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Discrete Dynamics in Nature and Society
Volume 2012, Article ID 373904, 9 pages
doi:10.1155/2012/373904
Research Article
Approximation of Homomorphisms and
Derivations on non-Archimedean Lie C∗ -Algebras
via Fixed Point Method
Yeol Je Cho,1 Reza Saadati,2 and Javad Vahidi2
1
Department of Mathematics Education and RINS, Gyeongsang National University,
Chinju 660-701, Republic of Korea
2
Department of Mathematics, Iran University of Science and Technology, Tehran, Iran
Correspondence should be addressed to Reza Saadati, rsaadati@eml.cc
Received 19 October 2011; Accepted 12 January 2012
Academic Editor: Yong Zhou
Copyright q 2012 Yeol Je Cho et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
Using fixed point methods, we prove the generalized Hyers-Ulam stability of homomorphisms in
C∗ -algebras and Lie C∗ -algebras and of derivations on non-Archimedean C∗ -algebras and NonArchimedean Lie C∗ -algebras for an m-variable additive functional equation.
1. Introduction and Preliminaries
By a non-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 any r ∈ K, and x ∈ X, rx |r|x;
iii the strong triangle inequality ultrametric holds; namely,
x y ≤ max x, y .
for all x, y ∈ X.
1.1
2
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Then X, · is called a non-Archimedean normed space. From the fact that
xn − xm ≤ max xj1 − xj : m ≤ j ≤ n − 1
n > m
1.2
holds, a sequence {xn } is a Cauchy sequence if and only if {xn1 − 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 number x, there exists a unique integer nx ∈ Z such that
x a/bpnx , where a and b are integers not divisible by p. Then |x|p : p−nx defines a nonArchimedean norm on Q. The completion of Q with respect to the metric dx, y |x − y|p is
denoted by Qp , which is called the p-adic number field.
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 refer the reader to 1, 2.
If U 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∗ αs∗ βt∗ ;
iii st∗ t∗ s∗ for s, t ∈ U.
If, in addition, t∗ t t2 for t ∈ U, then U is a non-Archimedean C∗ -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 > 0, does there exist a δ > 0 such that, if a mapping h : G1 → G2 satisfies
the inequality dhx ∗ y, hx hy < δ for all x, y ∈ G1 , then there is a homomorphism
H : G1 → G2 with dhx, Hx < for all x ∈ G1 ? If the answer is affirmative, we would
say that the equation of homomorphism Hx ∗ y Hx Hy is stable see also 4–6.
We recall a fundamental result in fixed point theory. Let Ω be a set. A function d :
Ω × Ω → 0, ∞ is called a generalized metric on Ω if d satisfies
1 dx, y 0 if and only if x y;
2 dx, y dy, x for all x, y ∈ Ω;
3 dx, z ≤ dx, y dy, 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
dJ n x, J n1 x ∞ for all nonnegative integers n or there exists a positive integer n0 such that
1 dJ n x, J n1 x < ∞ for all n ≥ n0 ;
2 the sequence {J n x} converges to a fixed point y∗ of J;
3 y∗ is the unique fixed point of J in the set Γ {y ∈ Ω | dJ n0 x, y < ∞};
4 dy, y∗ ≤ 1/1 − Ldy, Jy for all y ∈ Γ.
Discrete Dynamics in Nature and Society
3
In this paper, using the fixed point method, we prove the generalized Hyers-Ulam
stability of homomorphisms and derivations in non-Archimedean C∗ -algebras and nonArchimedean Lie C∗ -algebras for the following additive functional equation see 8:
⎛
⎞
m
m
m
m
f ⎝mxi xj ⎠ f
xi 2f
mxi
m ∈ N, m ≥ 2.
i1
j1,j /
i
i1
1.3
i1
2. Stability of Homomorphisms and Derivations in C∗ -Algebras
Throughout this section, assume that A is a non-Archimedean C∗ -algebra with norm · A
and that B is a non-Archimedean C∗ -algebra with norm · B .
For a given mapping f : A → B, we define
⎞
⎛
m
m
m
m
⎠
⎝
μf mxi xj f μ xi − 2f μ mxi
Dμ fx1 , . . . , xm :
i1
j1,j /i
i1
2.1
i1
for all μ ∈ T1 : {ν ∈ C : |ν| 1} and all x1 , . . . , xm ∈ A.
Note that a C-linear mapping H : A → B is called a homomorphism in nonArchimedean C∗ -algebras if H satisfies Hxy HxHy and Hx∗ Hx∗ for all
x, y ∈ A.
We prove the generalized Hyers-Ulam stability of homomorphisms in nonArchimedean C∗ -algebras for the functional equation Dμ fx1 , . . . , xm 0.
Theorem 2.1. Let f : 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
Dμ fx1 , . . . , xm ≤ ϕx1 , . . . , xm ,
B
f xy − fxf y ≤ ψ x, y ,
B
fx∗ − fx∗ ≤ ηx
B
2.2
2.3
2.4
for all μ ∈ T 1 and x1 , . . . , xm , x, y ∈ A. If there exists an L < 1 such that
ϕmx1 , . . . , mxm ≤ |m| Lϕx1 , . . . , xm ,
ψ mx, my ≤ |m|2 Lψ x, y ,
2.5
2.6
ηmx ≤ |m|Lηx
2.7
for all x, y, x1 , . . . , xm ∈ A, then there exists a unique homomorphism H : A → B such that
fx − Hx ≤
B
for all x ∈ A.
1
ϕx, 0, . . . , 0
|m| − |m|L
2.8
4
Discrete Dynamics in Nature and Society
Proof. It follows from 2.5, 2.6, 2.7 and L < 1 that
1
lim
n → ∞ |m|n
lim
ϕmn x1 , . . . , mn xm 0,
2.9
ψ mn x, mn y 0,
2.10
1
n → ∞ |m|2n
1
ηmn x 0
n → ∞ |m|n
lim
2.11
for all x, y, x1 , . . . , xm ∈ A.
Let us define Ω to be the set of all mappings g : A → B and introduce a generalized
metric on Ω as follows
d g, h inf k ∈ 0, ∞ : gx − hxB < kφx, 0, . . . , 0, ∀x ∈ A .
2.12
It is easy to show that Ω, d is a generalized complete metric space see 9.
Now we consider the function J : Ω → Ω defined by Jgx 1/mgmx for all
x ∈ A and g ∈ Ω. Note that for all g, h ∈ Ω we have
d g, h < k ⇒ gx − hxB < kφx, 0, . . . , 0
1
1
k
⇒ gmx − hmx
< |m| φmx, 0, . . . , 0
m
m
B
1
1
⇒ m gmx − m hmx < kLφmx, 0, . . . , 0
B
⇒ d Jg, Jh < kL.
2.13
From this it is easy to see that dJg, Jk ≤ Ldg, h for all g, h ∈ Ω, that is, J is a self-function
of Ω with the Lipschitz constant L.
Putting μ 1, x x1 and x2 x3 · · · xm 0 in 2.2, we have
for all x ∈ A. Then
fmx − mfx ≤ φx, 0, . . . , 0
B
2.14
fx − 1 fmx ≤ 1 φx, 0, . . . , 0
m
|m|
B
2.15
for all x ∈ A, that is, dJf, f ≤ 1/|m| < ∞. Now, from the fixed point alternative, it follows
that there exists a fixed point H of J in Ω such that
Hx lim
1
fm
n → ∞ |m|n
for all x ∈ A since limn → ∞ dJ n f, H 0.
n
x
2.16
Discrete Dynamics in Nature and Society
5
On the other hand, it follows from 2.2, 2.9, and 2.16 that
Dμ Hx1 , . . . , xm lim 1 Dfmn x1 , . . . , mn xm B
n → ∞ mn
B
1
≤ lim
φmn x1 , . . . , mn xm 0.
n → ∞ |m|n
2.17
By a similar method to the above, we get μHmx Hmμx for all μ ∈ T 1 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
n 1 2n
n
m
xy
−
fm
xf
m y f
n → ∞ |m|2n
B
H xy − HxH y lim
B
≤ lim
1
n → ∞ |m|2n
2.18
ψ m x, m y 0
n
n
for all x, y ∈ A. So Hxy HxHy 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 Hx∗ Hx∗ .
Corollary 2.2. Let r > 1 and θ be nonnegative real numbers, and let f : A → B be a mapping such
that
Dμ fx1 , . . . , xm ≤ θ · x1 r x2 r · · · xm r ,
A
A
A
B
f xy − fxf y ≤ θ · xr · yr ,
A
A
B
2.19
fx∗ − fx∗ ≤ θ · xr ,
A
B
for all μ ∈ T
such that
1
and x1 , . . . , xm , x, y ∈ A. Then there exists a unique homomorphism H : A → B
fx − Hx ≤
B
θ
xrA
|m| − |m|r
2.20
for all x ∈ A.
Proof. The proof follows from Theorem 2.1 by taking
ϕx1 , . . . , xm θ · x1 rA x2 rA · · · xm rA ,
r ψ x, y : θ · xrA · yA ,
ηx θ · xrA
for all x1 , . . . , xm , x, y ∈ A, L |m|r−1 and so we get the desired result.
2.21
6
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Note that a C-linear mapping δ : A → A is called a derivation on A if δ satisfies
δxy δxy xδy for all x, y ∈ A.
We prove the generalized Hyers-Ulam stability of derivations on non-Archimedean
C∗ -algebras for the functional equation Dμ fx1 , . . . , xm 0.
Theorem 2.3. Let f : 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
Dμ fx1 , . . . , xm ≤ ϕx1 , . . . , xm ,
A
f xy − fxy − xf y ≤ ψ x, y , fx∗ − fx∗ ≤ ηx
A
A
2.22
for all μ ∈ T 1 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
fx − δxA ≤
1
ϕx, 0, . . . , 0
|m| − |m|L
2.23
for all x ∈ A.
3. Stability of Homomorphisms and Derivations in
Non-Archimedean Lie C∗ -Algebras
A non-Archimedean C∗ -algebra C, endowed with the Lie product
xy − yx
x, y :
2
3.1
on C, is called a Lie non-Archimedean C∗ -algebra.
Definition 3.1. Let A and B be Lie C∗ -algebras. A C-linear mapping H : A → B is called a
non-Archimedean Lie C∗ -algebra homomorphism if Hx, y Hx, Hy for all x, y ∈ A.
Throughout this section, assume that A 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 nonArchimedean Lie C∗ -algebras for the functional equation Dμ fx1 , . . . , xm 0.
Theorem 3.2. Let f : 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
f x, y − fx, f y ≤ ψ x, y
B
3.2
for all μ ∈ T 1 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.
Discrete Dynamics in Nature and Society
7
Proof. By the same reasoning as in the proof of Theorem 2.1, we can find the mapping H :
A → B given by
fmn x
n → ∞ |m|n
Hx lim
3.3
for all x ∈ A. It follows from 2.6 and 3.3 that
n 1 2n
n
−
fm
m
x,
f
m y x,
y
f
n → ∞ |m|2n
B
H x, y − Hx, H y lim
B
≤ lim
1
n → ∞ |m|2n
3.4
ψ m x, m y 0
n
n
for all x, y ∈ A and so
H x, y Hx, H y ,
3.5
for all x, y ∈ A. Thus H : A → B is a Lie C∗ -algebra homomorphism satisfying 2.8, as
desired.
Corollary 3.3. Let r > 1 and θ be nonnegative real numbers, and let f : A → B be a mapping such
that
Dμ fx1 , . . . , xm ≤ θ x1 r x2 r · · · xm r ,
A
A
A
B
f x, y − fx, f y ≤ θ · xr · yr ,
A
B
A
fx∗ − fx∗ ≤ θ · xr
A
B
all μ ∈ T
that
1
3.6
and x1 , . . . , xm , x, y ∈ A. Then there exists a unique homomorphism H : A → B such
fx − Hx ≤
B
θ
xrA
|m| − |m|r
3.7
for all x ∈ A.
Proof. The proof follows from Theorem 3.2 and a method similar to Corollary 3.3.
Definition 3.4. Let A be a non-Archimedean Lie C∗ -algebra. A C-linear mapping δ : A → A
is called a Lie derivation if δx, y δx, y x, δy for all x, y ∈ A.
We prove the generalized Hyers-Ulam stability of derivations on non-Archimedean
Lie C∗ -algebras for the functional equation Dμ fx1 , . . . , xm 0.
8
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Theorem 3.5. Let f : 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
f x, y − fx, y − x, f y ≤ ψ x, y
A
3.8
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 Theorem 2.3, there exists a unique C-linear
mapping δ : A → A satisfying 2.8 and the mapping δ : A → A is given by
fmn x
n → ∞ |m|n
δx lim
3.9
for all x ∈ A.
It follows from 2.6 and 3.9 that
δ x, y − δx, y − x, δ y A
1 lim
f m2n x, y − fmn x, mn y − mn x, f mn y 2n
n → ∞ |m|
A
≤ lim
1
n → ∞ |m|2n
3.10
ψ mn x, mn y 0,
for all x, y ∈ A and so
δ x, y δx, y x, δ y
3.11
for all x, y ∈ A. Thus δ : A → A is a Lie derivation satisfying 2.8.
Acknowledgment
This paper was supported by Basic Science Research Program through the National Research
Foundation of Korea NRF funded by the Ministry of Education, Science and Technology
No. 2011-0021821.
References
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