# Nearly Derivations on Banach Algebras

Authors: M. Eshaghi Gordji; H. Khodaei; G. H. Kim
Journal: Discrete Dynamics in Nature and Society
 (2012)

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## Abstract

Letn be a fixed integer greater than 3 and let λ be a real number with λ≠(n2-n+4)/2. We investigate the Hyers-Ulam stability of derivations on Banach algebras related to the following generalized Cauchy functional inequality ∥∑1≤i<j≤n1≤kl≠i,j≤nf((xi+xj)/2+∑l=1n-2xkl)+f(∑i=2nxi)+f(x1)∥≤∥λf(∑i=1nxi)∥.


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## Body

## 1. Introduction and Preliminaries

Let𝒜 ba a Banach algebra and let X be a Banach 𝒜-bimodule. Then X*, the dual space of X, is also a Banach 𝒜-bimodule with module multiplications defined by
(1.1)〈x,a⋅x*〉=〈x⋅a,x*〉,〈x,x*⋅a〉=〈a⋅x,x*〉,(a∈A,x∈X,x*∈X*).
A bounded linear operator D:𝒜→X is called a derivation if(1.2)D(ab)=a⋅D(b)+D(a)⋅b(a,b∈A).Letx∈X. We define δx(a)=a·x-x·a for all a∈𝒜. δx is a derivation from 𝒜 into X, which is called inner derivation. A Banach algebra 𝒜 is amenable if every derivation from 𝒜 into every dual 𝒜-bimodule X* is inner. This definition was introduced by Johnson in [1]. A Banach algebra 𝒜 is weakly amenable if every derivation from 𝒜 into 𝒜* is inner. Bade et al. [2] have introduced the concept of weak amenability for commutative Banach algebras.The stability problem of functional equations originated from a question of Ulam [3, 4] concerning the stability of group homomorphisms.A famous talk presented by Ulam in 1940 triggered the study of stability problems for various functional equations.We are given a groupG1 and a metric group G2 with metric ρ(·,·). Given ϵ>0, does there exist a δ>0 such that if f:G1→G2 satisfies ρ(f(xy),f(x)f(y))<δ for all x,y∈G1, then a homomorphism h:G1→G2 exists with ρ(f(x),h(x))<ϵ for all x∈G1?In the following year, Hyers was able to give a partial solution to Ulam’s question that was the first significant breakthrough and step toward more solutions in this area (see [5]). Since then, a large number of papers have been published in connection with various generalizations of Ulam’s problem and Hyers’ theorem.Letn be a fixed integer greater than 3 and let λ be a real number with |λ|≠(n2-n+4)/2. We investigate the Hyers-Ulam stability of derivations on Banach algebras related to the following generalized Cauchy functional inequality:(1.3)‖∑1≤i<j≤n1≤kl≠i,j≤nf(xi+xj2+∑l=1n-2xkl)+f(∑i=2nxi)+f(x1)‖≤‖λf(∑i=1nxi)‖.


## 2. Main Results

LetA be a Banach algebra and let X be a Banach A-module. From now on, the sum of f(x) and f(-x) will be denoted by f̃(x). Also, f(ab)-f(a)b-af(b) will be denoted by Δf(a,b). In the following, we will use the Pascal formula:
(2.1)C(r,k)=C(r-1,k)+C(r-1,k-1)
here, C(r,k) denotes r!/k!(r-k)! Moreover, we assume that n0∈ℕ is a positive integer and suppose that 𝕋1/no1:={eiθ;0≤θ≤2π/no}.Lemma 2.1.
Letf:A→X be a mapping such that
(2.2)‖∑1≤i<j≤n1≤kl≠i,j≤nf(xi+xj2+∑l=1n-2xkl)+f(∑i=2nxi)+f(x1)‖≤‖λf(∑i=1nxi)‖
for all x1,…,xn∈A. Then f is Cauchy additive.Proof.
Substitutingx1,…,xn=0 in the functional inequality (2.2), we get
(2.3)‖(C(n,2)+2)f(0)‖≤‖λf(0)‖.
Since n≥3 and |λ|≠(n2-n+4)/2, f(0)=0. Letting x1=x, x2=-x and x3=⋯=xn=0 in (2.2) and using Pascal formula, we get
(2.4)‖(n-2)f̃(x2)+(C(n-2,2)+1)f(0)+f̃(x)‖≤‖λf(0)‖,
for all x∈A. Hence
(2.5)(n-2)f̃(x2)+f̃(x)=0
for all x∈A. Letting x1=2x, x2=-x, x3=-x and x4=⋯=xn=0 in (2.2), we get
(2.6)‖2f(-x2)+(n-3)f(-x)+f(x)+2(n-3)f(x2)+C(n-3,2)f(0)+f̃(2x)‖≤‖λf(0)‖
for all x∈A. Hence
(2.7)2f(-x2)+(n-3)f(-x)+f(x)+2(n-3)f(x2)+f̃(2x)=0,2f(x2)+(n-3)f(x)+f(-x)+2(n-3)f(-x2)+f̃(-2x)=0
for all x∈A. Since f̃(-x)=f̃(x), we obtain from (2.7) and (2.4) that
(2.8)2(n-2)f̃(x2)+(n-2)f̃(x)+2f̃(2x)=0
for all x∈A. It follows from (2.5) and (2.8) that
(2.9)2f̃(x2)-f̃(x)=0
for all x∈A. By using (2.5) and (2.9), we get nf̃(x/2)=0 and so f(-x)=-f(x) for all x∈A. Hence, we obtain from (2.7) that f(x/2)=(1/2)f(x) for all x∈A. Letting x1=x+y, x2=-x, x3=-y and x4=⋯=xn=0 in (2.2), we get
(2.10)‖f(-y2)+f(-x2)+(n-3)f(-x-y2)+f(x+y2)+(n-3)f(x2)+(n-3)f(y2)+C(n-3,2)f(0)+f̃(x+y)f(-y2)‖≤‖λf(0)‖
for all x,y∈A. Next, notice that, using oddness of f and f(x/2)=(1/2)f(x), we have
(2.11)f(x+y)=f(x)+f(y)
for all x,y∈A, as desired.We can prove the following lemma by the same reasoning as in the proof of Theorem  2.2 of [6].Lemma 2.2.
Letf:A→X be an additive mapping such that  f(μx)=μf(x) for all  μ∈T1/no1 and all  x∈A. Then the mapping  f is  ℂ-linear.Theorem 2.3.
Letf:A→X be a mapping satisfying  f(0)=0 and the inequality
(2.12)‖∑1≤i<j≤n1≤kl≠i,j≤nf(μxi+μxj2+∑l=1n-2μxkl)+f(∑i=2nμxi)+μf(x1)+Δf(a,b)‖≤‖λf(∑i=1nμxi)‖+δ
for some δ>0, for all μ∈T1/no1 and all a,b,x1,…,xn∈A. Then there exists a unique derivation 𝒟:A→X such that
(2.13)‖f(x)-D(x)‖≤13n-24n(n-4)δ
for all x∈A.Proof.
Lettinga=b=0, x1=2x, x2=-2x, x3=⋯=xn=0 and μ=1 in (2.12), we get
(2.14)‖(n-2)f̃(x)+f̃(2x)‖≤δ
for all x∈X. Letting a=b=0, x1=2x, x2=-x, x3=-x, x4=⋯=xn=0 and μ=1 in (2.12), we get
(2.15)‖2f(-x2)+(n-3)f(-x)+f(x)+2(n-3)f(x2)+f̃(2x)‖≤δ
for all x∈X. Letting a=b=0, x1=-2x, x2=x, x3=x, x4=⋯=xn=0 and μ=1 in (2.12), we get
(2.16)‖2f(x2)+(n-3)f(x)+f(-x)+2(n-3)f(-x2)+f̃(-2x)‖≤δ
for all x∈X. It follows from (2.15) and (2.16) that
(2.17)‖(n-2)f̃(x2)+(n-2)2f̃(x)+f̃(2x)‖≤δ
for all x∈X. It follows from (2.14) and (2.17) that
(2.18)‖f̃(x)‖≤6nδ
for all x∈X. It follows from (2.15) and (2.18) that
(2.19)‖2f̃(x2)+f̃(x)+(n-4)f(-x)+2(n-4)f(x2)‖≤n+6nδ
for all x∈X. From the last two inequalities, we have
(2.20)‖f(2x)+2f(-x)‖≤n+24n(n-4)δ
for all x∈X. It follows from (2.18) and (2.20) that
(2.21)‖f(x)-12f(2x)‖≤13n-242n(n-4)δ
for all x∈X. Hence
(2.22)‖12rf(2rx)-12mf(2mx)‖≤13n-242n(n-4)∑k=rm-1δ2k
for all x∈X and integers m>r≥0. Thus it follows that a sequence {(1/2m)f(2mx)} is Cauchy in Y and so it converges. Therefore we can define a mapping 𝒟:X→Y by 𝒟(x):=limm→∞(1/2m)f(2mx) for all x∈X. In addition it is clear from (2.12) that the following inequality:
(2.23)‖∑1≤i<j≤n1≤kl≠i,j≤nD(μxi+μxj2+∑l=1n-2μxkl)+D(∑i=2nμxi)+μD(x1)‖=lim⁡m→∞12m‖∑1≤i<j≤n1≤kl≠i,j≤nf(2m-1μ(xi+xj)+∑l=1n-22mμxkl)+f(∑i=2n2mμxi)+μf(2mx1)‖≤lim⁡m→∞12m‖λf(∑i=1n2mμxi)‖+lim⁡m→∞δ2m=‖λD(∑i=1nμxi)‖
holds for all μ∈T1/no1 and all x1,…,xn∈X. If we put μ=1 in the last inequality, then 𝒟 is additive by Lemma 2.1. Letting x1=x, x2=-x and x3=⋯=xn=0 in last inequality and using Lemma 2.1, we get
(2.24)(n-2)D̃(μx2)+D(-μx)+μD(x)=μD(x)-D(μx).
So 𝒟(μx)=μ𝒟(x) for all x∈X and all μ∈T1/no1. Now by using Lemmas 2.1 and 2.2, we infer that the mapping 𝒟:X→Y is ℂ-linear. Taking the limit as m→∞ in (2.22) with r=0, we get (2.13).
To prove the afore-mentioned uniqueness, we assume now that there is anotherℂ-linear mapping 𝔏:A→X which satisfies the inequality (2.13). Then we get
(2.25)‖12mf(2mx)-L(x)‖=12m‖f(2mx)-L(2mx)‖≤13n-242mn(n-4)δ
for all x∈A and integers m≥1. Thus from m→∞, one establishes
(2.26)D(x)-L(x)=0
for all x∈A, completing the proof of uniqueness.
Now, we have to show that𝒟 is a derivation. To this end, let x1=x2=⋯=xn=0 in (2.12), we get
(2.27)‖f(ab)-f(a)b-af(b)‖≤δ
for all a,b∈A. It follows from linearity of 𝒟 and (2.27) that
(2.28)‖D(ab)-D(a)b-aD(b)‖=‖12mD(2mab)-D(a)12m(2mb)-12m(2ma)D(b)‖=lim⁡m→∞‖14mf(4mab)-f(2ma)14m(2mb)-14m(2ma)f(2mb)‖=lim⁡m→∞14m‖f(2ma2mb)-f(2ma)(2mb)-(2ma)f(2mb)‖≤lim⁡m→∞14mδ=0
for all a,b∈A. This means that 𝒟 is a derivation from A into X. Therefore the mapping 𝒟:A→X is a unique derivation satisfying (2.13), as desired.Theorem 2.4.
LetA be an amenable Banach algebra and let  f:A→X* be a mapping such that  f(0)=0 and (2.12). If
(2.29)sup⁡{‖f(x)‖:‖x‖≤1}<∞,
then there exists x0∈X* such that
(2.30)‖f(a)-ax0-x0a‖≤13n-24n(n-4)δ
for all a∈A.Proof.
Letsup{∥f(x)∥:∥x∥≤1}=Mf. Then by (2.29), we have Mf<∞. By Theorem 2.3, there exists a derivation D:A→X* satisfying (2.13). Then we have
(2.31)sup⁡{‖D(x)‖:‖x‖≤1}≤Mf+13n-24n(n-4)δ.
This means that D is bounded, and hence D is continuous. On the other hand, A is amenable. Then every continuous derivation from A into X* is an inner derivation. It follows that D is and an inner derivation. In the other words, there exists x0∈X* such that D(a)=ax0-x0a for all a∈A. This completes the proof.We know that every nuclearC*-algebra is amenable (see [7]). Then we have the following result.Corollary 2.5.
LetA be a nuclear  C*-algebra and let  f:A→X* be a mapping such that  f(0)=0, and (2.12) and (2.29). Then there exists  x0∈X* such that
(2.32)‖f(a)-ax0-x0a‖≤13n-24n(n-4)δ
for all a∈A.Theorem 2.6.
LetA be a  C*-algebra and let  f:A→A* be a mapping such that  f(0)=0, and (2.12) and (2.29). Then there exists  a′∈A* such that
(2.33)‖f(a)(b)-a′(ba-ab)‖≤13n-24n(n-4)δ‖b‖
for all a,b∈A.Proof.
We know that everyC*-algebra is weakly amenable (see, e.g., [7]). Then every continuous derivation from A into A* is an inner derivation. By the same reasoning as in the proof of Theorem 2.4, there exists a a′∈A* such that D(a)=aa′-a′a for all a∈A, and
(2.34)‖f(a)-aa′-a′a‖≤13n-24n(n-4)δ
for all a∈A. By definition of mudule actions of A on A*, we have
(2.35)‖f(a)(b)-a′(ba-ab)‖≤13n-24n(n-4)δ‖b‖
for all a,b∈A.Corollary 2.7.
LetA be a commutative  C*-algebra and let  f:A→A* be a mapping such that  f(0)=0, and (2.12) and (2.29). Then
(2.36)lim⁡m→∞12mf(2ma)=0,‖f(a)‖≤13n-24n(n-4)δ
for all a∈A.



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