# Approximation of Homomorphisms and Derivations on non-Archimedean

Authors: Yeol Je Cho; Reza Saadati; Javad Vahidi
Journal: Discrete Dynamics in Nature and Society
 (2012)

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## Abstract

Using fixed point methods, we prove the generalized Hyers-Ulam stability
of homomorphisms inC∗-algebras and Lie C∗-algebras and of derivations on non-Archimedean C∗-algebras and Non-Archimedean Lie C∗-algebras for an m-variable additive functional equation.


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## Body

## 1. Introduction and Preliminaries

By anon-Archimedean field we mean a field K equipped with a function (valuation) |·| from K into [0,∞) such that |r|=0 if and only if r=0, |rs|=|r||s|, and |r+s|≤max{|r|,|s|} for all r,s∈K. Clearly |1|=|-1|=1 and |n|≤1 for all n∈ℕ. By the trivial valuation we mean the mapping |·| taking everything but 0 into 1 and |0|=0. Let X be a vector space over a field K with a non-Archimedean nontrivial valuation |·|. A function ∥·∥:X→[0,∞) is called a non-Archimedean norm if it satisfies the following conditions:(i)
∥x∥=0 if and only if x=0;(ii)
for anyr∈K, and x∈X, ∥rx∥=|r|∥x∥;(iii)
the strong triangle inequality (ultrametric) holds; namely,(1.1)‖x+y‖≤max⁡{‖x‖,‖y‖}.
for all x,y∈X.Then(X,∥·∥) is called a non-Archimedean normed space. From the fact that(1.2)‖xn-xm‖≤max⁡{‖xj+1-xj‖:m≤j≤n-1}(n>m)holds, a sequence{xn} is a Cauchy sequence if and only if {xn+1-xn} converges to zero in a non-Archimedean normed space. By a complete non-Archimedean normed space we mean one in which every Cauchy sequence is convergent.For any nonzero rational numberx, there exists a unique integer nx∈ℤ such that x=(a/b)pnx, where a and b are integers not divisible by p. Then |x|p∶=p-nx defines a non-Archimedean norm on ℚ. The completion of ℚ with respect to the metric d(x,y)=|x-y|p is denoted by ℚp, which is called the p-adic number field.A non-Archimedean Banach algebra is acomplete non-Archimedean algebra𝒜 which satisfies ∥ab∥≤∥a∥·∥b∥ for all a,b∈𝒜. For more detailed definitions of non-Archimedean Banach algebras, we refer the reader to [1, 2].If𝒰 is a non-Archimedean Banach algebra, then an involution on 𝒰 is a mapping t→t* from 𝒰 into 𝒰 which satisfies(i)
t**=t for t∈𝒰;(ii)
(αs+βt)*=α¯s*+β¯t*;(iii)
(st)*=t*s* for s,t∈𝒰.If, in addition,∥t*t∥=∥t∥2 for t∈𝒰, then 𝒰 is a non-ArchimedeanC*-algebra.The stability problem of functional equations was originated from a question of Ulam [3] concerning the stability of group homomorphisms: let (G1,*) be a group and let (G2,⋄,d) be a metric group (a metric which is defined on a set with group property) with the metric d(·,·). Given ϵ>0, does there exist a δ(ϵ)>0 such that, if a mapping h:G1→G2 satisfies the inequality d(h(x*y),h(x)⋄h(y))<δ for all x,y∈G1, then there is a homomorphism H:G1→G2 with d(h(x),H(x))<ϵ for all x∈G1? If the answer is affirmative, we would say that the equation of homomorphism H(x*y)=H(x)⋄H(y) is stable (see also [4–6]).We recall a fundamental result in fixed point theory. LetΩ be a set. A function d:Ω×Ω→[0,∞] is called a generalized metricon Ω if d satisfies(1)
d(x,y)=0 if and only if x=y;(2)
d(x,y)=d(y,x) for all x,y∈Ω;(3)
d(x,z)≤d(x,y)+d(y,z) for all x,y,z∈Ω.Theorem 1.1 (see [7]).
Let(Ω,d) be a complete generalized metric space and let J:Ω→Ω be a contractive mapping with Lipschitz constant L<1. Then for each given element x∈Ω, either d(Jnx,Jn+1x)=∞ for all nonnegative integers n or there exists a positive integer n0 such that(1)
d(Jnx,Jn+1x)<∞ for all n≥n0;(2)
the sequence{Jnx} converges to a fixed point y* of J;(3)
y* is the unique fixed point of J in the set Γ={y∈Ω∣d(Jn0x,y)<∞};(4)
d(y,y*)≤(1/(1-L))d(y,Jy) for all y∈Γ.In this paper, using the fixed point method, we prove the generalized Hyers-Ulam stability of homomorphisms and derivations in non-ArchimedeanC*-algebras and non-Archimedean Lie C*-algebras for the following additive functional equation (see [8]):(1.3)∑i=1mf(mxi+∑j=1,j≠imxj)+f(∑i=1mxi)=2f(∑i=1mmxi)(m∈N,m≥2).


## 2. Stability of Homomorphisms and Derivations inC*-Algebras

Throughout this section, assume that𝒜 is a non-Archimedean C*-algebra with norm ∥·∥𝒜 and that ℬ is a non-Archimedean C*-algebra with norm ∥·∥ℬ.For a given mappingf:𝒜→ℬ, we define(2.1)Dμf(x1,…,xm)∶=∑i=1mμf(mxi+∑j=1,j≠imxj)+f(μ∑i=1mxi)-2f(μ∑i=1mmxi)for allμ∈𝕋1∶={ν∈ℂ:|ν|=1} and all x1,…,xm∈𝒜.Note that aℂ-linear mapping H:𝒜→ℬ is called a homomorphismin non-Archimedean C*-algebras if H satisfies H(xy)=H(x)H(y) and H(x*)=H(x)* for all x,y∈𝒜.We prove the generalized Hyers-Ulam stability of homomorphisms in non-ArchimedeanC*-algebras for the functional equation Dμf(x1,…,xm)=0.Theorem 2.1.
Letf:𝒜→ℬ be a mapping for which there are functions φ:𝒜m→[0,∞),  ψ:𝒜2→[0,∞) and η:𝒜→[0,∞) such that |m|<1 is far from zero and
(2.2)‖Dμf(x1,…,xm)‖B≤φ(x1,…,xm),(2.3)‖f(xy)-f(x)f(y)‖B≤ψ(x,y),(2.4)‖f(x*)-f(x)*‖B≤η(x)
for all μ∈𝕋1 and x1,…,xm,x,y∈A. If there exists an L<1 such that
(2.5)φ(mx1,…,mxm)≤|m|Lφ(x1,…,xm),(2.6)ψ(mx,my)≤|m|2Lψ(x,y),(2.7)η(mx)≤|m|Lη(x)
for all x,y,x1,…,xm∈𝒜, then there exists a unique homomorphism H:𝒜→ℬ such that
(2.8)‖f(x)-H(x)‖B≤1|m|-|m|Lφ(x,0,…,0)
for all x∈𝒜.Proof.
It follows from (2.5), (2.6), (2.7) and L<1 that
(2.9)lim⁡n→∞1|m|nφ(mnx1,…,mnxm)=0,(2.10)lim⁡n→∞1|m|2nψ(mnx,mny)=0,(2.11)lim⁡n→∞1|m|nη(mnx)=0
for all x,y,x1,…,xm∈𝒜.
Let us defineΩ to be the set of all mappings g:𝒜→ℬ and introduce a generalized metric on Ω as follows(2.12)d(g,h)=inf⁡{k∈(0,∞):‖g(x)-h(x)‖B<kϕ(x,0,…,0),∀x∈A}.
It is easy to show that (Ω,d) is a generalized complete metric space (see [9]).
Now we consider the functionJ:Ω→Ω defined by Jg(x)=(1/m)g(mx) for all x∈𝒜 and g∈Ω. Note that for all g,h∈Ω we have
(2.13)d(g,h)<k⟹‖g(x)-h(x)‖B<kϕ(x,0,…,0)⟹‖1mg(mx)-1mh(mx)‖B<k|m|ϕ(mx,0,…,0)⟹‖1mg(mx)-1mh(mx)‖B<kLϕ(mx,0,…,0)⟹d(Jg,Jh)<kL.
From this it is easy to see that d(Jg,Jk)≤Ld(g,h) for all g,h∈Ω, that is, J is a self-function of Ω with the Lipschitz constant L.
Puttingμ=1, x=x1 and x2=x3=⋯=xm=0 in (2.2), we have(2.14)‖f(mx)-mf(x)‖B≤ϕ(x,0,…,0)
for all x∈𝒜. Then
(2.15)‖f(x)-1mf(mx)‖B≤1|m|ϕ(x,0,…,0)
for all x∈𝒜, that is, d(Jf,f)≤1/|m|<∞. Now, from the fixed point alternative, it follows that there exists a fixed point H of J in Ω such that
(2.16)H(x)=lim⁡n→∞1|m|nf(mnx)
for all x∈𝒜 since limn→∞d(Jnf,H)=0.
On the other hand, it follows from (2.2), (2.9), and (2.16) that(2.17)‖DμH(x1,…,xm)‖B=lim⁡n→∞‖1mnDf(mnx1,…,mnxm)‖B≤lim⁡n→∞1|m|nϕ(mnx1,…,mnxm)=0.
By a similar method to the above, we getμH(mx)=H(mμx) for all μ∈𝕋1 and x∈𝒜. Thus one can show that the mapping H:𝒜→ℬ is ℂ-linear.
It follows from (2.3), (2.10) and (2.16) that
(2.18)‖H(xy)-H(x)H(y)‖B=lim⁡n→∞1|m|2n‖f(m2nxy)-f(mnx)f(mny)‖B≤lim⁡n→∞1|m|2nψ(mnx,mny)=0for all x,y∈𝒜. So H(xy)=H(x)H(y) for all x,y∈𝒜. Thus H:𝒜→ℬ is a homomorphism, satisfying (2.8), as desired.
Also, by (2.4), (2.11), (2.16) and by a similar method, we have H(x*)=H(x)*.Corollary 2.2.
Letr>1 and θ be nonnegative real numbers, and let f:𝒜→ℬ be a mapping such that
(2.19)‖Dμf(x1,…,xm)‖B≤θ⋅(‖x1‖Ar+‖x2‖Ar+⋯+‖xm‖Ar),‖f(xy)-f(x)f(y)‖B≤θ⋅(‖x‖Ar⋅‖y‖Ar),‖f(x*)-f(x)*‖B≤θ⋅‖x‖Ar,
for all μ∈𝕋1 and x1,…,xm, x,y∈𝒜. Then there exists a unique homomorphism H:𝒜→ℬ such that
(2.20)‖f(x)-H(x)‖B≤θ|m|-|m|r‖x‖Ar
for all x∈𝒜.Proof.
The proof follows from Theorem2.1 by taking
(2.21)φ(x1,…,xm)=θ⋅(‖x1‖Ar+‖x2‖Ar+⋯+‖xm‖Ar),ψ(x,y)∶=θ⋅(‖x‖Ar⋅‖y‖Ar),η(x)=θ⋅‖x‖Arfor all x1,…,xm,x,y∈A, L=|m|r-1 and so we get the desired result.Note that aℂ-linear mapping δ:𝒜→𝒜 is called a derivationon 𝒜 if δ satisfies δ(xy)=δ(x)y+xδ(y) for all x,y∈𝒜.We prove the generalized Hyers-Ulam stability of derivations on non-ArchimedeanC*-algebras for the functional equation Dμf(x1,…,xm)=0.Theorem 2.3.
Letf:𝒜→𝒜 be a mapping for which there are functions φ:𝒜m→[0,∞),  ψ:𝒜2→[0,∞) and η:𝒜→[0,∞) such that |m|<1 is far from zero and
(2.22)‖Dμf(x1,…,xm)‖A≤φ(x1,…,xm),‖f(xy)-f(x)y-xf(y)‖A≤ψ(x,y),‖f(x*)-f(x)*‖A≤η(x)
for all μ∈𝕋1 and x1,…,xm,x,y∈A. If there exists an L<1 such that (2.5), (2.6) and (2.7) hold, then there exists a unique derivation δ:𝒜→𝒜 such that
(2.23)∥f(x)-δ(x)∥A≤1(|m|-|m|L)φ(x,0,…,0)
for all x∈𝒜.


## 3. Stability of Homomorphisms and Derivations in Non-Archimedean LieC*-Algebras

A non-ArchimedeanC*-algebra 𝒞, endowed with the Lie product(3.1)[x,y]∶=xy-yx2on𝒞, is called a Lie non-ArchimedeanC*-algebra.Definition 3.1.
Let𝒜 and ℬ be Lie C*-algebras. A ℂ-linear mapping H:𝒜→ℬ is called a non-Archimedean LieC*-algebra homomorphism if H([x,y])=[H(x),H(y)] for all x,y∈𝒜.
Throughout this section, assume that𝒜 is a non-Archimedean Lie C*-algebra with norm ∥·∥𝒜 and ℬ is a non-Archimedean Lie C*-algebra with norm ∥·∥ℬ.
We prove the generalized Hyers-Ulam stability of homomorphisms in non-Archimedean LieC*-algebras for the functional equation Dμf(x1,…,xm)=0.Theorem 3.2.
Letf:𝒜→ℬ be a mapping for which there are functions φ:𝒜m→[0,∞) and ψ:𝒜2→[0,∞) such that (2.2) and (2.4) hold and
(3.2)‖f([x,y])-[f(x),f(y)]‖B≤ψ(x,y)
for all μ∈𝕋1 and x,y∈𝒜. If there exists an L<1 and (2.5), (2.6), and (2.7) hold, then there exists a unique homomorphism H:𝒜→ℬ such that (2.8) holds.Proof.
By the same reasoning as in the proof of Theorem2.1, we can find the mapping H:𝒜→ℬ given by
(3.3)H(x)=lim⁡n→∞f(mnx)|m|n
for all x∈𝒜. It follows from (2.6) and (3.3) that
(3.4)‖H([x,y])-[H(x),H(y)]‖B=lim⁡n→∞1|m|2n‖f(m2n[x,y])-[f(mnx),f(mny)‖B≤lim⁡n→∞1|m|2nψ(mnx,mny)=0
for all x,y∈𝒜 and so
(3.5)H([x,y])=[H(x),H(y)],
for all x,y∈𝒜. Thus H:𝒜→ℬ is a Lie C*-algebra homomorphism satisfying (2.8), as desired.Corollary 3.3.
Letr>1 and θ be nonnegative real numbers, and let f:𝒜→ℬ be a mapping such that
(3.6)‖Dμf(x1,…,xm)‖B≤θ(‖x1‖Ar+‖x2‖Ar+⋯+‖xm‖Ar),‖f([x,y])-[f(x),f(y)]‖B≤θ⋅‖x‖Ar⋅‖y‖Ar,‖f(x*)-f(x)*‖B≤θ⋅‖x‖Ar
all μ∈𝕋1 and x1,…,xm,x,y∈𝒜. Then there exists a unique homomorphism H:𝒜→ℬ such that
(3.7)‖f(x)-H(x)‖B≤θ|m|-|m|r‖x‖Ar
for all x∈𝒜.Proof.
The proof follows from Theorem3.2 and a method similar to Corollary 3.3.Definition 3.4.
Let𝒜 be a non-Archimedean Lie C*-algebra. A ℂ-linear mapping δ:𝒜→𝒜 is called a Lie derivation if δ([x,y])=[δ(x),y]+[x,δ(y)] for all x,y∈𝒜.
We prove the generalized Hyers-Ulam stability of derivations on non-Archimedean LieC*-algebras for the functional equation Dμf(x1,…,xm)=0.Theorem 3.5.
Letf:𝒜→𝒜 be a mapping for which there are functions φ:Am→[0,∞) and ψ:A2→[0,∞) such that (2.2) and (2.4) hold and
(3.8)‖f([x,y])-[f(x),y]-[x,f(y)]‖A≤ψ(x,y)
for all x,y∈𝒜. If there exists an L<1 and (2.5), (2.6) and (2.7) hold, then there exists a unique Lie derivation δ:𝒜→𝒜 such that such that (2.8) holds.Proof.
By the same reasoning as the proof of Theorem2.3, there exists a unique ℂ-linear mapping δ:𝒜→𝒜 satisfying (2.8) and the mapping δ:𝒜→𝒜 is given by
(3.9)δ(x)=lim⁡n→∞f(mnx)|m|n
for all x∈𝒜.
It follows from (2.6) and (3.9) that
(3.10)‖δ([x,y])-[δ(x),y]-[x,δ(y)]‖A=lim⁡n→∞1|m|2n‖f(m2n[x,y])-[f(mnx),mny]-[mnx,f(mny)]‖A≤lim⁡n→∞1|m|2nψ(mnx,mny)=0,
for all x,y∈𝒜 and so
(3.11)δ([x,y])=[δ(x),y]+[x,δ(y)]
for all x,y∈𝒜. Thus δ:𝒜→𝒜 is a Lie derivation satisfying (2.8).



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