# Approximate Cubic ∗-Derivations on Banach ∗-Algebras

Authors: Seo Yoon Yang; Abasalt Bodaghi; Kamel Ariffin Mohd Atan
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We study the stability of cubic ∗-derivations on Banach ∗-algebras. We also prove the superstability of cubic ∗-derivations on a Banach ∗-algebraA, which is left approximately unital.


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## Body

## 1. Introduction

In [1], Ulam proposed the stability problems for functional equations concerning the stability of group homomorphisms. In fact, a functional equation is called stable if any approximately solution to the functional equation is near a true solution of that functional equation and is superstable if every approximate solution is an exact solution to it. In [2], Hyers considered the case of approximate additive mappings in Banach spaces and satisfying the well-known weak Hyers inequality controlled by a positive constant. Bourgin [3] was the second author to treat this problem for additive mappings (see also [4]). In [5], Rassias provided a generalization of Hyers Theorem, which allows the Cauchy difference to be unbounded. Ga˘vruţa then generalized the Rassias’ result in [6] for the unbounded Cauchy difference. Subsequently, various approaches to the problem have been studied by a number of authors (see, e.g., [7–11]).Recall that a Banach*-algebra is a Banach algebra (complete normed algebra) which has an isometric involution. For a locally compact group G, the algebraic group algebra L1(G) is a Banach *-algebra. The bounded operators on Hilbert space ℋ is also a Banach *-algebra. In general, all C*-algebras are Banach *-algebra. A left- (right-) bounded approximate identity for a normed algebra 𝒜 is a bounded net (ej)j in 𝒜 such that limjeja=a (limjaej=a) for each a∈𝒜. A bounded approximate identity for 𝒜 is a bounded net (ej)j, which is both a left- and a right-bounded approximate identity. Every group algebra and every C*-algebra has a bounded approximate identity.The stability of functional equations of*-derivations and of quadratic *-derivations with the Cauchy functional equation and the Jensen functional equation on Banach *-algebras is investigated in [12]. The author also proved the superstability of *-derivations and of quadratic *-derivations on C*-algebras.In 2003, Ca˘dariu and Radu employed the fixed point method to the investigation of the Jensen functional equation. They presented a short and a simple proof (different from the “direct method,” initiated by Hyers in 1941) for the Cauchy functional equation [13] and for the quadratic functional equation [14] (see also [15–18]).The functional equation(1.1)f(2x+y)+f(2x-y)=2f(x+y)+2f(x-y)+12f(x)
which is called cubic functional equation. In addition, every solution of functional equation (1.1) is said to be a cubic mapping. It is easy to check that function f(x)=ax3 is a solution of (1.1).In [19], Bodaghi et al. proved the generalized Hyers-Ulam stability and the superstability for the functional equation (1.1) by using the alternative fixed point (Theorem 3.1) under certain conditions on Banach algebras. Also, the stability and the superstability of homomorphisms on C*-algebras by using the same fixed point method was proved in [20]. The generalized Hyers-Ulam-Rassias stability of *-homomorphisms between unital C*-algebras associated with the Trif functional equation and of linear *-derivations on unital C*-algebras has earlier been proved by Park and Hou in [21].In this paper, we prove the stability and the superstability of cubic*-derivations on Banach *-algebras. We also show that these functional equations, under some mild conditions, are superstable. We also establish the stability and the superstability of cubic *-derivations on a Banach *-algebra with a left-bounded approximate identity.


## 2. Stability of Cubic*-Derivation

Throughout this paper, we assume thatA is a Banach *-algebra. A mapping D:A→A is a cubic derivation if D is a cubic homogeneous mapping, that is, D is cubic and D(μa)=μ3D(a) for all a∈A and μ∈ℂ, and D(ab)=D(a)b3+a3D(b) for all a,b∈A. In addition, if D satisfies in condition D(a*)=D(a)* for all a∈A, then it is called the cubic *-derivation. An example of cubic derivations on Banach algebras is given in [22].Letμ∈ℂ. For the given mapping f:A→A, we consider
(2.1)Dμf(a,b):=f(2μa+μb)+f(2μa-μb)-2μ3f(a+b)-2μ3f(a-b)-12μ3f(a),Df(a,b)=f(ab)-f(a)b3-a3f(b)
for all a,b∈A.Theorem 2.1.
Suppose thatf:A→A is a mapping with f(0)=0 for which there exists a function φ:A5→[0,∞) such that
(2.2)φ~(a,b,x,y,z):=∑k=0∞‍18kφ(2ka,2kb,2kx,2ky,2kz)<∞,(2.3)‖Dμf(a,b)‖≤φ(a,b,0,0,0),(2.4)‖Df(x,y)+f(z*)-f(z)*‖≤φ(0,0,x,y,z),
for all μ∈𝕋1/n01={eiθ:0≤θ≤2π/n0} and all a,b,x,y,z∈A in which n0∈ℕ. Also, if for each fixed a∈A the mapping t↦f(ta) from ℝ to A is continuous, then there exists a unique cubic *-derivation D on A satisfying
(2.5)‖f(a)-D(a)‖≤116ψ~(a),(a∈A),
in which ψ~(a)=φ~(a,0,0,0,0).Proof.
Puttingb=0 and μ=1 in (2.3), we have
(2.6)‖18f(2a)-f(a)‖≤116ψ(a)
for all a∈A in which ψ(a)=φ(a,0,0,0,0). We can use induction to show that
(2.7)‖f(2na)8n-f(2ma)8m‖≤116∑k=mn-1‍ψ(2ka)8k
for all a∈A and n>m≥0. On the other hand,
(2.8)‖f(2na)8n-f(a)‖≤116∑k=0n-1‍ψ(2ka)8k
for all a∈A and n>0. It follows from (2.2) and (2.7) that the sequence {f(2na)/8n} is a Cauchy sequence. Since A is a Banach algebra, this sequence converges to the map D, that is,
(2.9)lim⁡n→∞f(2na)8n=D(a).
Thus the inequalities (2.2) and (2.8) show that (2.5) holds. Substituting a,b by 2na,2nb, respectively, in (2.3), we get
(2.10)‖DμD(a,b)‖=lim⁡n→∞18n‖Dμf(2na,2nb)‖≤lim⁡n→∞φ(2na,2nb,0,0,0)8n=0
for all a,b∈A and μ∈𝕋1/n01. Since 𝒟1D(a,b)=0, the mapping D is cubic. The equality 𝒟μD(a,0)=0 implies that D(μa)=μ3D(a) for all a∈A and μ∈𝕋1/n01. Now, let μ∈𝕋1={λ∈ℂ:|λ|=1} such that μ=eiθ in which 0≤θ<2π. We set μ1=eiθ/n0, thus μ1 belongs to 𝕋1/n01 and D(μa)=D(μ1n0a)=μ13n0D(a)=μ3D(a) for all a∈A. Now, suppose that ℱ is any continuous linear functional on A and a is a fixed element in A. Define the mapping g:ℝ→ℝ via g(μ)=ℱ[D(μa)] for each μ∈ℝ. Obviously, g is a cubic function. Under the hypothesis that f(ta) is continuous in t∈ℝ for each fixed a∈A, the function g is the pointwise limit of the sequence of measurable functions {gn} in which gn(μ)=ℱ(2nμa)/8n, n∈ℕ, μ∈ℝ. Hence, g is a continuous function and has the form g(μ)=μ3g(1) for all μ∈ℝ. Therefore,
(2.11)F[D(μa)]=g(μ)=μ3g(1)=μ3F[D(a)]=F[μ3D(a)].
Since ℱ is an arbitrary continuous linear functional on A, D(μa)=μ3D(a) for all μ∈ℝ and a∈A. Thus
(2.12)D(μa)=D(μ|μ||μ|a)=μ3|μ|3D(|μ|a)=μ3|μ|3|μ|3D(a)=μ3D(a)
for all a∈A and μ∈ℂ(μ≠0). Therefore, D is a cubic homogeneous. If we replace x,y by 2nx,2ny, respectively, and put z=0 in (2.4), we have
(2.13)182n‖Df(2nx,2ny)‖≤φ(0,0,2nx,2ny,0)82n≤φ(0,0,2nx,2ny,0)8n
for all x,y∈A. Taking the limit as n tends to infinity, we get 𝒟D(x,y)=0, for all x,y∈A. Putting x=y=0 and substituting z by 2nz in (2.4) and then dividing the both sides of the obtained inequality by 8n, then we get
(2.14)‖f(2nz*)8n-f(2nz)*8n‖≤φ(0,0,0,0,2nz)8n
for all z∈A. Passing to the limit as n→∞ in (2.14), we get D(z*)=D(z)* for all z∈A. This shows that D is a cubic *-derivation.
Now, letD′:A→A be another cubic *-derivation satisfying (2.5). Then we have
(2.15)‖D(a)-D′(a)‖=18n‖D(2na)-D′(2na)‖≤18n(‖D(2na)-f(2na)‖+‖f(2na)-D′(2na)‖)≤18n+1ψ~(2na)=18∑k=n∞‍18kψ(2ka),
which tends to zero as n→∞ for all a∈A. So we can conclude that D(a)=D′(a) for all a∈A. This proves the uniqueness of D.We have the following theorem, which is analogous to Theorem2.1. Since the proof is similar, it is omitted.Theorem 2.2.
Suppose thatf:A→A is a mapping with f(0)=0 for which there exists a function φ:A5→[0,∞) satisfying (2.3), (2.4), and
(2.16)φ~(a,b,x,y,z):=∑k=1∞‍8kφ(2-ka,2-kb,2-kx,2-ky,2-kz)<∞
for all a,b,x,y,z∈A. Also, if for each fixed a∈A the mappings t↦f(ta) from ℝ to A is continuous, then there exists a unique cubic *-derivation D on A satisfying
(2.17)‖f(a)-D(a)‖≤116ψ~(a),(a∈A),
where ψ~(a)=φ~(a,0,0,0,0).Corollary 2.3.
Letθ,r be positive real numbers with r≠3, and let f:A→A be a mapping with f(0)=0 such that
(2.18)‖Dμf(a,b)‖≤θ(‖a‖r+‖b‖r),‖Df(x,y)+f(z*)-f(z)*‖≤θ(‖x‖r+‖y‖r+‖z‖r),
for all μ∈𝕋1/n01 and all a,b,x,y,z∈A. Then there exists a unique cubic *-derivation D on A satisfying
(2.19)‖f(a)-D(a)‖≤θ‖a‖r|16-2r+1|,
for all a∈A.Proof.
We can obtain the result from Theorem2.1 and Theorem 2.2 by taking
(2.20)φ(a,b,x,y,z)=θ(‖a‖r+‖b‖r+‖x‖r+‖y‖r+‖z‖r)
for all a,b,x,y,z∈A.In the next theorem, we investigate the superstability of cubic*-derivations of Banach *-algebras with a left-bounded approximate identity.Theorem 2.4.
Suppose thatA is a Banach *-algebra with a left-bounded approximate identity and s∈{-1,1}. Let f:A→A be a mapping for which there exists a function ψ:A×A→[0,∞) such that
(2.21)lim⁡n→∞n-3sψ(nsa,b)=lim⁡n→∞n-3sψ(a,nsb)=0,(2.22)‖a3f(b)-f(a)b3‖≤ψ(a,b),(2.23)‖f(c)(ab)3-c3[f(a)b3+a3f(b)]‖≤ψ(c,ab),(2.24)‖a3f(b*)-f(a)(b3)*‖≤ψ(a,b)
for all a,b,c∈A. Then f is a cubic *-derivation on A.Proof.
First, we show thatf is cubic. For each a,b,c∈A, we have
(2.25)‖c3[f(2a+b)+f(2a-b)-2f(a+b)-2f(a-b)-12f(a)]‖(a)=n-3s‖n3sc3f(2a+b)+n3sc3f(2a-b)-2n3sc3f(a+b)-2n3sc3f(a-b)-12n3sc3f(a)‖(a)≤n-3s[‖n3sc3f(2a+b)-f(n3sc3)(2a+b)3‖+‖n3sc3f(2a-b)-f(n3sc3)(2a-b)3‖+2‖n3sc3f(a+b)-f(n3sc3)(a+b)3‖+2‖n3sc3f(a-b)-f(n3sc3)(a-b)3‖+12‖n3sc3f(a)-f(n3sc3)a3‖](a)≤n-3s[ψ(nsc,2a+b)+ψ(nsc,2a-b)+2ψ(nsc,a+b)+2ψ(nsc,a-b)+12ψ(nsc,a)].
Taking the limit from the right side asn tends to infinity and using (2.21), we get
(2.26)c3[f(2a+b)+f(2a-b)-2f(a+b)-2f(a-b)-12f(a)]=0
for all a,b,c∈A. If (ej) is a left-bounded approximate identity for A, then so is (ej3). Now, it follows from (2.26) that f is cubic. For being cubic homogeneous of f, we have
(2.27)‖n3sb3[f(μa)-μ3f(a)]‖≤‖n3sb3f(μa)-f(nsb)(μa)3‖+‖(μa)3f(nsb)-n3s(μb)3f(a)‖≤ψ(nsb,μa)+|μ|3ψ(nsb,a).
Thus ∥b3[f(μa)-μ3f(a)]∥≤n-3sψ(nsb,μa)+n-3s|μ|3ψ(nsb,a). By the same reasoning as in the above, f is cubic homogeneous. For each a,b,c∈A, we have
(2.28)‖c3[f(ab)-f(a)b3-a3f(b)]‖=n-3s‖n3sc3[f(ab)-f(a)b3-a3f(b)]‖≤n-3s‖n3sc3f(ab)-f(nsc)(ab)3‖+n-3s‖f(nsc)(ab)3-n3sc3f(a)b3-n3sc3a3f(b)‖≤2n-3sψ(nsc,ab).
The above inequality and (2.21), (2.22), and (2.23) show that f(ab)=f(a)b3+a3f(b) for all a,b∈A. Finally, we have
(2.29)‖b3[f(a*)-f(a)*]‖=n-3s‖n3sb3f(a*)-n3sb3f(a)*‖≤n-3s‖n3sb3f(a*)-f(nsb)(a3)*‖+n-3s‖f(nsb)(a3)*-n3sb3f(a)*‖≤n-3sψ(nsb,a*)+n-3sψ(nsb,a)
for all a,b∈A. Note that in the last inequality we have used (2.22) and (2.24). This completes the proof.Corollary 2.5.
Letr,δ be the nonnegative real numbers with r≠3, and let A be a Banach *-algebra with a left bounded approximate identity. Suppose that f:A→A is a mapping satisfying
(2.30)‖a3f(b)-f(a)b3‖≤δ(‖a‖r‖b‖r),‖f(c)(ab)3-c3[f(a)b3+a3f(b)]‖≤δ(‖ab‖r‖c‖r),‖a3f(b*)-f(a)(b3)*‖≤δ(‖a‖r‖b‖r)
for all all a,b,c∈A. Then f is a cubic *-derivation on A.Proof.
Using Theorem2.4 with ψ(a,b)=δ(∥a∥r∥b∥r), we get the desired result.


## 3. A Fixed Point Approach

Before proceeding to the main results in this section, we bring the upcoming theorem, which is useful to our purpose (For an extension of the result see [23]).Theorem 3.1 (The fixed point alternative [24]).
Let(Ω,d) be a complete generalized metric space and 𝒯:Ω→Ω a mapping with Lipschitz constant L<1. Then, for each element α∈Ω, either d(𝒯nα,𝒯n+1α)=∞ for all n≥0, or there exists a natural number n0 such that:(i)
d(𝒯nα,𝒯n+1α)<∞  for all n≥n0;(ii)
the sequence{𝒯nα} is convergent to a fixed point β* of 𝒯;(iii)
β* is the unique fixed point of 𝒯 in the set Λ={β∈Ω:d(𝒯n0α,β)<∞};(iv)
d(β,β*)≤1/(1-L)d(β,𝒯β) for all β∈Λ.Theorem 3.2.
Letf:A→A be a continuous mapping with f(0)=0, and let φ:A4→[0,∞) be a continuous function such that
(3.1)‖Dμf(a,b)+Df(c,d)‖≤φ(a,b,c,d),(3.2)‖f(a*)-f(a)*‖≤φ(a,a,a,a)
for all μ∈𝕋1/n01 and all a,b,c,d∈A. If there exists a constant k∈(0,1) such that
(3.3)φ(2a,2b,2c,2d)≤8kφ(a,b,c,d)
for all a,b,c,d∈A, then there exists a unique cubic *-derivation D on A satisfying
(3.4)‖f(a)-D(a)‖≤116(1-k)φ~(a)(a∈A),
in which φ~(a)=φ(a,0,0,0).Proof.
First, we wish to provide the conditions of Theorem3.1. We consider the set
(3.5)Ω={g:A⟶A∣g(0)=0}
and define the mapping d on Ω×Ω as follows:
(3.6)d(g1,g2):=inf⁡{C∈(0,∞):‖g1(a)-g2(a)‖≤Cφ~(a),(∀a∈A)}
if there exist such constant C and d(g1,g2)=∞, otherwise. It is easy to check that d(g,g)=0 and d(g1,g2)=d(g2,g1), for all g,g1,g2∈Ω. For each g1,g2,g3∈Ω, we have
(3.7)inf⁡{C∈(0,∞):‖g1(a)-g2(a)‖≤Cφ~(a)∀a∈A}(0,∞)≤inf⁡{C∈(0,∞):‖g1(a)-g3(a)‖≤Cφ~(a)∀a∈A}+inf⁡{C∈(0,∞):‖g3(a)-g2(a)‖≤Cφ~(a)∀a∈A}.
Henced(g1,g2)≤d(g1,g3)+d(g3,g2). If d(g1,g2)=0, then for every fixed a0∈A, we have ∥g1(a0)-g2(a0)∥≤Cφ~(a0) for all C>0. This implies g1=g2. Let {gn} be a d-Cauchy sequence in Ω. Then d(gm,gn)→0, and thus ∥gm(a)-gn(a)∥→0 for all a∈A. Since A is complete, then there exists g∈Ω such that gn→dg in Ω. Therefore, d is a generalized metric on Ω and the metric space (Ω,d) is complete. Now, we define the mapping 𝒯:Ω→Ω by
(3.8)Tg(a)=18g(2a),(a∈A).
Ifg1,g2∈Ω such that d(g1,g2)<C, by definition of d and 𝒯, we have
(3.9)‖18g1(2a)-18g2(2a)‖≤18Cφ(2a,0,0,0)
for all a∈A. By using (3.3), we get
(3.10)‖18g1(2a)-18g2(2a)‖≤Ckφ(a,0,0,0)
for all a∈A. The above inequality shows that d(𝒯g1,𝒯g2)≤kd(g1,g2) for all g1,g2∈Ω. Hence, 𝒯 is a strictly contractive mapping on Ω with a Lipschitz constant k. To achieve inequality (3.4), we prove that d(𝒯f,f)<∞. Putting b=c=d=0 and μ=1 in (3.1), we obtain
(3.11)‖2f(2a)-16f(a)‖≤φ~(a)
for all a∈A. Hence
(3.12)‖18f(2a)-f(a)‖≤116φ~(a)
for all a∈A. We conclude from (3.12) that d(𝒯f,f)≤1/16. It follows from Theorem 3.1 that d(𝒯ng,𝒯n+1g)<∞ for all n≥0, and thus in this theorem we have n0=0. Therefore, the parts (iii) and (iv) of Theorem 3.1 hold on the whole Ω. Hence there exists a unique mapping D:A→A such that D is a fixed point of 𝒯 and that 𝒯nf→D as n→∞. Thus
(3.13)lim⁡n→∞f(2na)8n=D(a)
for all a∈A, hence
(3.14)d(f,D)≤11-kd(Tf,f)≤116(1-k).
The above equalities show that (3.4) is true for all a∈A. It follows from (3.3) that
(3.15)lim⁡n→∞φ(2na,2nb,2nc,2nd)8n=0.
Putting c=d=0 and substituting a,b by 2na,2nb, respectively, in (3.1), we get
(3.16)18n‖Dμf(2na,2nb)‖≤φ(2na,2nb,0,0)8n.
Taking the limit as n tend to infinity, we obtain 𝒟μD(a,b)=0 for all a,b∈A and all μ∈𝕋1/n01. Similar to the proof of Theorem 2.1, we have D(μa)=μ3D(a) for all a∈A and μ∈𝕋1. Since 𝒟1D(a,b)=0, we can show that D(ra)=r3D(a) for any rational number r. The continuity of f and φ imply that D(μa)=μ3D(a), for all a∈A and μ∈ℝ. Hence D(μa)=μ3D(a), for all a∈A and μ∈ℂ(μ≠0). Therefore, D is a cubic homogeneous. If we put a=b=0 and replace c,d by 2nc,2nd, respectively, in (3.1), we have
(3.17)182n‖Df(2nc,2nd)‖≤φ(0,0,2nc,2nd)82n≤φ(0,0,2nc,2nd)8n
for all c,d∈A. By letting n→∞ in the preceding inequality, we find 𝒟D(c,d)=0 for all c,d∈A. Substituting a by 2na in (3.2) and then dividing the both sides of the obtained inequality by 8n, we get
(3.18)‖f(2na*)8n-f(2na)*8n‖≤φ(2na,2na,2na,2na)8n
for all a∈A. Passing to the limit as n→∞ in (3.18) and applying (3.13), we conclude that D(a*)=D(a)* for all a∈A. This shows that D is a unique cubic *-derivation.Corollary 3.3.
Letθ,r be positive real numbers with r<3, and let f:A→A be a mapping with f(0)=0 such that
(3.19)‖Dμf(a,b)+Df(c,d)‖≤θ(‖a‖r+‖b‖r+‖c‖r+‖d‖r),‖f(a*)-f(a)*‖≤4θ‖a‖r
for all μ∈𝕋1/n01 and all a,b,c,d∈A. Then there exists a unique cubic *-derivation D on A satisfying
(3.20)‖f(a)-D(a)‖≤θ2(8-2r)‖a‖r
for all a∈A.Proof.
The result follows from Theorem3.2 by letting
(3.21)φ(a,b,c,d)=θ(∥a∥r+∥b∥r+∥c∥r+∥d∥r).In the following corollary, we show the superstability for cubic*-derivations.Corollary 3.4.
Letrj(1≤j≤4)θ be nonnegative real numbers with 0<∑j=14‍rj≠3, and let f:A→A be a mapping such that
(3.22)‖Dμf(a,b)+Df(c,d)‖≤θ(‖a‖r1‖b‖r2‖c‖r3‖d‖r4),(3.23)‖f(a*)-f(a)*‖≤θ‖a‖∑j=14rj
for all μ∈𝕋1/n01 and all a,b,c,d∈A. Then f is a cubic *-derivation on A.Proof.
Puttinga=b=c=d=0 in (3.22), we get f(0)=0. Now, if we put b=c=d=0, μ=1 in (3.22), then we have f(2a)=8f(a) for all a∈A. It is easy to see by induction that f(2na)=8nf(a), and thus f(a)=f(2na)/8n for all a∈A and n∈ℕ. It follows from Theorem 3.2 that f is a cubic mapping. Now, by putting φ(a,b,c,d)=θ(∥a∥r1∥b∥r2∥c∥r3∥d∥r4) in Theorem 3.2, we can obtain the desired result.



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