# On the Stability ofα,β,γ-Derivations and Lie C*-Algebra Homomorphisms on Lie C*-Algebras: A Fixed Points Method

Authors: Seong Sik Kim; Ga Ya Kim; Soo Hwan Kim
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

We investigate new generalized Hyers-Ulam stability results for(α,β,γ)-derivations and Lie C*-algebra homomorphisms on Lie C*-algebras associated with the additive functional equation: ∑i=1n‍fnxi+∑j=1,j≠in‍xj+f∑i=1n‍xi=2f∑i=1n‍nxi.


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## Body

## 1.   Introduction

The theory of finite dimensional complex Lie algebras is an important part of Lie theory. It has several applications in physics and connections with other parts of mathematics. With an increasing amount of theory and applications concerning Lie algebras of various dimensions, it is becoming necessary to ascertain which tools are applicable for handling them. The miscellaneous characteristics of Lie algebras constitute such tools and have also found applications: Casimir operators [1], derived, lower central and upper central sequences, and the Lie algebra of derivations, radicals, nilradicals, ideals, subalgebras [2, 3], and recently megaideals [4]. These characteristics are particularly crucial when considering possible affinities among Lie algebras. Physically motivated relations between two Lie algebras, namely, contractions and deformations, have been extensively studied [5]. Moreover, in modern industry, various analytical approaches for solving mathematical equations are widely applied in analysis of problems in packaging engineering, and so mathematical modeling and computation methods by using mathematical equations play an important role in application of packaging engineering. From now, we wish to note that mathematical equations for stability properties in this paper can have applications to Packaging Engineering.AC*-algebra 𝒜 endowed with the Lie product [x,y]=(xy-yx)/2 on 𝒜 is called a LieC*-algebra. A ℂ-linear mapping δ:𝒜→𝒜 is called a (α,β,γ)-derivation of 𝒜 if there exist α,β, and γ∈C such that αδ[x,y]=β[δ(x),y]+γ[x,δ(y)] for all x,y∈𝒜 [6]. Let 𝒜 and ℬ be Lie C*-algebras. A ℂ-linear mapping H:𝒜→ℬ is called a LieC*-algebra homomorphism if H([x,y])=[H(x),H(y)] for all x,y∈𝒜.The stability problem of functional equations was originated from a question of Ulam [7, 8] concerning the stability of group homomorphisms as follows.Let(G1,*)be a group and(G2,⋄,d)be a metric group with the metricd(·,·). Givenϵ>0, does there exist aδ(ϵ)>0such that if a mappingh:G1→G2satisfies the inequalityd(h(x*y),h(x)⋄h(y))<ϵfor allx,y∈G1, then there is a homomorphismH:G1→G2withd(h(x),H(x))<ϵfor allx∈G1?If the answer is affirmative, we would say that the equation of a homomorphismH(x*y)=H(x)⋄H(y) is stable (see also [9–14]). Cădariu and Radu [15] applied the fixed point method to investigation of the stability of a functional equation. In 2008, Novotný and Hrivnák [6] investigated generalizing the concept of Lie derivations via certain complex parameters, obtained various Lie, and established the structure and properties of (α,β,γ)-derivations of Lie algebras. Recently, the generalized Hyers-Ulam stability of problems on C*-algebras associated with functional equations has been investigated by using a fixed point method (see [16–20]). The following fixed point theorem will play an important role in proving our main theorem.Theorem 1 (see [21]).
Suppose that we are given a complete generalized metric space(Ω,d) and a strictly contractive mapping T:Ω→Ω with Lipschitz constant L. Then for each given x∈Ω, either d(Tnx,Tn+1x)=∞ for all nonnegative integers n≥0 or there exists a natural number no such that (1)
d(Tnx,Tn+1x)<∞ for all n≥no;(2)
the sequence{Tnx} is convergent to a fixed point y* of T;(3)
y* is the unique fixed point of T in the set Λ={y∈Ω:d(Tnox,y)<∞};(4)
d(y,y*)≤(1/(1-L))d(y,Ty) for all y∈Λ.Letk be a fixed positive integer. We recall that a mapping ρ:A→B having a domain A and a codomain (B,≤) that are both closed under addition is called a contractively subadditive mapping if there exists a constant L with 0<L<1 such that ρ(x+y)≤L(ρ(x)+ρ(y)) and an expansively superadditive mapping if there exists a constant L with 0<L<1 such that ρ(x+y)≥(1/L)(ρ(x)+ρ(y)) for all x,y∈A. A mapping ρ:A→B is called a homogeneous of degreek if ρ(λx)=λkρ(x) for all x,y∈A. Also, if there exists a constant L with 0<L<1 such that a mapping ρ:An→B satisfies ρ(x1,…,λℓx,…,xn)≤λℓLρ(x1,…,x,…,xn) for all x,y∈A and positive integer λ>1, then we say that ρ is a  n-contractively subhomogeneous mapping if ℓ=1, and ρ is an expansively superhomogeneous mapping if ℓ=-1.Now, we consider a mappingf:X→Y satisfying the following functional equation [22]:
(1)∑i=1n‍f(nxi+∑j=1,j≠in‍xj)+f(∑i=1nxi)=2f(∑i=1n‍nxi)
for all xi,xj∈X, where n∈ℤ+ is a fixed integer with n≥2. Note that the mapping f(x)=ax is a solution of the functional equation (1). Eskandni [22] investigated the Hyers-Ulam-Rassias stability of the functional equation (1) in quasi-Banach spaces using the direct method.In this paper, using some strategies from [15, 18, 22], we investigate new generalized Hyers-Ulam stability for (α,β,γ)-derivations and Lie C*-algebra homomorphisms on Lie C*-algebras associated with the functional equation (1) via the fixed point method.Throughout this paper, let𝒜 and ℬ be Lie C*-algebras and T1/no1={eiθ:0≤θ≤2π/no}. For any mapping f:𝒜→ℬ, we define
(2)𝒟μf(x1,…,xn)=∑i=1n‍f(nμxi+∑j=1,j≠in‍μxj)+f(∑i=1n‍μxi)-2f(∑i=1n‍nμxi),𝒟α,β,γf(x,y)=αf[x,y]-β[f(x),y]-γ[x,f(y)]
for all x1,…,xn,  x,y∈X  (n≥2),  μ∈T1/no1,  and some α,β,γ∈ℂ.


## 2. Main Results

In this section, we point out the stability of the functional equation (1) on Lie C*-algebras using a fixed point method. Let us recall that a mapping d:X2→[0,∞) is called a generalized metric on a nonempty set X if (i)  d(x,y)=0 if and only if x=y, (ii)  d(x,y)=d(y,x), and (iii)  d(x,z)≤d(x,y)+d(y,z) for all x,y, and z∈X. We need the following lemma to prove the main result in this paper.Lemma 2 (see [22]).
LetX,Y be linear spaces and n≥2 a fixed positive integer. A mapping f:X→Y satisfies the functional equation (1) if and only if f is additive.Theorem 3.
Assume that there exist a contractively subadditive mappingϕ:𝒜n→[0,∞) and a 2-contractively subhomogeneous mapping ψ:𝒜2→[0,∞) with a constant 0<L<1 such that a mapping f:𝒜→𝒜 satisfies
(3)∥𝒟μf(x1,…xn)∥≤ϕ(x1,…,xn),(4)∥𝒟α,β,γf(x,y)∥≤ψ(x,y)
for all x1,…,xn,  x,y∈𝒜, μ∈T1/no1, and some α,β, and γ∈ℂ. Then there exists a unique (α,β,γ)-derivation δ:𝒜→𝒜 which satisfies (1) and the inequality
(5)∥f(x)-δ(x)∥≤1n(1-L)ϕ(x,0,…,0)
for all x∈𝒜.Proof.
LetΩ be a set of all mappings from 𝒜 into 𝒜, and introduce a generalized metric on Ω as follows:
(6)d(g,h)=inf{C∈ℝ+:∥g(x)-h(x)∥≤Cϕ(x,0,…,0)∀x∈𝒜}.
Then (Ω,d) is a generalized complete metric space. Now, we consider the mapping T:Ω→Ω defined by
(7)Tg(x)=1ng(nx)
for all g∈Ω, x∈𝒜, and n∈ℤ+. Let g,h∈Ω and C∈(0,∞) be an arbitrary constant with d(g,h)≤C. Then we have ∥g(x)-h(x)∥≤Cϕ(x,0,…,0) and
(8)∥Tg(x)-Th(x)∥≤Cnϕ(nx,0,…,0)≤LCϕ(x,0,…,0)
for all x∈𝒜, which means that d(Tg,Th)≤Ld(g,h) for all g,h∈Ω. Thus T is a strictly contractive self-mapping on Ω with the Lipschitz constant L.
Lettingx1=x, x2=x3=⋯=xn=0, and μ=1 in (3), we have
(9)∥f(nx)-nf(x)∥≤ϕ(x,0,…,0),
which gives
(10)∥f(x)-1nf(nx)∥≤1nϕ(x,0,…,0)
for all x∈𝒜 and n∈ℤ+ with n≥2. Then we have d(Tf,f)≤1/n<∞. From Theorem 1, there is a mapping δ which is a unique fixed point of T in the set Ω1={g∈Ω:d(f,g)<∞} such that
(11)δ(x)=limm→∞1nmf(nmx)
for all x∈𝒜, since limm→∞d(Tmf,δ)=0. Again by Theorem 1, we have
(12)d(f,δ)≤11-Ld(Tf,f)≤1n(1-L)
for all x∈𝒜. Thus, inequality (5) holds.
It follows from (3), (11), and the contractively subadditive mapping of ϕ that
(13)∥𝒟μδ(x1,…,xn)∥≤limm→∞1nm∥𝒟μf(nmx1,…,nmxn)∥≤limm→∞Lmϕ(x1,…,xn)=0,
which gives 𝒟μδ(x1,…,xn)=0 for all x1,…,xn∈𝒜 and μ∈T1/no1. If we put μ=1 in 𝒟μδ(x1,…,xn)=0; then δ satisfies the functional equation (1), and so δ is additive by Lemma 2. Also, we let x1=x and x2=⋯=xn=0, then δ(μx)=μδ(x). By the same reasoning as that of the proof of Theorem 2.1 of [23], the mapping δ is ℂ-linear. So, it follows from the 2-contractively subhomogeneous of ψ, (4), and (11) that
(14)∥𝒟α,β,γδ(x,y)∥≤limm→∞1n2mψ(nmx,nmy)≤limm→∞L2mψ(x,y)=0
for all x,y∈𝒜 and some α,β, and γ∈ℂ. Then we have
(15)αδ[x,y]=β[δ(x),y]+γ[x,δ(y)]
for all x,y∈𝒜 and some α,β, and γ∈ℂ. Thus, δ is a unique Lie (α,β,γ)-derivation on Lie C*-algebra 𝒜 satisfying (5). This completes the proof.Corollary 4.
Let0<r<1,  0<s<2, and θ be nonnegative real numbers. Suppose that a mapping f:𝒜→𝒜 satisfies
(16)∥𝒟μf(x1,…xn)∥≤θ∑i=1n‍∥xi∥r,∥𝒟α,β,γf(x,y)∥≤θ(∥x∥s+∥y∥s)
for all x1,…,xn,  x,y∈𝒜,  μ∈T1/no1, and some α,β, and γ∈ℂ. Then there exists a unique (α,β,γ)-derivation δ:𝒜→𝒜 such that
(17)∥f(x)-δ(x)∥≤θn-nr∥x∥r,
for all x∈𝒜 and n∈ℤ+ with n≥2.Proof.
The proof follows from Theorem3 by taking ϕ(x1,…,xn)=θ∑i=1n‍∥xi∥r,  ψ(x,y)=θ(∥x∥s+∥y∥s) for all x1,…,xn,  x,y∈𝒜. Then we can choose L=nr-1, and we obtain the desired result. This completes the proof.Theorem 5.
Assume that there exist an expansively superadditive mappingϕ:𝒜n→[0,∞) and a 2-expansively superhomogeneous mapping ψ:𝒜2→[0,∞) with a constant 0<L<1 such that a mapping f:𝒜→𝒜 satisfies (3) and (4). Then there exists a unique (α,β,γ)-derivation δ:𝒜→𝒜 which satisfies (1) and the inequality
(18)∥f(x)-δ(x)∥≤Ln(1-L)ϕ(x,0,…,0)
for all x∈𝒜.Proof.
LetΩ and d be as in the proof of Theorem 3. Then (Ω,d) becomes a generalized complete metric space, and we consider the mapping T:Ω→Ω defined by (Tg)(x)=ng(x/n) for all g∈Ω and x∈𝒜. So, d(Tg,Th)≤Ld(g,h) for all g,h∈Ω. It follows from (9) that
(19)∥f(x)-nf(xn)∥≤Lnϕ(x,0,…,0)
for all x∈𝒜 and n∈ℤ+ with n≥2. Then, d(Tf,f)≤L/n<∞. From Theorem 1, there exists a unique mapping δ which is a unique fixed point of T in the set Ω1={g∈Ω:d(f,g)<∞} such that δ(x)=limm→∞nmf(x/nm) for all x∈𝒜. Thus, we have
(20)d(f,δ)≤11-Ld(Tf,f)≤Ln(1-L),
which implies that (18) holds. The remaining assertion goes through in similar method to the corresponding part of Theorem 3. This completes the proof.Corollary 6.
Letr>1,  s>2 and θ be nonnegative real numbers. Suppose that a mapping f:𝒜→𝒜 satisfies (16). Then there exists a unique (α,β,γ)-derivation δ:𝒜→𝒜 such that
(21)∥f(x)-δ(x)∥≤θnr-n∥x∥r,
for all x∈𝒜 and n∈ℤ+ with n≥2.Proof.
The proof follows from Theorem5 by taking ϕ(x1,…,xn)=θ∑i=1n‍∥xi∥r,  ψ(x,y)=θ(∥x∥s+∥y∥s) for all x1,…,xn,  x,y∈𝒜. Then we can choose L=n1-r, and we obtain the desired result. This completes the proof.Next, we establish another theorem about the stability of the functional equation (1).Theorem 7.
Assume that there exists a contractively subadditive mappingφ:𝒜n+2→[0,∞) with a constant L<1 such that a mapping f:𝒜→𝒜 satisfies
(22)∥𝒟μf(x1,…,xn)+𝒟α,β,γf(x,y)∥≤φ(x1,…,xn,x,y)
for all x1,…,xn,  x,y∈𝒜, μ∈T1/no1, and some α,β, and γ∈ℂ. Then there exists a unique (α,β,γ)-derivation δ:𝒜→𝒜 which satisfies (1) and the inequality
(23)∥f(x)-δ(x)∥≤1n(1-L)φ(x,0,…,0,0,0)
for all x∈𝒜.Proof.
Substitutingx1=a, x2=⋯,xn=x=y=0, and μ=1 in (22), we obtain
(24)∥1nf(na)-f(a)∥≤1nφ(a,0,…,0,0,0)
for all a∈𝒜 and a positive integer n≥2. Let Ω and d be as in the proof of Theorem 3 such that (Ω,d) becomes a generalized complete metric space. Let a mapping T:Ω→Ω defined by (Tg)(x)=(1/n)g(nx) for all x∈𝒜,  g∈Ω. Then, we have d(Tg,Th)≤Ld(g,h) for all g,h∈Ω. It follows from (24) that d(Tf,f)≤1/n<∞. The remaining assertion goes through in a similar way to the corresponding part of Theorem 3. This completes the proof.Corollary 8.
Assume that there exists an expansively superadditive mappingφ:𝒜n+2→[0,∞) with a constant L<1 such that a mapping f:𝒜→𝒜 satisfies (22). Then there exists a unique (α,β,γ)-derivation δ:𝒜→𝒜 which satisfies (1) and the inequality
(25)∥f(x)-δ(x)∥≤Ln(1-L)φ(x,0,…,0,0,0)
for all x∈𝒜.Next, we investigate the LieC*-algebra homomorphisms on Lie C*-algebras associated with the functional equation (1). The results in Theorems 9 and 10 are similar to those in [24].Theorem 9.
Assume that there exist a contractively subadditive mappingϕ:𝒜n→[0,∞) and a 2-contractively subhomogeneous mapping ψ:𝒜2→[0,∞) with a constant 0<L<1 such that a mapping f:𝒜→ℬ satisfies (3) and
(26)∥f([x,y])-[f(x),f(y)]∥≤ψ(x,y)
for all x,y∈𝒜 and μ∈T1/no1. Then there exists a unique Lie C*-algebra homomorphism H:𝒜→ℬ satisfying
(27)∥f(x)-H(x)∥≤1n(1-L)ϕ(x,0,…,0)
for all x∈𝒜.Proof.
By the same method as in Theorem3, we obtain a ℂ-linear mapping H:𝒜→ℬ satisfying (27). The mapping is given by H(x)=limm→∞(1/nm)f(nmx) for all x∈𝒜. It follows from (25) that
(28)∥H[x,y]-[H(x),H(y)]∥=limm→∞1n2m∥f(n2m[x,y])-[f(nmx),f(nmy)]∥≤limm→∞L2mψ(x,y)=0
for all x,y∈𝒜. Thus, H is a Lie C*-algebra homomorphism. This completes the proof.Theorem 10.
Assume that there exist an expansively superadditive mappingϕ:𝒜n→[0,∞) and a 2-expansively superhomogeneous mapping ψ:𝒜2→[0,∞) with a constant 0<L<1 such that a mapping f:𝒜→ℬ satisfies (3) and (26). Then there exists a unique Lie C*-algebra homomorphism H:𝒜→ℬ satisfying
(29)∥f(x)-H(x)∥≤Ln(1-L)ϕ(x,0,…,0)
for all x∈𝒜.Proof.
The proof is similar to the proofs of Theorems5 and 9.Corollary 11.
Letr<1,  s<2, and θ be nonnegative real numbers. Suppose that a mapping f:𝒜→ℬ satisfies (30)∥𝒟μf(x1,…,xn)∥≤θ∑i=1n‍∥xi∥r,∥f([x,y])-[f(x),f(y)]∥≤θ(∥x∥s+∥y∥s)
for all x1,…,xn,  x,y∈𝒜,  μ∈T1/no1. Then there exists a unique Lie C*-algebra homomorphism H:𝒜→ℬ satisfying
(31)∥f(x)-H(x)∥≤θn-nr∥x∥r,
for all x∈𝒜 and n∈ℤ+ with n≥2.



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