# Homomorphisms and Derivations inC∗-Ternary Algebras

Authors: Abbas Najati; Choonkil Park; Jung Rye Lee
Journal: Abstract and Applied Analysis
 (2009)

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## Abstract

In 2006, C. Park proved the stability of homomorphisms inC∗-ternary algebras and of derivations on C∗-ternary algebras for the following
generalized Cauchy-Jensen additive mapping: 2f((∑j=1pxj/2)+∑j=1dyj)=∑j=1pf(xj)+2∑j=1df(yj). In this note, we improve and generalize some results concerning this functional equation.


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## Body

## 1. Introduction and Preliminaries

The stability problem of functional equations is  originated from a question of Ulam [1] concerning the stability of group homomorphisms. Hyers [2] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ theorem was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by considering an unbounded Cauchy difference.Theorem 1.1 (Th. M. Rassias).
Letf:E→E' be a mapping from a normed vector space E into a Banach space E' subject to the inequality
(1.1)∥f(x+y)-f(x)-f(y)∥≤ε(∥x∥p+∥y∥p)
for all x,y∈E, where ε and p are constants with ε>0 and p<1. Then the limit
(1.2)L(x)=limn→∞f(2nx)2n
exists for all x∈E, and L:E→E' is the unique additive mapping which satisfies
(1.3)∥f(x)-L(x)∥≤2ε2-2p∥x∥p
for all x∈E. If p<0, then inequality (1.1) holds for x,y≠0 and (1.3) for x≠0. Also, if for each x∈E the mapping f(tx) is continuous in t∈ℝ, then L is linear.It was shown by Gajda [5] as well as by Rassias and Šemrl [6] that one cannot prove a Rassias’s type theorem when p=1. The counter examples of Gajda [5] as well as of Rassias and Šemrl [6] have stimulated several mathematicians to invent new definitions of approximately additive or approximately linear mappings; compare Găvruţa [7] and Jung [8], who among others studied the stability of functional equations. Theorem 1.1 provided a lot of influence in the development of a generalization of the Hyers-Ulam stability concept. This new concept is known as Hyers-Ulam-Rassiasstability of functional equations (cf. the books of Czerwik [9], Hyers et al. [10]).Theorem 1.2 (Rassias [11–13]).
LetX be a real normed linear space and Y a real Banach space. Assume that f:X→Y is a mapping for which there exist constants θ≥0 and p,q∈ℝ such that r=p+q≠1 and f satisfies the functional inequality (Cauchy-Găvruţa-Rassias inequality)(1.4)∥f(x+y)-f(x)-f(y)∥≤θ∥x∥p∥y∥q
for all x,y∈X. Then there exists a unique additive mapping L:X→Y satisfying
(1.5)∥f(x)-L(x)∥≤θ|2r-2|∥x∥r
for all x∈X. If, in addition, f:X→Y is a mapping such that the transformation t→f(tx) is continuous in t∈ℝ for each fixed x∈X, then L is linear.For the caser=1, a counter example has been given by Găvruţa [14]. The stability in Theorem 1.2 involving a product of different powers of norms is called Ulam-Găvruţa-Rassias stability (see [15–17]). In 1994, a generalization of Theorems 1.1 and 1.2 was obtained by Găvruţa [7], who replaced the bounds ε(∥x∥p+∥y∥p) and θ∥x∥p∥y∥q by a general control function φ(x,y). During past few years several mathematicians have published on various generalizations and applications of generalized Hyers-Ulam stability to a number of functional equations and mappings (see [16–44]).Following the terminology of [45], a nonempty set G with a ternary operation [·,·,·]:G×G×G→G is called a ternary groupoid and is denoted by (G,[·,·,·]). The ternary groupoid (G,[·,·,·]) is called commutative if [x1,x2,x3]=[xσ(1),xσ(2),xσ(3)] for all x1,x2,x3∈G and all permutations σ of {1,2,3}.If a binary operation∘ is defined on G such that [x,y,z]=(x∘y)∘z for all x,y,z∈G, then we say that [·,·,·] is derived from ∘. We say that (G,[·,·,·]) is a ternary semigroup if the operation [·,·,·] is associative, that is, if [[x,y,z],u,v]=[x,[y,z,u],v]=[x,y,[z,u,v]] holds for all x,y,z,u,v∈G (see [46]).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 are 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 (see [45, 47]). Every left Hilbert C*-module is a C*-ternary algebra via the ternary product [x,y,z]:=〈x,y〉z.If aC*-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.Aℂ-linear mapping H:A→B is called a C*-ternary algebra homomorphism if
(1.6)H([x,y,z])=[H(x),H(y),H(z)]
for all x,y,z∈A. If, in addition, the mapping H is bijective, then the mapping H:A→B is called a C*-ternary algebra isomorphism. A ℂ-linear mapping δ:A→A is called a C*-ternary derivation if
(1.7)δ([x,y,z])=[δ(x),y,z]+[x,δ(y),z]+[x,y,δ(z)]
for all x,y,z∈A (see [23, 45, 48]).Let(A,∘) be a C*-algebra and [x,y,z]:=x∘y*∘z for all x,y,z∈A. The mapping H:A→A defined by H(x)=-ix is a C*-ternary algebra isomorphism. Let a∈A with a*=a. The mapping δa:A→A defined by δa(x)=i(ax-xa) is a C*-ternary derivation. There are some applications, although still hypothetical, in the fractional quantum Hall effect, the nonstandard statistics, supersymmetric theory, and Yang-Baxter equation (cf. [49–51]).Throughout this paper, assume thatp, d are nonnegative integers with p+d≥3, and that A and B are C*-ternary algebras.


## 2. Stability of Homomorphisms inC*-Ternary Algebras

The stability of homomorphisms inC*-ternary algebras has been investigated in [31] (see also [37]). In this note, we improve some results in [31]. For a given mapping f:A→B, we define
(2.1)Cμf(x1,…,xp,y1,…,yd):=2f(∑j=1pμxj2+∑j=1dμyj)-∑j=1pμf(xj)-2∑j=1dμf(yj)
for all μ∈𝕋1:={λ∈ℂ:|λ|=1} and all x1,…,xp,y1,…,yd∈A.One can easily show that a mappingf:A→B satisfies
(2.2)Cμf(x1,…,xp,y1,…,yd)=0
for all μ∈𝕋1 and all x1,…,xp,y1,…,yd∈A if and only if
(2.3)f(μx+λy)=μf(x)+λf(y)
for all μ,λ∈𝕋1 and all x,y∈A.We will use the following lemmas in this paper.Lemma 2.1 (see [30]).
Letf:A→B be an additive mapping such that f(μx)=μf(x) for all x∈A and all μ∈𝕋1. Then the mapping f is ℂ-linear.Lemma 2.2.
Let{xn}n,{yn}n and {zn}n be convergent sequences in A. Then the sequence {[xn,yn,zn]}n is convergent in A.Proof.
Letx,y,z∈A such that
(2.4)limn→∞xn=x,limn→∞yn=y,limn→∞zn=z.
Since
(2.5)[xn,yn,zn]-[x,y,z]=[xn-x,yn-y,zn-z]+[xn-x,yn,z]+[x,yn-y,zn]+[xn,y,zn-z]
for all n, we get
(2.6)∥[xn,yn,zn]-[x,y,z]∥≤∥xn-x∥∥yn-y∥∥zn-z∥+∥xn-x∥∥yn∥∥z∥+∥x∥∥yn-y∥∥zn∥+∥xn∥∥y∥∥zn-z∥
for all n. So
(2.7)limn→∞[xn,yn,zn]=[x,y,z].
This completes the proof.Theorem 2.3 (see [31]).
Letr and θ be nonnegative real numbers such that r∉[1,3], and let f:A→B be a mapping such that
(2.8)∥Cμf(x1,…,xp,y1,…,yd)∥B≤θ(∑j=1p∥xj∥Ar+∑j=1d∥yj∥Ar),(2.9)∥f([x,y,z])-[f(x),f(y),f(z)]∥B≤θ(∥x∥Ar+∥y∥Ar+∥z∥Ar)
for all μ∈𝕋1 and all x,y,z,x1,…,xp,y1,…,yd∈A. Then there exists a unique C*-ternary algebra homomorphism H:A→B such that
(2.10)∥f(x)-H(x)∥B≤2r(p+d)θ|2(p+2d)r-(p+2d)2r|∥x∥Ar
for all x∈A.In the following theorem we have an alternative result of Theorem2.3.Theorem 2.4.
Letr, s, and θ be nonnegative real numbers such that 0<r<1, 0<s<3 (resp., r>1, s>3), and let d≥2. Suppose that f:A→B is a mapping with f(0)=0, satisfying (2.8) and
(2.11)∥f([x,y,z])-[f(x),f(y),f(z)]∥B≤θ(∥x∥As+∥y∥As+∥z∥As)
for all μ∈𝕋1 and all x,y,z∈A. Then there exists a unique C*-ternary algebra homomorphism H:A→B such that
(2.12)∥f(x)-H(x)∥B≤dθ2|d-dr|∥x∥Ar
for all x∈A.Proof.
We prove the theorem in two cases.Case 1.
0<r<1 and 0<s<3.
Lettingμ=1, x1=⋯=xp=0 and y1=⋯=yd=x in (2.8), we get
(2.13)∥f(dx)-df(x)∥B≤dθ2∥x∥Ar
for all x∈A. If we replace x by dnx in (2.13) and divide both sides of (2.13) to dn+1, we get
(2.14)∥1dn+1f(dn+1x)-1dnf(dnx)∥B≤θ2d(r-1)n∥x∥Ar
for all x∈A and all nonnegative integers n. Therefore,
(2.15)∥1dn+1f(dn+1x)-1dmf(dmx)∥B≤θ2∑i=mnd(r-1)i∥x∥Ar
for all x∈A and all nonnegative integers n≥m. From this it follows that the sequence {(1/dn)f(dnx)} is Cauchy for all x∈A. Since B is complete, the sequence {(1/dn)f(dnx)} converges. Thus one can define the mapping H:A→B by
(2.16)H(x):=limn→∞1dnf(dnx)
for all x∈A. Moreover, letting m=0 and passing the limit n→∞ in (2.15), we get (2.12). It follows from (2.8) that
(2.17)∥2H(∑j=1pμxj2+∑j=1dμyj)-∑j=1pμH(xj)-2∑j=1dμH(yj)∥B=limn→∞1dn∥2f(dn∑j=1pμxj2+dn∑j=1dμyj)-∑j=1pμf(dnxj)-2∑j=1dμf(dnyj)∥B≤limn→∞dnrdnθ(∑j=1p∥xj∥Ar+∑j=1d∥yj∥Ar)=0
for all μ∈𝕋1 and all x1,…,xp,y1,…,yd∈A. Hence
(2.18)2H(∑j=1pμxj2+∑j=1dμyj)=∑j=1pμH(xj)+2∑j=1dμH(yj)
for all μ∈𝕋1 and all x1,…,xp,y1,…yd∈A. So H(λx+μy)=λH(x)+μH(y) for all λ,μ∈𝕋1 and all x,y∈A. Therefore by Lemma 2.1 the mapping H:A→B is ℂ-linear.
It follows from Lemma2.2 and (2.11) that
(2.19)∥H([x,y,z])-[H(x),H(y),H(z)]∥B=limn→∞1d3n∥f([dnx,dny,dnz])-[f(dnx),f(dny),f(dnz)]∥B=θlimn→∞dnsd3n(∥x∥As+∥y∥As+∥z∥As)=0
for all x,y,z∈A. Thus
(2.20)H([x,y,z])=[H(x),H(y),H(z)]
for all x,y,z∈A. Therefore the mapping H is a C*-ternary algebra homomorphism.
Now letT:A→B be another C*-ternary algebra homomorphism satisfying (2.12). Then we have
(2.21)∥H(x)-T(x)∥B=limn→∞1dn∥f(dnx)-T(dnx)∥B≤dθ2|d-dr|limn→∞dnrdn∥x∥Ar=0
for all x∈A. So we can conclude that H(x)=T(x) for all x∈A. This proves the uniqueness of H. Thus the mapping H:A→B is a unique C*-ternary algebra homomorphism satisfying (2.12), as desired.Case 2.
r>1 and s>3.
Similar to the proof of Case1, we conclude that the sequence {dnf(d-nx)} is a Cauchy sequence in B. So we can define the mapping H:A→B by
(2.22)H(x):=limn→∞dnf(d-nx)
for all x∈A. The rest of the proof is similar to the proof of Case 1.Theorem 2.5 (see [31]).
Letr and θ be nonnegative real numbers such that r∉[1/(p+d),1], and let f:A→B be a mapping such that
(2.23)∥Cμf(x1,…,xp,y1,…,yd)∥B≤θ∏j=1p∥xj∥Ar·∏j=1d∥yj∥Ar,(2.24)∥f([x,y,z])-[f(x),f(y),f(z)]∥B≤θ∥x∥Ar∥y∥Ar∥z∥Ar
for all μ∈𝕋1 and all x,y,z,x1,…,xp,y1,…,yd∈A. Then there exists a unique C*-ternary algebra homomorphism H:A→B such that
(2.25)∥f(x)-H(x)∥B≤2(p+d)rθ|2(p+2d)(p+d)r-2(p+d)r(p+2d)|∥x∥A(p+d)r
for all x∈A.The following theorem shows that the mappingf:A→B in Theorem 2.5 is a C*-ternary algebra homomorphism when r>0.Theorem 2.6.
Letr,s,q,r1,…,rp,s1,…,sd, and θ be nonnegative real numbers such that r+s+q≠3 and rk>0(sk>0) for some 1≤k≤p, p≥2(1≤k≤d,d≥2).
Letf:A→B be a mapping satisfying
(2.26)∥Cμf(x1,…,xp,y1,…,yd)∥B≤θ∏j=1p∥xj∥Arj·∏j=1d∥yj∥Asj,(2.27)∥f([x,y,z])-[f(x),f(y),f(z)]∥B≤θ∥x∥Ar∥y∥As∥z∥Aq
for all μ∈𝕋1 and all x,y,z,x1,…,xp,y1,…,yd∈A. Then the mapping f:A→B is a C*-ternary algebra homomorphism. (We put ∥·∥A0=1).Proof.
Letrk>0 for some 1≤k≤p (we have similar proof when sk>0 for some 1≤k≤d). We now assume, without loss of generality, that r1>0. Letting x1=⋯=xp=y1=⋯=yd=0 in (2.26), we get that f(0)=0. Letting x2=2x and x1=x3=⋯=xp=y1=⋯=yd=0 in (2.26), we get
(2.28)μf(2x)=2f(μx)
for all μ∈𝕋1 and all x∈A. Setting μ=1 in (2.28), we get that f(2x)=2f(x) for all x∈A. Therefore,
(2.29)f(μx)=μf(x),f(2μx)=2μf(x)
for all μ∈𝕋1 and all x∈A. If we put x2=2x and y1=y and x1=x3=⋯=xp=y2=⋯=yd=0 in (2.26), we get
(2.30)2f(μx+μy)=μf(2x)+2μf(y)
for all μ∈𝕋1 and all x∈A. It follows from (2.29) and (2.30) that
(2.31)f(μx+λy)=μf(x)+λf(y)
for all λ,μ∈𝕋1 and all x,y∈A. Therefore, by Lemma 2.1 the mapping f:A→B is ℂ-linear. Let r+s+q>3. Then it follows from (2.27) that
(2.32)∥f([x,y,z])-[f(x),f(y),f(z)]∥B=limn→∞8n∥f([x2n,y2n,z2n])-[f(x2n),f(y2n),f(z2n)]∥B≤θ∥x∥Ar∥y∥As∥z∥Aqlimn→∞(82r+s+q)n=0
for all x,y,z∈A. Therefore,
(2.33)f([x,y,z])=[f(x),f(y),f(z)]
for all x,y,z∈A. Similarly, for r+s+q<3, we get (2.33).In the rest of this section, assume thatA is a unital C*-ternary algebra with norm ∥·∥A and unit e, and that B is a unital C*-ternary algebra with norm ∥·∥B and unit e'.We investigate homomorphisms inC*-ternary algebras associated with the functional equation Cμf(x1,…,xp,y1,…,yd)=0.Theorem 2.7 (see [31]).
Letr>1(r<1) and θ be nonnegative real numbers, and let f:A→B be a bijective mapping satisfying (2.8) such that
(2.34)f([x,y,z])=[f(x),f(y),f(z)]
for all x,y,z∈A. If limn→∞((p+2d)n/2n)f(2ne/(p+2d)n)=e'(limn→∞(2n/(p+2d)n)f((p+2d)n/2n)e=e'), then the mapping f:A→B is a C*-ternary algebra isomorphism.In the following theorems we have alternative results of Theorem2.7.Theorem 2.8.
Letr<1,s<2 and θ be nonnegative real numbers, and let f:A→B be a mapping satisfying (2.8) and (2.11). If there exist a real number λ>1(0<λ<1) and an element x0∈A such that limn→∞(1/λn)f(λnx0)=e'(limn→∞λnf(x0/λn)=e'), then the mapping f:A→B is a C*-ternary algebra homomorphism.Proof.
By using the proof of Theorem2.4, there exists a unique C*-ternary algebra homomorphism H:A→B satisfying (2.12). It follows from (2.12) that
(2.35)H(x)=limn→∞1λnf(λnx),(H(x)=limn→∞λnf(xλn))
for all x∈A and all real numbers λ>1(0<λ<1). Therefore, by the assumption we get that H(x0)=e'. Let λ>1 and limn→∞(1/λn)f(λnx0)=e'. It follows from (2.11) that
(2.36)∥[H(x),H(y),H(z)]-[H(x),H(y),f(z)]∥B=∥H[x,y,z]-[H(x),H(y),f(z)]∥B=limn→∞1λ2n∥f([λnx,λny,z])-[f(λnx),f(λny),f(z)]∥B≤θlimn→∞1λ2n(λns∥x∥As+λns∥y∥As+∥z∥As)=0
for all x∈A. So [H(x),H(y),H(z)]=[H(x),H(y),f(z)] for all x,y,z∈A. Letting x=y=x0 in the last equality, we get f(z)=H(z) for all z∈A. Similarly, one can shows that H(x)=f(x) for all x∈A when 0<λ<1 and limn→∞λnf(x0/λn)=e'. Therefore, the mapping f:A→B is a C*-ternary algebra homomorphism.


## 3. Derivations onC*-Ternary Algebras

Throughout this section, assume thatA is a C*-ternary algebra with norm ∥·∥A.Park [31] proved the Hyers-Ulam-Rassias stability and Ulam-Găvruţa-Rassias stability of derivations on C*-ternary algebras for the following functional equation:
(3.1)Cμf(x1,…,xp,y1,…,yd)=0.For a given mappingf:A→A, let
(3.2)Df(x,y,z)=f([x,y,z])-[f(x),y,z]-[x,f(y),z]-[x,y,f(z)]
for all x,y,z∈A.Theorem 3.1 (see [31]).
Letr and θ be nonnegative real numbers such that r∉[1,3], and let f:A→A a mapping satisfying (2.8) and
(3.3)∥Df(x,y,z)∥A≤θ(∥x∥Ar+∥y∥Ar+∥z∥Ar)
for all x,y,z∈A. Then there exists a unique C*-ternary derivation δ:A→A such that
(3.4)∥f(x)-δ(x)∥A≤2r(p+d)|2(p+2d)r-(p+2d)2r|θ∥x∥Ar
for all x∈A.Theorem 3.2 (see [31]).
Letr and θ be nonnegative real numbers such that r∉[1/(p+d),1], and let f:A→A be a mapping satisfying (2.23) and
(3.5)∥Df(x,y,z)∥A≤θ∥x∥Ar∥y∥Ar∥z∥Ar
for all x,y,z∈A. Then there exists a unique C*-ternary derivation δ:A→A such that
(3.6)∥f(x)-δ(x)∥A≤2(p+d)r|2(p+2d)(p+d)r-(p+2d)2(p+d)r|θ∥x∥A(p+d)r
for all x∈A.In the following theorems we generalize and improve the results in Theorems 3.1 and 3.2.Theorem 3.3.
Letφ:Ap+d→[0,∞) and ψ:A3→[0,∞) be functions such that
(3.7)φ̃(x):=∑n=0∞γ-nφ(γnx,…,γnx)<∞,(3.8)limn→∞γ-nφ(γnx1,…,γnxp,γny1,…,γnyd)=0,(3.9)limn→∞γ-3nψ(γnx,γny,γnz)=0,limn→∞γ-2nψ(γnx,γny,z)=0
for all x,y,z,x1,…,xp,y1,…,yd∈A where γ=(p+2d)/2. Suppose that f:A→A is a mapping satisfying
(3.10)∥Cμf(x1,…,xp,y1,…,yd)∥A≤φ(x1,…,xp,y1,…,yd),(3.11)∥Df(x,y,z)∥A≤ψ(x,y,z)
for all μ∈𝕋1 and all x,y,z,x1,…,xp,y1,…,yd∈A. Then the mapping f:A→A is a C*-ternary derivation.Proof.
Let us assumeμ=1 and x1=⋯=xp=y1=⋯=yd=x in (3.10). Then we get
(3.12)∥2f(p+2d2x)-(p+2d)f(x)∥A≤φ(x,…,x)
for all x∈A. If we replace x in (3.12) by γnx and divide both sides of (3.12) to γn+1, then we get
(3.13)∥1γn+1f(γn+1x)-1γnf(γnx)∥A≤12γn+1φ(γnx,…,γnx)
for all x∈A and all integers n≥0. Hence
(3.14)∥1γn+1f(γn+1x)-1γmf(γmx)∥A≤12γ∑i=mn1γiφ(γix,…,γix)
for all x∈A and all integers n≥m≥0. From this it follows that the sequence {(1/γn)f(γnx)} is Cauchy for all x∈A. Since A is complete, the sequence {(1/γn)f(γnx)} converges. Thus we can define the mapping δ:A→A by
(3.15)δ(x):=limn→∞1γnf(γnx)
for all x∈A. Moreover, letting m=0 and passing the limit n→∞ in (3.14), we get
(3.16)∥δ(x)-f(x)∥A≤12γφ̃(x)
for all x∈A. It follows from (3.8) and (3.10) that
(3.17)∥Cμδ(x1,…,xp,y1,…,yd)∥A=limn→∞1γn∥Cμf(γnx1,…,γnxp,γny1,…,γnyd)∥A≤limn→∞1γnφ(γnx1,…,γnxp,γny1,…,γnyd)=0
for all μ∈𝕋1 and all x,y,z,x1,…,xp,y1,…,yd∈A. Hence
(3.18)2δ(∑j=1pμxj2+∑j=1dμyj)=∑j=1pμδ(xj)+2∑j=1dμδ(yj)
for all μ∈𝕋1 and all x1,…,xp,y1,…,yd∈A. So δ(λx+μy)=λδ(x)+μδ(y) for all λ,μ∈𝕋1 and all x,y∈A. Therefore, by Lemma 2.1 the mapping δ:A→A is ℂ-linear.
It follows from (3.9) and (3.11) that
(3.19)∥Dδ(x,y,z)∥A=limn→∞1γ3n∥Df(γnx,γny,γnz)∥A≤limn→∞1γ3nψ(γnx,γny,γnz)=0
for all x,y,z∈A. Hence
(3.20)δ([x,y,z])=[δ(x),y,z]+[x,δ(y),z]+[x,y,δ(z)]
for all x,y,z∈A. So the mapping δ:A→A is a C*-ternary derivation.
It follows from (3.9) and (3.11)
(3.21)∥δ[x,y,z]-[δ(x),y,z]-[x,δ(y),z]-[x,y,f(z)]∥A=limn→∞1γ2n∥f[γnx,γny,z]-[f(γnx),γny,z]-[γnx,f(γny),z]-[γnx,γny,f(z)]∥A≤limn→∞1γ2nψ(γnx,γny,z)=0
for all x,y,z∈A. Thus
(3.22)δ[x,y,z]=[δ(x),y,z]+[x,δ(y),z]+[x,y,f(z)]
for all x,y,z∈A. Hence we get from (3.20) and (3.22) that
(3.23)[x,y,δ(z)]=[x,y,f(z)]
for all x,y,z∈A. Letting x=y=f(z)-δ(z) in (3.23), we get
(3.24)∥f(z)-δ(z)∥A3=∥[f(z)-δ(z),f(z)-δ(z),f(z)-δ(z)]∥A=0
for all z∈A. Hence f(z)=δ(z) for all z∈A. So the mapping f:A→A is a C*-ternary derivation, as desired.Corollary 3.4.
Letr<1,s<2, and θ be nonnegative real numbers, and let f:A→A be a mapping satisfying (2.8) and
(3.25)∥Df(x,y,z)∥A≤θ(∥x∥As+∥y∥As+∥z∥As)
for all x,y,z∈A. Then the mapping f:A→A is a C*-ternary derivation.Proof.
Define(3.26)φ(x1,…,xp,y1,…,yd)=θ(∑j=1p‍∥xj∥Ar+∑j=1d‍∥yj∥Ar),ψ(x,y,z)=θ(∥x∥As+∥y∥As+∥z∥As)
for all x,y,z,x1,…,xp,y1,…,yd∈A, and apply Theorem 3.3.Corollary 3.5.
Letr,s, and θ be nonnegative real numbers such that s,r(p+d)<1, and let f:A→A be a mapping satisfying (2.23) and
(3.27)∥Df(x,y,z)∥A≤θ∥x∥As∥y∥As∥z∥As
for all x,y,z∈A. Then the mapping f:A→A is a C*-ternary derivation.Proof.
Define(3.28)φ(x1,…,xp,y1,…,yd)=θ∏j=1p∥xj∥Ar∏j=1d∥yj∥Ar,ψ(x,y,z)=θ∥x∥As∥y∥As∥z∥As
for all x,y,z,x1,…,xp,y1,…,yd∈A, and apply Theorem 3.3.Theorem 3.6.
Letφ:Ap+d→[0,∞) and ψ:A3→[0,∞) be functions such that
(3.29)φ̃(x):=∑n=1∞γnφ(xγn,…,xγn)<∞,limn→∞γnφ(x1γn,…,xpγn,y1γn,…,ydγn)=0,limn→∞γ3nψ(xγn,yγn,zγn)=0,limn→∞γ2nψ(xγn,yγn,z)=0
for all x,y,z,x1,…,xp,y1,…,yd∈A where γ=(p+2d)/2. Suppose that f:A→A is a mapping satisfying (3.10) and (3.11). Then the mapping f:A→A is a C*-ternary derivation.Proof.
If we replacex in (3.12) by x/γn+1 and multiply both sides of (3.12) by γn, then we get
(3.30)∥γn+1f(xγn+1)-γnf(xγn)∥A≤γn2φ(xγn+1,…,xγn+1)
for all x∈A and all integers n≥0. Hence
(3.31)∥γn+1f(xγn+1)-γmf(xγm)∥A≤12γ∑i=m+1n+1γiφ(xγi,…,xγi)
for all x∈A and all integers n≥m≥0. From this it follows that the sequence {γnf(x/γn)} is Cauchy for all x∈A. Since A is complete, the sequence {γnf(x/γn)} converges. Thus we can define the mapping δ:A→A by
(3.32)δ(x):=limn→∞γnf(xγn)
for all x∈A. The rest of the proof is similar to the proof of Theorem 3.3, and we omit it.Corollary.
Letr,s, and θ be nonnegative real numbers such that s,r(p+d)>1, and let f:A→A be a mapping satisfying (2.23) and (3.27). Then the mapping f:A→A is a C*-ternary derivation.



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*Source: 612392-2009-06-14.xml*