# Nearly Ring Homomorphisms and Nearly Ring Derivations on Non-Archimedean Banach Algebras

Authors: Madjid Eshaghi Gordji
Journal: Abstract and Applied Analysis
 (2010)

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## Abstract

We prove the generalized Hyers-Ulam stability of homomorphisms and derivations
on non-Archimedean Banach algebras. Moreover, we prove the superstability of homomorphisms
on unital non-Archimedean Banach algebras and we investigate the superstability of derivations in
non-Archimedean Banach algebras with bounded approximate identity.


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## Body

## 1. Introduction and Preliminaries

In 1897, Hensel [1] has introduced a normed space which does not have the Archimedean property.During the last three decades theory of non-Archimedean spaces has gained the interest of physicists for their research in particular in problems coming from quantum physics, p-adic strings, and superstrings [2]. Although many results in the classical normed space theory have a non-Archimedean counterpart,  their proofs are essentially different and require an entirely new kind of intuition [3–9].Let𝕂 be a field. A non-Archimedean absolute value on 𝕂 is a function |·|:𝕂→ℝ such that for any a,b∈𝕂 we have  (i)
|a|≥0 and equality holds if and only if a=0,(ii)
|ab|=|a||b|,(iii)
|a+b|≤max{|a|,|b|}.Condition (iii) is called the strict triangle inequality. By (ii), we have|1|=|-1|=1. Thus, by induction, it follows from (iii) that |n|≤1 for each integer n. We always assume in addition that |·| is non trivial, that is, that there is an a0∈𝕂 such that |a0|∉{0,1}.LetX be a linear space over a scalar field 𝕂 with a non-Archimedean nontrivial valuation |·|. A function ∥·∥:X→ℝ is a non-Archimedean norm (valuation) if it satisfies the following conditions:  (NA1)
∥x∥=0 if and only if x=0;(NA2)
∥rx∥=|r|∥x∥ for all r∈𝕂 and x∈X;(NA3)
the strong triangle inequality (ultrametric), namely,(1.1)‖x+y‖≤max⁡{‖x‖,‖y‖}(x,y∈X).Then(X,∥·∥) is called a non-Archimedean space.It follows from (NA3) that(1.2)‖xm-xl‖≤max⁡{‖x+1-x‖:l≤≤m-1}(m>l),
therefore a sequence {xm} is Cauchy in X if and only if {xm+1-xm} converges to zero in a non-Archimedean space. By a complete non-Archimedean space we mean one in which every Cauchy sequence is convergent. A non-Archimedean Banach algebra is a complete non-Archimedean algebra 𝒜 which satisfies ∥ab∥≤∥a∥∥b∥ for all a,b∈𝒜. For more detailed definitions of non-Archimedean Banach algebras, we can refer to [10].The first stability problem concerning group homomorphisms was raised by Ulam [11] in 1960 and affirmatively solved by Hyers [12]. Perhaps Aoki was the first author who has generalized the theorem of Hyers (see [13]).T. M. Rassias [14] provided a generalization of Hyers' Theorem which allows the Cauchy difference to be unbounded.Theorem 1.1 (T. M. Rassias).
Letf:E→E′ be a mapping from a normed vector space E into a Banach space E′ subject to the inequality
(1.3)‖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
(1.4)L(x)=lim⁡n→∞f(2nx)2n
exists for all x∈E and L:E→E′ is the unique additive mapping which satisfies
(1.5)‖f(x)-L(x)‖≤2ϵ2-2p‖x‖p
for all x∈E. Also, if for each x∈E the mapping f(tx) is continuous in t∈ℝ, then L is ℝ-linear.Moreover, Bourgin [15] and Găvruţa [16] have considered the stability problem with unbounded Cauchy differences (see also [17–27]).On the other hand, J. M. Rassias [28–33] considered the Cauchy difference controlled by a product of different powers of norm. However, there was a singular case; for this singularity a counterexample was given by Găvruţa [34]. This stability phenomenon is called the Ulam-Găvruta-Rassias stability (see also [35]).Theorem 1.2 (J. M. Rassias [28]).
LetX be a real normed linear space and Y a real complete normed linear space. Assume 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 the inequality
(1.6)‖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
(1.7)‖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 an ℝ-linear mapping.Very recently, Ravi et al. [36] in the inequality (1.6) replaced the bound by a mixed one involving the product and sum of powers of norms, that is, θ{∥x∥p∥y∥p+(∥x∥2p+∥y∥2p)}.For more details about the results concerning such problems and mixed product-sum stability (J. M.-Rassias Stability) the reader is referred to [37–49].Khodaei and T. M. Rassias [50] have established the general solution and investigated the Hyers-Ulam-Rassias stability of the following n-dimensional additive functional equation:(1.8)∑k=2n(∑i1=2k∑i2=i1+1k+1⋯∑in-k+1=in-k+1n)f(∑i=1,i≠i1,…,in-k+1naixi-∑r=1n-k+1airxir)+f(∑i=1naixi)=2n-1a1f(x1),
where a1,…,an∈ℤ-{0} with a1≠±1.In this paper, we investigate the Hyers-Ulam stability of homomorphisms and derivations associated with functional equation (1.8).


## 2. Main Results

Before taking up the main subject, for a givenf:𝒜→ℬ between vector spaces, we define the difference operator (2.1)Df(x1,…,xn)∶=∑k=2n(∑i1=2k∑i2=i1+1k+1⋯∑in-k+1=in-k+1n)f(∑i=1,i≠i1,…,in-k+1naixi-∑r=1n-k+1airxir)+f(∑i=1naixi)-2n-1a1f(x1).Theorem 2.1.
Let𝒜,ℬ be two non-Archimedean Banach algebras and let ψ:𝒜n→[0,∞), ϕ:𝒜2→[0,∞) be functions such that
(2.2)lim⁡m→∞1|a1|mψ(a1mx1,…,a1mxn)=lim⁡k→∞1kϕ(kx,y)=0
for all x1,…,xn∈𝒜, and the limit
(2.3)ψ̃(x)∶=lim⁡m→∞max⁡{1|a1|lψ(a1lx,0,…,0):0≤l<m}
exists and limk→∞(1/k)ψ̃(kx)=0 for all x∈𝒜. Suppose that f:𝒜→ℬ is a function satisfying
(2.4)‖Df(x1,…,xn)‖≤ψ(x1,…,xn),‖f(xy)-f(x)f(y)‖≤ϕ(x,y)
for all x1,…,xn,x,y∈𝒜. Then there exists a ring homomorphism H:𝒜→ℬ such that
(2.5)‖f(x)-H(x)‖≤1|2n-1a1|ψ̃(x)
for all x∈𝒜 and
(2.6)H(x)(H(y)-f(y))=(f(x)-H(x))H(y)=0
for all x,y∈𝒜. Moreover, if
(2.7)lim⁡→∞lim⁡m→∞max⁡{1|a1|lψ(a1lx,0,…,0):≤l<m+}=0,
then H is the unique ring homomorphism satisfying (2.5).Proof.
By [35, Theorem   4.4], there exists an additive functionH:𝒜→ℬ which satisfies (2.5). We have
(2.8)H(x)∶=lim⁡m→∞a1mf(xa1m)
for all x∈𝒜. Now we show that H is a multiplicative function. It follows from (2.5) that
(2.9)‖f(kx)-H(kx)‖≤1|2n-1a1|ψ̃(kx)
for all x∈𝒜 and all k∈ℕ. On the other hand H is additive then we have
(2.10)‖1kf(kx)-H(x)‖≤1|2n-1a1|kψ̃(kx)
for all x∈𝒜 and all k∈ℕ. If k→∞, then by (2.3), the right hand side of above inequality tends to zero. It follows that
(2.11)H(x)=lim⁡k→∞1kf(kx)
for all x∈𝒜. Applying (2.3), (2.4), and (2.11) we have
(2.12)H(xy)-H(x)f(y)=lim⁡k→∞1k(f(kxy)-f(kx)f(y))=0
for all x,y∈𝒜. This means that
(2.13)H(xy)=H(x)f(y)
for all x,y∈𝒜. From (2.13) and additivity of H we have
(2.14)H(x)H(y)=H(x)lim⁡k→∞1kf(ky)=lim⁡k→∞1k(H(x)f(ky))=lim⁡k→∞1kH(x(ky))=H(xy)
for all x,y∈𝒜. In other words, H is multiplicative.It follows from (2.13) and (2.14) that
(2.15)H(x)(H(y)-f(y))=0
for all x,y∈𝒜. Similarly, we can show that
(2.16)(f(x)-H(x))H(y)=0
for all x,y∈𝒜. To prove the uniqueness property of H, let T:𝒜→ℬ be another ring homomorphism which satisfies (2.5). Applying (2.11) and (2.5) we have
(2.17)‖H(x)-T(x)‖=lim⁡k→∞1k‖f(kxy)-T(kx)‖≤lim⁡k→∞1k1|2n-1a1|ψ̃(kx)=0
for all x∈𝒜 which is the desired conclusion.Now, we establish the superstability of homomorphisms as follows.Corollary 2.2.
Let𝒜,ℬ be two unital non-Archimedean Banach algebras, and let ψ:𝒜n→[0,∞),ϕ:𝒜2→[0,∞),f:𝒜→ℬ be functions with conditions of Theorem 2.1. Suppose that
(2.18)lim⁡m→∞a1mf(1Aa1m)=1B.
Then the mapping f:𝒜→ℬ is a ring homomorphism.Proof.
It follows from (2.6) and (2.18) that f=H in Theorem 2.1. Hence, f is a ring homomorphism.Corollary 2.3.
Letη:[0,∞)→[0,∞) be a function satisfying (i)
η(|a1|t)≤η(|a1|)η(t) for all t≥0;(ii)
η(|a1|)<|a1|;(iii)
limk→∞(1/k)η(k|a1|)=0. Suppose that ɛ>0, and let f:𝒜→ℬ satisfying
(2.19)‖Df(x1,…,xn)‖+‖f(xy)-f(x)f(y)‖≤ɛMin⁡{∑i=1nη(‖xi‖),η(‖x‖)η(‖y‖)}
for all x1,…,xn,x,y∈𝒜. Then there exists a unique ring homomorphism H:𝒜→ℬ such that
(2.20)‖f(x)-H(x)‖≤ɛ|2n-1a1|η(‖x‖)
for all x∈𝒜.Proof.
Definingψ:𝒜n→[0,∞) and ϕ:𝒜2→[0,∞) by
(2.21)ψ(x1,…,xn)∶=ɛ∑i=1nη(‖xi‖),ϕ(x,y)∶=η(‖x‖)η(‖y‖),
respectively, we have
(2.22)lim⁡m→∞1|a1|mψ(a1mx1,…,a1mxn)≤lim⁡m→∞(η(|a1|)|a1|)mψ(x1,…,xn)=0
for all x1,…,xn∈𝒜. Hence
(2.23)ψ̃(x)∶=lim⁡m→∞max⁡{1|a1|lψ(a1lx,0,…,0):0≤l<m}=ψ(x,0,…,0),lim⁡→∞lim⁡m→∞max⁡{1|a1|lψ(a1lx,0,…,0):≤l<m+}=lim⁡→∞1|a1|ψ(a1x,0,…,0)=0
for all x∈𝒜. On the other hand
(2.24)lim⁡k→∞1kϕ(kx,y)=lim⁡k→∞1kη(k‖x‖)η(‖y‖)=0
for all x,y∈𝒜. The conclusion follows from Theorem 2.1.Remark 2.4.
The classical example of the functionη is the function η(t)=tp for all t∈[0,∞), where p>1 with the further assumption that |a1|<1.Now, we prove the stability of derivations non-Archimedean Banach algebras by using Theorem2.1.Theorem 2.5.
Let𝒜 be a non-Archimedean Banach algebra, and let 𝒳 be a non-Archimedean Banach 𝒜-module. Let ψ:𝒜n→[0,∞),ϕ:𝒜2→[0,∞) be a function such that
(2.25)lim⁡m→∞1|a1|mψ(a1mx1,…,a1mxn)=lim⁡k→∞1kϕ(kx,y)=0
for all x1,…,xn∈𝒜, and the limit
(2.26)ψ̃(x)∶=lim⁡m→∞max⁡{1|a1|lψ(a1lx,0,…,0):0≤l<m}
exists and limk→∞(1/k)ψ̃(kx)=0 for all x∈𝒜. Suppose that f:𝒜→𝒳 is a function satisfying
(2.27)‖Df(x1,…,xn)‖≤ψ(x1,…,xn),‖f(xy)-f(x)y-xf(y)‖≤ϕ(x,y)
for all x1,…,xn,x,y∈𝒜. Then there exists a ring derivation D:𝒜→𝒳 such that
(2.28)‖f(x)-D(x)‖≤1|2n-1a1|ψ̃(x)
for all x∈𝒜.Proof.
It is easy to see that𝒳⊕1𝒜 is a non-Archimedean Banach algebra equipped with the product
(2.29)(x1,a1)(x2,a2)=(x1⋅a2+a1⋅x2,a1a2)(a1,a2∈A,x1,x2∈X)
and with the following ℓ1-norm:
(2.30)‖(x,a)‖=‖x‖+‖a‖(a∈A,x∈X).
Let us define the mapping φf:𝒜→𝒳⊕1𝒜 by a↦(f(a),a). It is easy to see that φf:𝒜→𝒳⊕1𝒜 satisfies the conditions of Theorem 2.1. By Theorem 2.1, there exists a unique ring homomorphism H:𝒜→𝒳⊕1𝒜 such that
(2.31)‖H(a)-φf(a)‖≤1|2n-1a1|ψ̃(a)(a∈A).
We define projection maps π1:𝒳⊕1𝒜→𝒳 and π2:𝒳⊕1𝒜→𝒜 by (x,b)↦x and (x,b)↦b, respectively.
It follows from (2.31) that
(2.32)‖(π2∘φf)(ka)-(π2∘H)(ka)‖≤‖φf(ka)-H(ka)‖≤1|2n-1a1|ψ̃(ka)(k∈N,a∈A).
By the additivity of mappings under consideration
(2.33)(π2∘φ)(ka)=k(π2∘φ)(a),(π2∘φf)(ka)=π2(f(ka),ka)=ka,
whence, by (2.32),
(2.34)‖a-(π2∘H)(a)‖≤1k1|2n-1a1|ψ̃(ka)
for all k∈ℕ, a∈𝒜. By letting k tend to ∞ in (2.34), we obtain by (2.25) that
(2.35)(π2∘H)(a)=a(a∈A).
Put D∶=π1∘H. Then we have
(2.36)((π1∘H)(ab),ab)=(π1(H(ab)),π2(H(ab)))=H(ab)=H(a)H(b)=(π1(H(a)),π2(H(a)))(π1(H(b)),π2(H(b)))=(π1(H(a)),a)(π1(H(b)),b)=(aπ1(H(b))+π1(H(a))b,ab)
for all a,b∈𝒜. It follows that D is a derivation. On the other hand, by (2.31) we have
(2.37)‖D(a)-f(a)‖=‖π1(H(a))-π1(φf(a))‖≤‖H(a)-φf(a)‖≤1|2n-1a1|ψ̃(a)
for all a∈𝒜.
To prove the uniqueness property ofD, assume that D* is another derivation from 𝒜 into 𝒳 satisfying
(2.38)‖D*(a)-f(a)‖≤1|2n-1a1|ψ̃(a)(a∈A).
Then by (2.25), we have
(2.39)‖D(a)-D*(a)‖=lim⁡k→∞1k‖D(ka)-D*(ka)‖≤lim⁡k→∞(1k‖D*(a)-f(a)‖+1k‖D(a)-f(a)‖)≤lim⁡k→∞2k1|2n-1a1|ψ̃(ka)=0
for all a∈𝒜. This means that D(a)=D*(a) for all a∈𝒜.Corollary 2.6.
Letη:[0,∞)→[0,∞) be a function satisfying (i)
η(|a1|t)≤η(|a1|)η(t) for all t≥0;(ii)
η(|a1|)<|a1|;(iii)
limk→∞(1/k)η(k|a1|)=0.Suppose that ε>0, and let f:𝒜→𝒳 satisfying
(2.40)‖Df(x1,…,xn)‖+‖f(xy)-f(x)y-xf(y)‖≤εMin⁡{∑i=1nη(‖xi‖),η(‖x‖)η(‖y‖)}
for all x1,…,xn,x,y∈𝒜. Then there exists a unique ring derivation D:𝒜→𝒳 such that
(2.41)‖f(x)-D(x)‖≤ε|2n-1a1|η(‖x‖)
for all x∈𝒜.Now, we would like to prove the superstability of derivations on non-Archimedean Banach algebras.Theorem 2.7.
Let𝒜 be a non-Archimedean Banach algebra with bounded approximate identity. Let  ψ:𝒜n→[0,∞), ϕ:𝒜2→[0,∞), f:𝒜→𝒜 be functions satisfying the conditions of Theorem 2.5. Then  f:𝒜→𝒜 is a ring derivation.Proof.
In the proof of Theorem2.5, we can see that
(2.42)H(b)(H(a)-φf(a))=(H(a)-φf(a))H(b)=0
for all a,b∈𝒜(2.43)(f(a)-D(a))b=π1((f(a)-D(a))b,0)=π1((f(a)-D(a),0)(D(b),b))=π1((π1(H(a)-φf(a)),0)(π1(H(b)),b))=π1((π1(H(a)-φf(a)),0)H(b))=π1(((π1(H(a)),a)-(π1(φf(a)),a))H(b))=π1(0,0)(by(2.42))=0
for all a,b∈𝒜. Since 𝒜 has a bounded approximate identity, then by above equation, we have f(a)=D(a) for all a∈𝒜. f is a ring derivation on 𝒜.



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