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Discrete Dynamics in Nature and Society
Volume 2012, Article ID 864340, 8 pages
doi:10.1155/2012/864340
Research Article
Nearly Derivations on Banach Algebras
M. Eshaghi Gordji,1 H. Khodaei,1 and G. H. Kim2
1
2
Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran
Department of Mathematics, Kangnam University, Yongin, Gyeonggi 446-702, Republic of Korea
Correspondence should be addressed to M. Eshaghi Gordji, madjid.eshaghi@gmail.com
and G. H. Kim, ghkim@kangnam.ac.kr
Received 11 December 2011; Accepted 23 January 2012
Academic Editor: John Rassias
Copyright q 2012 M. Eshaghi Gordji 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.
Let n 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
n−2
n
generalized Cauchy functional inequality 1≤i<j≤n fxi xj /2 l1 xkl f i2 xi 1≤kl /
i,j≤n
fx1 ≤ λf ni1 xi .
1. Introduction and Preliminaries
Let A 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
x, a · x∗ x · a, x∗ ,
x, x∗ · a a · x, x∗ ,
a ∈ A, x ∈ X, x∗ ∈ X ∗ .
1.1
A bounded linear operator D : A → X is called a derivation if
Dab a · Db Da · b
a, b ∈ A.
1.2
Let x ∈ 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.
2
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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 group G1 and a metric group G2 with metric ρ·, ·. Given > 0, does
there exist a δ > 0 such that if f : G1 → G2 satisfies ρfxy, fxfy < δ for all x, y ∈ G1 ,
then a homomorphism h : G1 → G2 exists with ρfx, hx < 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.
Let n 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:
n−2
n
n
xi xj ≤ λf
xkl f
f
xi fx1 xi .
2
1≤i<j≤n
i2
i1
l1
1≤k i,j≤n
l
1.3
/
2. Main Results
Let A be a Banach algebra and let X be a Banach A-module. From now on, the sum of fx
and f−x will be denoted by fx.
Also, fab − fab − afb will be denoted by Δfa, b.
In the following, we will use the Pascal formula:
Cr, k Cr − 1, k Cr − 1, k − 1
2.1
here, Cr, k denotes r!/k!r − k! Moreover, we assume that n0 ∈ N is a positive integer and
suppose that T11/no : {eiθ ; 0 ≤ θ ≤ 2π/no }.
Lemma 2.1. Let f : A → X be a mapping such that
n−2
n
xi xj xkl f
f
xi fx1 2
1≤i<j≤n
i2
l1
1≤k i,j≤n
l
/
n
≤ λf
x
i
i1
2.2
for all x1 , . . . , xn ∈ A. Then f is Cauchy additive.
Proof. Substituting x1 , . . . , xn 0 in the functional inequality 2.2, we get
Cn, 2 2f0 ≤ λf0.
2.3
Discrete Dynamics in Nature and Society
3
Since n ≥ 3 and |λ| /
n2 − n 4/2, f0 0. Letting x1 x, x2 −x and x3 · · · xn 0 in
2.2 and using Pascal formula, we get
x
2.4
Cn − 2, 2 1f0 fx
≤ λf0,
n − 2f
2
for all x ∈ A. Hence
x
fx
0
n − 2f
2
2.5
for all x ∈ A. Letting x1 2x, x2 −x, x3 −x and x4 · · · xn 0 in 2.2, we get
≤ λf0
2f −x n − 3f−x fx 2n − 3f x Cn − 3, 2f0 f2x
2
2
2.6
for all x ∈ A. Hence
x
−x
2f
n − 3f−x fx 2n − 3f
f2x
0,
2
2
x
−x
2f
n − 3fx f−x 2n − 3f
f−2x
0
2
2
2.7
for all x ∈ A. Since f−x
fx,
we obtain from 2.7 and 2.4 that
x
2n − 2f
n − 2fx
2f2x
0
2
2.8
for all x ∈ A. It follows from 2.5 and 2.8 that
x
2f
− fx
0
2
2.9
for all x ∈ A. By using 2.5 and 2.9, we get nfx/2
0 and so f−x −fx for all x ∈ A.
Hence, we obtain from 2.7 that fx/2 1/2fx for all x ∈ A. Letting x1 x y, x2 −x,
x3 −y and x4 · · · xn 0 in 2.2, we get
y
x
−y
−x − y
xy
−x
f
n − 3f
f
−
3f
f
n − 3f
n
2
2
2
2
2
2
2.10
Cn − 3, 2f0 f x y ≤ λf0
for all x, y ∈ A. Next, notice that, using oddness of f and fx/2 1/2fx, we have
f x y fx f y
for all x, y ∈ A, as desired.
2.11
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We can prove the following lemma by the same reasoning as in the proof of
Theorem 2.2 of 6.
1
and
Lemma 2.2. Let f : A → X be an additive mapping such that fμx μfx for all μ ∈ T1/n
o
all x ∈ A. Then the mapping f is C-linear.
Theorem 2.3. Let f : A → X be a mapping satisfying f0 0 and the inequality
n−2
n
n
μxi μxj λf
≤
f
μfx
f
μx
μx
μx
Δfa,
b
kl
i
1
i δ
2
1≤i<j≤n
i2
i1
l1
1≤k i,j≤n
l
/
2.12
1
for some δ > 0, for all μ ∈ T1/n
and all a, b, x1 , . . . , xn ∈ A. Then there exists a unique derivation
o
D : A → X such that
fx − Dx ≤ 13n − 24 δ
nn − 4
2.13
for all x ∈ A.
Proof. Letting a b 0, x1 2x, x2 −2x, x3 · · · xn 0 and μ 1 in 2.12, we get
f2x
≤δ
n − 2fx
2.14
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
≤δ
2f −x n − 3f−x fx 2n − 3f x f2x
2
2
2.15
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
2f x n − 3fx f−x 2n − 3f −x f−2x
≤δ
2
2
2.16
for all x ∈ X. It follows from 2.15 and 2.16 that
n − 2f x n − 2 fx
≤δ
f2x
2
2
2.17
for all x ∈ X. It follows from 2.14 and 2.17 that
6
fx ≤ δ
n
2.18
Discrete Dynamics in Nature and Society
5
for all x ∈ X. It follows from 2.15 and 2.18 that
x
x n 6
fx
n − 4f−x 2n − 4f
δ
2f
≤
2
2
n
2.19
for all x ∈ X. From the last two inequalities, we have
f2x 2f−x ≤ n 24 δ
nn − 4
2.20
for all x ∈ X. It follows from 2.18 and 2.20 that
fx − 1 f2x ≤ 13n − 24 δ
2nn − 4
2
2.21
m−1
δ
1
f2r x − 1 f2m x ≤ 13n − 24
2r
2m
2nn − 4 kr 2k
2.22
for all x ∈ X. Hence
for all x ∈ X and integers m > r ≥ 0. Thus it follows that a sequence {1/2m f2m x} is
Cauchy in Y and so it converges. Therefore we can define a mapping D : X → Y by Dx :
limm → ∞ 1/2m f2m x for all x ∈ X. In addition it is clear from 2.12 that the following
inequality:
n−2
n
μxi μxj D
μDx
D
μx
μx
kl
i
1 2
1≤i<j≤n
i2
l1
1≤k i,j≤n
l
/
n−2
n
1 m−1
m
m
m
lim m f
μf2
f
2
μ
x
x
2
μx
2
μx
x
i
j
kl
i
1 m→∞2 2.23
i2
1≤i<j≤n
l1
1≤k i,j≤n
l
/
n
1 δ
m
≤ lim m λf
2 μxi lim m
m→∞2 m→∞2
i1
n
λD
μxi i1
1
holds for all μ ∈ T1/n
and all x1 , . . . , xn ∈ X. If we put μ 1 in the last inequality, then D is
o
additive by Lemma 2.1. Letting x1 x, x2 −x and x3 · · · xn 0 in last inequality and
using Lemma 2.1, we get
n − 2D
μx 2
D −μx μDx μDx − D μx .
2.24
6
Discrete Dynamics in Nature and Society
1
. Now by using Lemmas 2.1 and 2.2, we infer
So Dμx μDx for all x ∈ X and all μ ∈ T1/n
o
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 another Clinear mapping L : A → X which satisfies the inequality 2.13. Then we get
1
f2m x − Lx 1 f2m x − L2m x ≤ 13n − 24 δ
2m
2m
2m nn − 4
2.25
for all x ∈ A and integers m ≥ 1. Thus from m → ∞, one establishes
Dx − Lx 0
2.26
for all x ∈ A, completing the proof of uniqueness.
Now, we have to show that D is a derivation. To this end, let x1 x2 · · · xn 0 in
2.12, we get
fab − fab − afb ≤ δ
2.27
for all a, b ∈ A. It follows from linearity of D and 2.27 that
1
1 m
1 m
m
Dab − Dab − aDb m D2 ab − Da m 2 b − m 2 aDb
2
2
2
1
1 m
1 m
m
m
m lim m f4 ab − f2 a m 2 b − m 2 af2 b
m→∞ 4
4
4
lim
1 f2m a2m b − f2m a2m b − 2m af2m b
2.28
m → ∞ 4m
1
δ
m → ∞ 4m
≤ lim
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. Let A be an amenable Banach algebra and let f : A → X ∗ be a mapping such that
f0 0 and 2.12. If
sup fx : x ≤ 1 < ∞,
2.29
fa − ax0 − x0 a ≤ 13n − 24 δ
nn − 4
2.30
then there exists x0 ∈ X ∗ such that
for all a ∈ A.
Discrete Dynamics in Nature and Society
7
Proof. Let sup{fx : 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
sup{Dx : x ≤ 1} ≤ Mf 13n − 24
δ.
nn − 4
2.31
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 Da ax0 − x0 a
for all a ∈ A. This completes the proof.
We know that every nuclear C∗ -algebra is amenable see 7. Then we have the
following result.
Corollary 2.5. Let A be a nuclear C∗ -algebra and let f : A → X ∗ be a mapping such that f0 0,
and 2.12 and 2.29. Then there exists x0 ∈ X ∗ such that
fa − ax0 − x0 a ≤ 13n − 24 δ
nn − 4
2.32
for all a ∈ A.
Theorem 2.6. Let A be a C∗ -algebra and let f : A → A∗ be a mapping such that f0 0, and
2.12 and 2.29. Then there exists a ∈ A∗ such that
fab − a ba − ab ≤ 13n − 24 δb
nn − 4
2.33
for all a, b ∈ A.
Proof. We know that every C∗ -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 Da aa − a a for all a ∈ A, and
fa − aa − a a ≤ 13n − 24 δ
nn − 4
2.34
for all a ∈ A. By definition of mudule actions of A on A∗ , we have
fab − a ba − ab ≤ 13n − 24 δb
nn − 4
for all a, b ∈ A.
2.35
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Discrete Dynamics in Nature and Society
Corollary 2.7. Let A be a commutative C∗ -algebra and let f : A → A∗ be a mapping such that
f0 0, and 2.12 and 2.29. Then
lim
1
m → ∞ 2m
f2m a 0,
fa ≤ 13n − 24 δ
nn − 4
2.36
for all a ∈ A.
References
1 B. E. Johnson, “Cohomology in Banach algebras,” Memoirs of the American Mathematical Society, vol.
127, 1972.
2 W. G. Bade, P. G. Curtis, and H. G. Dales, “Amenability and weak amenability for Beurling and
Lipschits algebra,” Proceedings London Mathematical Society, vol. 55, no. 3, pp. 359–377, 1987.
3 S. M. Ulam, A Collection ofMathematical Problems, Interscience Publisher, New York, NY, USA, 1960.
4 S. M. Ulam, Problems in Modern Mathematics, vol. 6, Wiley-Interscience, New York, 1964.
5 D.H. Hyers, “On the stability of the linear functional equation,” Proceedings of the National Academy of
Sciences of the United States of America, vol. 27, pp. 222–224, 1941.
6 M. E. Gordji, “Nearly involutions onBanach algebras; A fixed point approach,” Fixed Point Theory. In
press.
7 H. G. Dales, Banach Algebras and Automatic Continuity, vol. 24 of London Mathematical Society Monographs, New Series, Oxford Science Publications, The Clarendon Press, Oxford University Press, New
York, NY, USA, 2000.
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