# Jordan∗-Derivations on C∗-Algebras and JC∗-Algebras

Authors: Jong Su An; Jianlian Cui; Choonkil Park
Journal: Abstract and Applied Analysis
 (2008)

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## Abstract

We investigate Jordan∗-derivations on C∗-algebras and Jordan ∗-derivations on JC∗-algebras associated with the following functional inequality ∥f(x)+f(y)+kf(z)∥≤∥kf((x+y)/k+z)∥ for some integer k greater than 1. We moreover prove the generalized Hyers-Ulam stability of Jordan ∗-derivations on C∗-algebras and of Jordan ∗-derivations on JC∗-algebras associated with the following functional equation f((x+y)/k+z)=(f(x)+f(y))/k+f(z) for some integer k greater than 1.


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## Body

## 1. Introduction and preliminaries

The stability problem of functional equations 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 Th. M. Rassias [4] for linear mappings by
considering an unbounded Cauchy difference. In 1982–1994, a generalization of
Hyers-Ulam stability result was proved by J. M. Rassias [5–9]. This author assumed that
Cauchy-Găvruţa-Rassias inequality(1.1)∥f(x+y)−f(x)−f(y)∥≤θ∥x∥p⋅∥y∥qis
controlled by a product of different powers of norms, where θ≥0 and p,q∈ℝ such that r=p+q≠1,
and retained the condition of continuity of f(tx) in t∈ℝ for each fixed x.
In 1999, Găvruţa [10] studied the singular case r=p+q=1,
by constructing a nice counterexample to the above pertinent Ulam stability
problem. Also J. M. Rassias [5–9, 11–13] investigated other conditions and still obtained
stability results. In all these cases, the approach to the existence question
was to prove asymptotic type formulas of the form(1.2)L(x)=limn→∞⁢f(2nx)2norL(x)=limn→∞⁢2nf(x2n).Theorem 1.1  (see [5–7, 9]).
LetX be a real normed vector space and Y a real complete normed vector space. Assume in
addition that f:X→Y is an approximately additive mapping for which
there exist constants θ≥0 and p,q∈ℝ such that r=p+q≠1 and f satisfies(1.3)∥f(x+y)−f(x)−f(y)∥≤θ∥x∥p⋅∥y∥qfor all x,y∈X.
Then, there exists a unique additive mapping L:X→Y satisfying(1.4)∥f(x)−L(x)∥≤θ|2−2r|∥x∥rfor 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 an ℝ-linear mapping.Theorem 1.2  (see [4]).
Letf:E→E′ be a mapping from a normed vector space E into a Banach space E′ subject to the inequality(1.5)∥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(1.6)L(x)=limn→∞⁢f(2nx)2nexists for all x∈E,
and L:E→E′ is the unique additive mapping which
satisfies(1.7)∥f(x)−L(x)∥≤2ε2−2p∥x∥pfor all x∈E.
Also, if for each x∈E the mapping f(tx) is continuous in t∈ℝ,
then L is linear.Th. M. Rassias [14], during the 27th International Symposium on
Functional Equations, asked the question whether such a theorem can also be
proved for p≥1.
Gajda [15] following
the same approach as in Th. M. Rassias [4] gave an affirmative solution to this question for p>1.
It was shown by Gajda [15]
as well as by Th. M. Rassias and Šemrl [16] that one cannot prove a
Th. M. Rassias’ type theorem when p=1.
The counterexamples of Gajda [15] as well as of Th. M. Rassias and  Šemrl [16] have stimulated several
mathematicians to invent new definitions of approximately additive or approximately linear mappings, compare
Găvruţa [17] and Jung [18], who among others studied
the Hyers-Ulam stability of functional equations. Theorem 1.2 that was
introduced for the first time by Th. M. Rassias [4] provided a lot of influence in the development of a
generalization of the Hyers-Ulam stability concept. This new concept is known
as generalized Hyers-Ulam stability of functional equations (cf. the books of Czerwik [19], Hyers et al. [20]).Găvruţa [17] provided a further generalization of Th. M. Rassias’
theorem. Isac and Th. M. Rassias [21] applied the Hyers-Ulam stability theory to prove
fixed point theorems and study some new applications in nonlinear analysis. In
[22], Hyers et al. studied the asymptoticity aspect of Hyers-Ulam stability of
mappings. Beginning around the year 1980, the topic of approximate
homomorphisms and their stability theory in the field of functional equations
and inequalities was taken up by several mathematicians (see [3–18, 21–51]).Gilányi [26] showed that if f satisfies the functional
inequality(1.8)∥2f(x)+2f(y)−f(x−y)∥≤∥f(x+y)∥,then f satisfies the Jordan-von Neumann functional
equation(1.9)2f(x)+2f(y)=f(x+y)+f(x−y).See also [52]. Fechner [53] and Gilányi [27] proved the generalized
Hyers-Ulam stability of the functional inequality  (1.8). Park et al. [42] introduced and investigated
3-variable Cauchy-Jensen functional inequalities and proved the generalized
Hyers-Ulam stability of the 3-variable Cauchy-Jensen functional inequalities.Definition 1.3.
LetA be a C*-algebra. A ℂ-linear mapping δ:A→A is called a
Jordan *-derivation if (1.10)δ(a2)=aδ(a)+δ(a*)a*for all a∈A.AC*-algebra A,
endowed with the Jordan product a∘b:=(ab+ba)/2 on A,
is called a JC*-algebra (see [38, 39]).Definition 1.4.
LetA be a JC*-algebra. A ℂ-linear mapping δ:A→A is called a
Jordan *-derivation if (1.11)δ(a2)=a∘δ(a)+δ(a*)∘a*for all a∈A.This paper is organized as follows. In Section2, we
investigate Jordan *-derivations on C*-algebras associated with the functional
inequality(1.12)∥f(x)+f(y)+kf(z)∥≤∥kf(x+yk+z)∥,and prove the generalized
Hyers-Ulam stability of Jordan *-derivations on C*-algebras associated with the functional
equation(1.13)f(x+yk+z)=f(x)+f(y)k+f(z).In Section3, we investigate Jordan *-derivations on JC*-algebras associated with the functional
inequality  (1.12), and prove the generalized
Hyers-Ulam stability of Jordan *-derivations on JC*-algebras associated with the functional
equation  (1.13).Throughout this paper, letk be an integer greater than 1.


## 2. Jordan*-derivations on C*-algebras

Throughout this section, assume thatA is a C*-algebra with norm ∥·∥.Lemma 2.1.
Letf:A→A be a mapping such that (2.1)∥f(a)+f(b)+kf(c)∥≤∥kf(a+bk+c)∥for all a,b,c∈A.
Then, f is Cauchy additive, that is, f(a+b)=f(a)+f(b).Proof.
Lettinga=b=c=0 in
(2.1), we get(2.2)∥(k+2)f(0)∥≤∥kf(0)∥.So f(0)=0.
Lettingc=0 and b=−a in
(2.1), we get(2.3)∥f(a)+f(−a)∥≤∥kf(0)∥=0for all a∈A.
Hence f(−a)=−f(a) for all a∈A.
Lettingc=−(a+b)/k in
(2.1), we get(2.4)∥f(a)+f(b)−kf(a+bk)∥=∥f(a)+f(b)+kf(−a+bk)∥≤∥kf(0)∥=0for all a,b∈A.
Thus(2.5)kf(a+bk)=f(a)+f(b)for all a,b∈A.
Letting b=0 in
(2.5), we get kf(a/k)=f(a) for all a∈A.
So(2.6)f(a+b)=kf(a+bk)=f(a)+f(b)for all a,b∈A,
as desired.Theorem 2.2.
Letr>1 and θ be a nonnegative real number, and let f:A→A be a mapping such that(2.7)∥μf(a)+f(b)+kf(c)∥≤∥kf(μa+bk+c)∥,(2.8)∥f(a2)−af(a)−f(a*)a*∥≤θ∥a∥2r for all μ∈𝕋:={λ∈ℂ∣|λ|=1} and all a,b,c∈A.
Then, the mapping f:A→A is a Jordan *-derivation.Proof.
Letμ=1 in
(2.7). By Lemma 2.1, the mapping f:A→A is Cauchy additive. So f(0)=0 and f(x)=limn→∞2nf(a/2n) for all a∈A.
Lettingb=−μa and c=0,
we get(2.9)∥μf(a)+f(−μa)∥≤∥kf(0)∥=0for all a∈A and all μ∈𝕋.
So(2.10)μf(a)−f(μa)=μf(a)+f(−μa)=0for all a∈A,
and all μ∈𝕋.
Hence, f(μa)=μf(a) for all a∈A and all μ∈𝕋.
By the same reasoning as in the proof of [39, Theorem 2.1], the mapping f:A→A is ℂ-linear.
It follows from
(2.8) that(2.11)∥f(a2)−af(a)−f(a*)a*∥=limn→∞⁢4n∥f(a22n⋅2n)−a2nf(a2n)−f(a*2n)a*2n∥≤limn→∞⁢4nθ4nr∥a∥2r=0for all a∈A.
Thus(2.12)f(a2)=af(a)+f(a*)a*for all a∈A.
Hence the mappingf:A→A is a Jordan *-derivation.Theorem 2.3.
Letr<1 and θ be a nonnegative real number, and let f:A→A be a mapping satisfying  (2.7) and
(2.8). Then, the mapping f:A→A is a Jordan *-derivation.Proof.
The proof is similar to the proof of
Theorem2.2.We prove the generalized Hyers-Ulam stability of
Jordan*-derivations on C*-algebras.Theorem 2.4.
Suppose thatf:A→A is a mapping with f(0)=0 for which there exists a function φ:A×A×A→[0,∞) such that(2.13)limn→∞⁢k−nφ(kna,knb,knc)=0,(2.14)φ˜(a):=∑n=1∞k−n+1φ(kna,0,0)<∞,(2.15)∥f(λa+λbk+c2)−λkf(a)−λkf(b)−cf(c)−f(c*)c*∥≤φ(a,b,c)for all λ∈𝕋,
and all a,b,c∈A.
Then, there exists a unique Jordan *-derivation δ:A→A such that(2.16)∥f(a)−δ(a)∥≤φ˜(a)for all a∈A.Proof.
Puttingb=c=0 and λ=1,
and replacing a by ka in
(2.15), we get(2.17)∥f(a)−f(ka)k∥≤φ(ka,0,0)(∀a∈A).
Using the induction method, we have(2.18)∥f(kna)kn−f(kma)km∥≤∑j=m+1nk−j+1φ(kja,0,0)for all n>m≥0 and all a∈A.
It follows that for every a∈A,
the sequence {f(kna)/kn} is Cauchy, and hence it is convergent since A is complete. Set(2.19)δ(a):=limn→∞⁢f(kna)kn(a∈A).Let c=0 and replace a and b by kna and knb,
respectively, in  (2.15), we get(2.20)∥k−nf(knλa+λbk)−λkk−nf(kna)−λkk−nf(knb)∥≤k−nφ(kna,knb,0).Taking the limit as n→∞,
we obtain(2.21)δ(λa+λbk)=λkδ(a)+λkδ(b)for all a,b∈A and all λ∈𝕋.
Letting b=0 and λ=1 in
(2.21), we get(2.22)δ(ak)=1kδ(a)for all a∈A.
Hence,(2.23)λδ(a)+λδ(b)=kδ(λa+λbk)=δ(λa+λb)for all a,b∈A,
in particular, δ:A→A is additive. Now similar to the discussion in
[54, Theorem 2.1], we
show that δ:A→A is ℂ-linear. Letting m=0 in
(2.18), we get(2.24)∥f(a)−f(kna)kn∥≤∑j=1nk−j+1φ(kja,0,0).Taking the limit as n→∞,
we have(2.25)∥f(a)−δ(a)∥≤φ˜(a)for all a∈A.
It follows from [55]
that δ:A→A is unique.
Lettingλ=1, a=b=0 and replacing c by knc in
(2.15), we obtain(2.26)∥f(k2n(c2))−kncf(knc)−knf(knc*)c*∥≤φ(0,0,knc).So(2.27)∥k−2nf(k2n(c2))−k−ncf(knc)−k−nf(knc*)c*∥≤k−2nφ(0,0,knc)for all c∈A.
Letting n tend to infinity, we have(2.28)δ(c2)=cδ(c)+δ(c*)c*for all c∈A.
Hence δ:A→A is a Jordan *-derivation.Corollary 2.5.
Suppose thatf:A→A is a mapping with f(0)=0 for which there exist constant β≥0 and p1,p2,p3∈(−∞,1) such that(2.29)∥f(λa+λbk+c2)−λkf(a)−λkf(b)−cf(c)−f(c*)c*∥≤β(∥a∥p1+∥b∥p2+∥c∥p3)for all λ∈𝕋 and all a,b,c∈A.
Then, there exists a unique Jordan *-derivation δ:A→A such that(2.30)∥f(a)−δ(a)∥≤kβ∥a∥p1k1−p1−1for all a∈A.Proof.
Lettingφ(a,b,c):=β(∥a∥p1+∥b∥p2+∥c∥p3) in Theorem 2.4, we obtain the result.Theorem 2.6.
Suppose thatf:A→A is a mapping with f(0)=0 for which there exists a function φ:A×A×A→[0,∞) satisfying
(2.15) such that(2.31)limn→∞⁢knφ(k−na,k−nb,k−nc)=0,φ˜(a):=∑n=1∞kn+1φ(k−na,0,0)<∞for all a,b,c∈A.
Then, there exists a unique Jordan *-derivation δ:A→A such that(2.32)∥f(a)−δ(a)∥≤φ˜(a)for all a∈A.Proof.
Lettingb=c=0 and λ=1 in
(2.15), we get(2.33)∥kf(ak)−f(a)∥≤kφ(a,0,0)(∀a∈A).One can apply the induction
method to prove that(2.34)∥knf(k−na)−kmf(k−mb)∥≤∑j=mn−1kj+1φ(k−ja,0,0)for all n>m≥0 and a∈A.
It follows that for every a∈A,
the sequence {knf(k−na)} is Cauchy, and hence it is convergent since A is complete. Set(2.35)δ(a):=limn→∞⁢knf(k−na)(a∈A).Letting c=0 and replacing a and b by k−na and k−nb,
respectively, in  (2.15), we
get(2.36)∥knf(k−n(a+b)k)−knkf(k−na)−knkf(k−nb)∥≤knφ(k−na,k−nb,0)for all a,b∈A.
Taking the limit as n→∞,
we obtain(2.37)δ(a+bk)=δ(a)k+δ(b)kfor all a,b∈A.
Hence δ:A→A is additive.
The rest of the proof is similar to the proof of
Theorem2.4.Corollary 2.7.
Suppose thatf:A→A is a mapping with f(0)=0 for which there exist constant β≥0 and p1,p2,p3∈(1,∞) such that(2.38)∥f(λa+λbk+c2)−λkf(a)−λkf(b)−cf(c)−f(c*)c*∥≤β(∥a∥p1+∥b∥p2+∥c∥p3)for all λ∈𝕋 and all a,b,c∈A.
Then, there exists a unique Jordan *-derivation δ:A→A such that(2.39)∥f(a)−δ(a)∥≤kβ∥a∥p1k1−p1−1for all a∈A.Proof.
Lettingφ(a,b,c):=β(∥a∥p1+∥b∥p2+∥c∥p3) in Theorem 2.6, we obtain the result.


## 3. Jordan*-derivations on JC*-algebras

Throughout this section, assume thatA is a JC*-algebra with norm ∥·∥.Theorem 3.1.
Letr>1 and θ be a nonnegative real number, and let f:A→A be a mapping satisfying  (2.7) such that(3.1)∥f(a2)−a∘f(a)−f(a*)∘a*∥≤θ∥a∥2rfor all a∈A.
Then, the mapping f:A→A is a Jordan *-derivation.Proof.
By the same reasoning as in the
proof of Theorem2.2, the mapping f:A→A is ℂ-linear.
It follows from
(3.1) that(3.2)∥f(a2)−a∘f(a)−f(a*)∘a*∥=limn→∞⁢4n∥f(a22n⋅2n)−a2n∘f(a2n)−f(a*2n)∘a*2n∥≤limn→∞⁢4nθ4nr∥a∥2r=0for all a∈A.
Thus,(3.3)f(a2)=a∘f(a)+f(a*)∘a*for all a∈A.
Hence, the mappingf:A→A is a Jordan *-derivation.Theorem 3.2.
Letr<1 and θ be a nonnegative real number, and let f:A→A be a mapping satisfying  (2.7) and
(3.1). Then, the mapping f:A→A is a Jordan *-derivation.Proof.
The proof is similar to the proofs
of Theorems2.2 and 3.1.We prove the generalized Hyers-Ulam stability of
Jordan*-derivations on JC*-algebras.Theorem 3.3.
Suppose thatf:A→A is a mapping with f(0)=0 for which there exists a function φ:A×A×A→[0,∞) satisfying
(2.13) and  (2.14) such
that(3.4)∥f(λa+λbk+c2)−λkf(a)−λkf(b)−c∘f(c)−f(c*)∘c*∥≤φ(a,b,c)for all λ∈𝕋 and all a,b,c∈A.
Then, there exists a unique Jordan *-derivation δ:A→A such that(3.5)∥f(a)−δ(a)∥≤φ˜(a)for all a∈A.Proof.
By the same reasoning as in the
proof of Theorem2.4, there exists a unique ℂ-linear mapping δ:A→A such that(3.6)∥f(a)−δ(a)∥≤φ˜(a)for all a∈A.
The mapping δ:A→A is given by(3.7)δ(a):=limn→∞⁢f(kna)kn(a∈A).
Lettingλ=1, a=b=0 and replacing c by knc in
(3.4), we obtain(3.8)∥f(k2nc2)−knc∘f(knc)−knf(knc*)∘c*∥≤φ(0,0,knc).So(3.9)∥k−2nf(k2nc2)−k−nc∘f(knc)−k−nf(knc*)∘c*∥≤k−2nφ(0,0,knc)for all c∈A.
Letting n tend to infinity, we have(3.10)δ(c2)=c∘δ(c)+δ(c*)∘c*for all c∈A.
Hence, δ:A→A is a Jordan *-derivation.Corollary 3.4.
Suppose thatf:A→A is a mapping with f(0)=0 for which there exist constant β≥0 and p1,p2,p3∈(−∞,1) such that(3.11)∥f(λa+λbk+c2)−λkf(a)−λkf(b)−c∘f(c)−f(c*)∘c*∥≤β(∥a∥p1+∥b∥p2+∥c∥p3)for all λ∈𝕋 and all a,b,c∈A.
Then, there exists a unique Jordan *-derivation δ:A→A such that(3.12)∥f(a)−δ(a)∥≤kβ∥a∥p1k1−p1−1for all a∈A.Proof.
Lettingφ(a,b,c):=β(∥a∥p1+∥b∥p2+∥c∥p3) in Theorem 3.3, we obtain the result.Theorem 3.5.
Suppose thatf:A→A is a mapping with f(0)=0 for which there exists a function φ:A×A×A→[0,∞) satisfying
(2.31) and  (3.4). Then, there exists a unique Jordan *-derivation δ:A→A such that(3.13)∥f(a)−δ(a)∥≤φ˜(a)for all a∈A.Proof.
The rest of the proof is similar to
the proofs of Theorems2.4 and 3.3.Corollary 3.6.
Suppose thatf:A→A is a mapping with f(0)=0 for which there exist constant β≥0 and p1,p2,p3∈(1,∞) such that(3.14)∥f(λa+λbk+c2)−λkf(a)−λkf(b)−c∘f(c)−f(c*)∘c*∥≤β(∥a∥p1+∥b∥p2+∥c∥p3)for all λ∈𝕋 and all a,b,c∈A.
Then, there exists a unique Jordan *-derivation δ:A→A such that(3.15)∥f(a)−δ(a)∥≤kβ∥a∥p1k1−p1−1for all a∈A.Proof.
Lettingφ(a,b,c,d):=β(∥a∥p1+∥b∥p2+∥c∥p3) in Theorem 3.5, we obtain the result.



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