ulam stability ofC*-ternary homomorphisms and C*-ternary derivations on C*-ternary algebras by the direct method.Theorem 4.
Letp>1 and θ be nonnegative real numbers, and let f:A→B be a mapping satisfying (6) and
(8)∥f(μx2-x13)+f(x1-3μx33)ai+μf(3x1+3x3-x23)-μf(x1)∥≤θ(∥x1∥p+∥x2∥p+∥x3∥p)
for all μ∈𝕋1/n01 and all x1,x2,x3∈A. Then, there exists a unique C*-ternary homomorphism H:A→B such that
(9)∥H(x1)-f(x1)∥≤3p(1+2p)θ∥x1∥p3p-3
for all x1∈A.Proof.
Lettingμ=1, x2=2x1, and x3=0 in (8), we get
(10)∥3f(x13)-f(x1)∥≤(1+2p)θ∥x1∥p
for all x1∈A. By induction, we have
(11)∥3nf(x13n)-f(x1)∥≤(1+2p)θ∥x1∥p∑i=0n-1‍3i(1-p)
for all x1∈A. Hence,
(12)∥3n+mf(x13n+m)-3mf(x13m)∥≤(1+2p)θ∥x1∥p∑i=0n-1‍3(i+m)(1-p)≤(1+2p)θ∥x1∥p∑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
(13)H(x1):=limn→∞3nf(x13n)
for all x1∈A. Moreover, letting m=0 and passing the limit n→∞ in (12), we get (9). It follows from (8) that
(14)∥H(μx2-x13)+H(x1-3μx33)+μH(3x1+3x3-x23)-μH(x1)∥=limn→∞3n[∥f(μx2-x13n+1)+f(x1-3μx33n+1)+μf(3x1+3x3-x23n+1)-μf(x13n)∥]≤limn→∞3nθ3np(∥x1∥p+∥x2∥p+∥x3∥p)=0
for all μ∈𝕋1/n01 and all x1,x2,x3∈A. So
(15)H(μx2-x13)+H(x1-3μx33)+μH(3x1+3x3-x23)=μH(x1)
for all μ∈𝕋1/n01 and all x1,x2,x3∈A. Put μ=1 in (15). Then the mapping H:A→B satisfies the inequality (4), and thus, the mapping H:A→B is additive. Letting x1=x2=0 in (15), we get H(-3μx3/3)+μH(3x3/3)=0 and so H(μx3)=μH(x3) for all μ∈𝕋1/n01 and all x3∈A. By the same reasoning as in the proof of [14, Theorem 2.2], the mapping H is ℂ-linear. Now, let H′:A→B be another additive mapping satisfying (9). Then, we have
(16)∥H(x1)-H′(x1)∥=3n∥H(x13n)-H′(x13n)∥≤3n(∥H(x13n)-f(x13n)∥+∥H′(x13n)-f(x13n)∥)≤2·3n(1+2p)3p3np(3p-3)θ∥x∥p,
which tends to zero as n→∞ for all x1∈A. Thus, we can conclude that H(x1)=H′(x1) for all x1∈A. This shows the uniqueness of H. It follows from (6) that
(17)∥H([x1,x2,x3])-[H(x1),H(x2),H(x3)]∥=limn→∞27n∥f([x1,x2,x3]27n)i-[f(x13n),f(x23n),f(x33n)]([x1,x2,x3]27n)∥≤limn→∞27nθ27np(∥x1∥3p+∥x2∥3p+∥x3∥3p)=0
for all x1,x2,x3∈A. Therefore, the mapping H is a unique C*-ternary homomorphism satisfying (9).Theorem 5.
Letp<1 and θ be nonnegative real numbers, and let f:A→B be a mapping satisfying (6) and (8). Then, there exists a unique C*-ternary homomorphism H:A→B such that
(18)∥H(x1)-f(x1)∥≤3p(1+2p)θ∥x1∥p3-3p
for all x1∈A.Proof.
The proof is similar to the proof of Theorem4.In the following theorem, we prove the Hyers-Ulam stability of derivations onC*-ternary algebras via the direct method.Theorem 6.
Letp>1 and θ be nonnegative real numbers, and let f:A→A be a mapping satisfying (7) and
(19)∥f(μx2-x13)+f(x1-3μx33)+μf(3x1+3x3-x23)-μf(x1)∥≤θ(∥x1∥p+∥x2∥p+∥x3∥p)
for all μ∈𝕋1/n01 and all x1,x2,x3∈A. Then, there exists a unique C*-ternary derivation D:A→A such that
(20)∥D(x1)-f(x1)∥≤3p(1+2p)θ∥x1∥p3p-3
for all x1∈A.Proof.
By the same reasoning as in the proof of Theorem4, there exists a unique ℂ-linear mapping D:A→A satisfying (20) which is defined by
(21)D(x1):=limn→∞3nf(x13n)
for all x1∈A. The inequality (7) implies that
(22)∥D([x1,x2,x3])-[D(x1),x2,x3]-[x1,D(x2),x3]-[x1,x2,D(x3)]∥=limn→∞27n(∥f([x1,x2,x3]27n)-[f(x13n),x23n,x33n]i-[x13n,f(x23n),x33n]-[x13n,x23n,f(x33n)]([x1,x2,x3]27n)∥)≤limn→∞27nθ27np(∥x1∥3p+∥x2∥3p+∥x3∥3p)=0
for all x1,x2,x3∈A. So
(23)D([x1,x2,x3])=[D(x1),x2,x3]+[x1,D(x2),x3]+[x1,x2,D(x3)]
for all x1,x2,x3∈A. Consequently, the mapping D is a unique C*-ternary derivation satisfying (20).The following consequence is analogous to Theorem4 for C*-ternary derivations and its proof is similar to the proof of Theorems 4 and 6.Theorem 7.
Letp<1 and θ be nonnegative real numbers, and let f:A→A be a mapping satisfying (7) and (19). Then, there exists a unique C*-ternary derivation D:A→A such that
(24)∥D(x1)-f(x1)∥≤3p(1+2p)θ∥x1∥p3-3p
for all x1∈A.


## 4. Superstability: A Fixed Point Approach

In this section, we prove the superstability ofC*-ternary homomorphisms and of C*-ternary derivations on C*-ternary algebras by using the fixed point method (Theorem 8).LetX be a set. A function d:X×X→[0,∞] is called a generalized metricon X if d satisfies (1)
d(x,y)=0 if and only if x=y;(2)
d(x,y)=d(y,x) for all x,y∈X;(3)
d(x,z)≤d(x,y)+d(y,z) for all x,y,z∈X.We recall a fundamental result in the fixed point theory from [15] which is a useful tool to achieve our purposes in the sequel.Theorem 8.
Let(X,d) be a complete generalized metric space, and let J:X→X be a strictly contractive mapping with the Lipschitz constant α<1. Then, for each given element x∈X, either
(25)d(Jnx,Jn+1x)=∞
for all nonnegative integers n or there exists a positive integer n0 such that(i)
d(Jnx,Jn+1x)<∞, for all n≥n0;(ii)
the sequence{Jnx} converges to a fixed point y* of J;(iii)
y* is the unique fixed point of J in the set Y={y∈X∣d(Jn0x,y)<∞};(iv)
d(y,y*)≤(1/(1-α))d(y,Jy) for all y∈Y.In 1996, Isac and Rassias [16] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. In 2003, Cădariu and Radu applied a fixed point method to the investigation of the Jensen functional equation [17]. They presented a short and a simple proof for the Cauchy functional equation and the quadratic functional equation in [18, 19], respectively. By using the fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors. For instance, the Hyers-Ulam stability and the superstability of a ternary quadratic derivation on ternary Banach algebras and C*-ternary rings by using Theorem 8 are investigated in [20]. Recently, in [21], Park and Bodaghi proved the stability and the superstability of *-derivations associated with the Cauchy functional equation and the Jensen functional equation by the mentioned theorem (for more applications, see [22–28]).From now on, we denoteA×A×⋯×A︷n-times by An. We prove the superstability of C*-ternary homomorphism on C*-ternary algebras by employing Theorem 8 as follows.Theorem 9.
Letφ:A3→[0,∞) be a function such that there exists an α<1 with
(26)φ(x1,x2,x3)≤27αφ(x13,x23,x23)
for all x1,x2,x3∈A. Let f:A→B be a mapping satisfying (5) and
(27)∥f([x1,x2,x3])-[f(x1),f(x2),f(x3)]∥≤φ(x1,x2,x3)
for all x1,x2,x3∈A. Then, the mapping f:A→B is a C*-ternary homomorphism.Proof.
Since the proof is similar to the proof of [13, Theorem 2.2], we only show some parts of it. From the proof of [13, Theorem 2.2], one can show that the mapping f:A→B is ℂ-linear. The inequality (26) implies that
(28)limn→∞127nφ(3nx1,3nx2,3nx3)=0
for all x1,x2,x3∈A. Since f is additive, it follows from (27) and (28) that
(29)f([x1,x2,x3])=[f(x1),f(x2),f(x3)]
for all x1,x2,x3∈A. Thus, the mapping f:A→B is a C*-ternary homomorphism.Theorem 10.
Letφ:A3→[0,∞) be a function such that there exists an α<1 with
(30)φ(x1,x2,x3)≤α3φ(3x1,3x2,3x3)
for all x1,x2,x3∈A. Let f:A→B be a mapping satisfying (5) and (27). Then, the mapping f:A→B is a C*-ternary homomorphism.Proof.
Similar to the proof of Theorem9, the mapping f:A→B is ℂ-linear. It also follows from (30) that
(31)limn→∞3nφ(x13n,x23n,x33n)=0
for all x1,x2,x3∈A. Since f is additive, we can deduce from (27) and (31) that
(32)f([x1,x2,x3])=[f(x1),f(x2),f(x3)]
for all x1,x2,x3∈A. Therefore, the mapping f is a C*-ternary homomorphism.Remark 11.
Theorem2 follows from Theorems 9 and 10 by taking φ(x1,x2,x3)=θ(∥x1∥3p+∥x2∥3p+∥x3∥3p) for all x1,x2,x3∈A.In analogy with Theorems9 and 10, we have the following theorems for the superstability of C*-ternary derivations on C*-ternary algebras.Theorem 12.
Letφ:A3→[0,∞) be a function satisfying (26). Let f:A→A be a mapping satisfying (5) and
(33)∥f([x1,x2,x3])-[f(x1),x2,x3]-[x1,f(x2),x3]-[x1,x2,f(x3)]∥≤φ(x1,x2,x3)
for all x1,x2,x3∈A. Then, the mapping f:A→A is a C*-ternary derivation.Proof.
The proof is similar to the proof of Theorem9.Theorem 13.
Letφ:A3→[0,∞) be a function satisfying (30). Let f:A→A be a mapping satisfying (5) and (33). Then, the mapping f:A→A is a C*-ternary derivation.Proof.
Refer to the proof of Theorem10.Note that Theorem3 follows immediately from Theorems 12 and 13 by putting φ(x1,x2,x3)=θ(∥x1∥3p+∥x2∥3p+∥x3∥3p) for all x1,x2,x3∈A.


## 5. Hyers-Ulam Stability: Fixed Point Method

In this section, we apply Theorem8 to prove the Hyers-Ulam stability of C*-ternary homomorphisms and C*-ternary derivations on C*-ternary algebras.Theorem 14.
Letφ:A3→[0,∞) be a function satisfying (30). Let f:A→B be a mapping satisfying (27) and
(34)∥f(μx2-x13)+f(x1-3μx33)+μf(3x1+3x3-x23)-μf(x1)∥≤φ(x1,x2,x3)
for all μ∈𝕋1/n01 and all x1,x2,x3∈A. Then, there exists a unique C*-ternary homomorphism H:A→B such that
(35)∥H(x1)-f(x1)∥≤α1-αφ(x1,2x1,0)
for all x1∈A.Proof.
Lettingμ=1, x2=2x1, and x3=0 in (34), we get
(36)∥3f(x13)-f(x1)∥≤(x1,2x1,0)
for all x1∈A. Consider the set
(37)S:={h:A⟶B}
and introduce the generalized metric on S as follows:
(38)d(g,h)=inf{μ∈ℝ+:∥g(x)-h(x)∥≤μφ(x,2x,0),∀x∈A},
where, as usual, infϕ=+∞. Similar to the proof of [29, Theorem 2.2], we can show that d is a generalized metric on S and the metric space (S,d) is complete. We now define the linear mapping J:S→S via Jg(x):=(1/3)g(3x) for all x∈A. Let g,h∈S be given such that d(g,h)=ε. Then
(39)∥g(x)-h(x)∥≤φ(x,2x,0)
for all x∈A. Hence
(40)∥Jg(x)-Jh(x)∥=∥13g(3x)-13h(3x)∥≤αφ(x,2x,0)
for all x∈A. Thus, d(g,h)=ε implies that d(Jg,Jh)≤αε. This means that
(41)d(Jg,Jh)≤αd(g,h)
for all g,h∈S. It follows from (36) that
(42)∥f(x1)-13f(3x1)∥≤13φ(3x1,6x1,0)≤αφ(x1,2x1,0)
for all x1∈A. So d(f,Jf)≤α. By Theorem 8, there exists a mapping H:A→B satisfies the following:(1)
H is a fixed point of J, that is,
(43)H(3x)=3H(x)
for all x∈A. Indeed, the mapping H is a unique fixed point of J in the set M={g∈S:d(h,g)<∞}. This implies that H satisfying (43) such that there exists a μ∈(0,∞) satisfying
(44)∥f(x)-H(x)∥≤μφ(x,2x,0)
for all x∈A;(2)
d(Jnf,H)→0 as n→∞, and thus, we have the following equality:
(45)limn→∞13nf(3nx)=H(x)(x∈A);(3)
d(f,H)≤(1/(1-α))d(f,Jf), which implies the followin inequality:
(46)d(f,H)≤α1-α.This shows that the inequality (35) holds. The rest of the proof is similar to the proof of Theorem 4.Theorem 15.
Letφ:A3→[0,∞) be a function satisfying (26). Let f:A→B be a mapping satisfying (27) and (34). Then, there exists a unique C*-ternary homomorphism H:A→B such that
(47)∥H(x1)-f(x1)∥≤11-αφ(x1,2x1,0)
for all x1∈A.Proof.
Let(S,d) be the generalized metric space defined in the proof of Theorem 14. Consider the linear mapping J:S→S such that
(48)Jg(x):=3g(x3)
for all x∈X. The inequality (36) implies that d(f,Jf)≤1. So d(f,H)≤1/(1-α). Thus, we obtain the inequality (47). The rest of the proof is similar to the proofs of Theorems 4 and 14.The following parallel results for the Hyers-Ulam stability of derivations onC*-ternary algebras can be proved in similar ways to the proofs of Theorems 6 and 14, and so we omit their proofs.Theorem 16.
Letφ:A3→[0,∞) be a function satisfying (30). Let f:A→A be a mapping satisfying (33) and
(49)∥f(μx2-x13)+f(x1-3μx33)+μf(3x1+3x3-x23)-μf(x1)∥≤φ(x1,x2,x3)
for all μ∈𝕋1/n01 and all x1,x2,x3∈A. Then, there exists a unique C*-ternary derivation D:A→A such that
(50)∥D(x1)-f(x1)∥≤α1-αφ(x1,2x1,0)
for all x1∈A.Theorem 17.
Letφ:A3→[0,∞) be a function satisfying (26). Let f:A→A be a mapping satisfying (33) and (49). Then, there exists a unique C*-ternary derivation D:A→A such that
(51)∥D(x1)-f(x1)∥≤11-αφ(x1,2x1,0)
for all x1∈A.Remark 18.
All results of Section3 are the direct consequences of the results of this section as follows: (i)
Theorem4 follows from Theorem 15 by taking φ(x1,x2,x3)=θ(∥x1∥3p+∥x2∥3p+∥x3∥3p) for all x1,x2,x3∈A, and α=31-p;(ii)
we can obtain Theorem5 from Theorem 14 by letting φ(x1,x2,x3)=θ(∥x1∥3p+∥x2∥3p+∥x3∥3p) for all x1,x2,x3∈A, and α=3p-1;(iii)
if we putφ(x1,x2,x3)=θ(∥x1∥3p+∥x2∥3p+∥x3∥3p) for all x1,x2,x3∈A, and α=31-p in Theorem 17, then we conclude Theorem 6;(iv)
puttingφ(x1,x2,x3)=θ(∥x1∥3p+∥x2∥3p+∥x3∥3p) for all x1,x2,x3∈A, and α=3p-1 in Theorem 16, we get Theorem 7.



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