Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2012, Article ID 835893, 12 pages
doi:10.1155/2012/835893
Research Article
Stability of a Bi-Additive Functional Equation in
Banach Modules Over a C -Algebra
Won-Gil Park1 and Jae-Hyeong Bae2
1
Department of Mathematics Education, College of Education, Mokwon University,
Daejeon 302-729, Republic of Korea
2
Graduate School of Education, Kyung Hee University, Yongin 446-701, Republic of Korea
Correspondence should be addressed to Jae-Hyeong Bae, jhbae@khu.ac.kr
Received 6 April 2012; Accepted 30 May 2012
Academic Editor: Baodong Zheng
Copyright q 2012 W.-G. Park and J.-H. Bae. 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.
We solve the bi-additive functional equation fx y, z − w fx − y, z w 2fx, z − 2fy, w
and prove that every biadditive Borel function is bilinear. And we investigate the stability of a
biadditive functional equation in Banach modules over a unital C -algebra.
1. Introduction
In 1940, Ulam proposed the stability problem see 1.
Let G1 be a group, and let G2 be a metric group with the metric d·, ·. Given ε >
0, does there exist a δ > 0 such that if a mapping h : G1 → G2 satisfies the inequality
dhxy, hxhy < δ for all x, y ∈ G1 then there is a homomorphism H : G1 → G2 with
dhx, Hx < ε for all x ∈ G1 ?
In 1941, this problem was solved by Hyers 2 in the case of Banach space. Thereafter,
many authors investigated solutions or stability of various functional equations see 3–21.
Let X and Y be real or complex vector spaces. In 1989, Aczél and Dhombres 22
proved that a mapping g : X → Y satisfies the quadratic functional equation
g x y g x − y 2gx 2g y
1.1
if and only if there exists a symmetric bi-additive mapping S : X × X → Y such that gx Sx, x, where
1 S x, y : g x y − g x − y
4
1.2
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Discrete Dynamics in Nature and Society
for all x, y ∈ X. For a mapping f : X × X → Y , consider the bi-additive functional equation:
f x y, z − w f x − y, z w 2fx, z − 2f y, w .
1.3
For a mapping g : X → Y satisfying 1.1, the Aczél’s bi-additive mapping S : X × X → Y
given by 1.2 is a solution of 1.3.
In this paper, we find out the general solution of the bi-additive functional equation
1.3 and investigate the linearity of bi-additive Borel functions. And we investigate the
stability of 1.3 in Banach modules over a unital C -algebra.
2. Solution of the bi-additive Functional Equation 1.3
The general solution of the bi-additive functional equation 1.3 is as follows.
Theorem 2.1. A mapping f : X × X → Y satisfies 1.3 if and only if the mapping f is bi-additive.
Proof. Assume that the mapping f satisfies 1.3. Letting x y z w 0 in 1.3, we gain
f0, 0 0. Putting w z in 1.3, we get
f x y, 0 f x − y, 2z 2fx, z − 2f y, z
2.1
for all x, y, z ∈ X. Setting y x in 2.1, we have
fx, 0 −f0, z
2.2
for all x, z ∈ X. Taking z 0 resp., x 0 in the above equation, we obtain
fx, 0 0 resp., f0, z 0
2.3
for all x ∈ X resp., for all z ∈ X. Letting x w 0 in 1.3 and using 2.3, we gain
f −y, z −f y, z
2.4
for all y, z ∈ X. Putting y 0 in 2.1 and using 2.3, we get
fx, 2z 2fx, z
2.5
for all x, z ∈ X. Replacing y by −y in 2.1 and using 2.3, 2.4, and 2.5 and the above
equation, we see that fx y, z fx, z fy, z for all x, y, z ∈ X.
On the other hand, letting y x in 1.3 and using 2.3, we gain
f2x, z − w 2fx, z − 2fx, w
2.6
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for all x, z, w ∈ X. Putting y z 0 in 1.3 and using 2.3, we get
fx, −w −fx, w
2.7
for all x, w ∈ X. Setting w 0 in 2.6 and using 2.3, we have
f2x, z 2fx, z
2.8
for all x, z ∈ X. Replacing w by −w in 2.6 and using 2.7 and 2.8, we obtain that fx, z w fx, z fx, w for all x, z, w ∈ X.
The converse is trivial.
1.1.
The bi-additive functional equation 1.3 is related to the quadratic functional equation
If f : X × X → Y is a mapping satisfying 1.3 and g : X → Y is the mapping given
by gx : fx, x for all x ∈ X, then one can easily obtain that g satisfies 1.1.
Let a ∈ R and g : X → Y be a mapping satisfying 1.1. If f : X × X → Y is the
mapping given by fx, y : a/4gx y − gx − y for all x, y ∈ X, then one can easily
prove that f satisfies 1.3. Furthermore, gx fx, x holds for all x ∈ X if a 1.
The following is a result on bi-additive Borel functions.
Theorem 2.2. Let ψ : R×R → R be a bi-additive Borel function; then it is bilinear, that is, it satisfies
ψs, t stψ1, 1 for all s, t ∈ R.
Proof. Since the function ψ is bi-additive, we gain
ψ pu, qv pqψu, v
2.9
for all p, q ∈ Q and all u, v ∈ R. Letting p v 1 in equality 2.9, we get
ψ u, q qψu, 1
2.10
for all q ∈ Q and all u ∈ R. Putting u v 1 in equality 2.9 again, we have
ψ p, q pqψ1, 1
2.11
for all p, q ∈ Q. Note that the function v → ψu, v is measurable for each fixed u ∈ R see 23,
Proposition 2.34. Since the function v → ψu, v is additive for each fixed u ∈ R, by 24, it
is continuous for each fixed u ∈ R. By the same reasoning, the function u → ψu, v is also
continuous for each fixed v ∈ R. Let s, t ∈ R be fixed. Since ψ is measurable, by 25, Theorem
7.14.26, for every m ∈ N there is a closed set Fm ⊂ s, s 1 such that μs, s 1 \ Fm < 1/m
and ψ|Fm ×R is continuous. Since μFm → 1, one can choose um ∈ Fm satisfying um → s.
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Discrete Dynamics in Nature and Society
Take a sequence {qn } in Q converging to t. For each fixed m ∈ N, take a sequence {pn } in Q
converging to um . By equalities 2.10 and 2.11, we see that
ψum , t ψ um , lim qn lim ψ um , qn lim qn ψum , 1 tψum , 1
n→∞
tψ
n→∞
n→∞
lim pn , 1 t lim ψ pn , 1 t lim pn ψ1, 1 tum ψ1, 1
n→∞
n→∞
2.12
n→∞
for all m ∈ N. Hence we obtain that
ψs, t ψ
lim um , t
m→∞
lim ψum , t lim tum ψ1, 1 stψ1, 1,
m→∞
m→∞
2.13
as desired.
3. Stability of the bi-additive Functional Equation 1.3
From now on, let X be a normed space, Y a complete normed space, and r /
2 a nonnegative
real number. In this section, we investigate the stability of the bi-additive functional equation
1.3.
Lemma 3.1. Let f : X × X → Y be a mapping such that
f x y, z − w f x − y, z w − 2fx, z 2f y, w 4ε,
r 0,
r
≤
r
r
r
ε x y z w , 0 < r /
2
3.1
for all x, y, z, w ∈ X. Then there exists a unique bi-additive mapping F : X × X → Y satisfying 1.3
such that
⎧
⎪
2ε f0, 0,
⎪
⎪
⎪
⎪
⎪
⎪
⎨ 3ε xr yr ,
r
f x, y − F x, y ≤ 4 − 2
⎪
⎪
⎪
r ⎪
3ε ⎪
r
,
⎪
⎪
⎩ 2r − 4 x y
r 0,
0 < r < 2,
3.2
r > 2
for all x, y ∈ X. The mapping F is given by
⎧
1 j
⎪
⎪
f 2 x, 2j y ,
⎨jlim
→ ∞ 4j
F x, y :
x y
⎪
⎪
⎩ lim 4j f j , j ,
j →∞
2 2
for all x, y ∈ X.
0 ≤ r < 2,
r > 2
3.3
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Proof. Consider the case r ∈ 0, 2. Letting y x and w −z in 3.1, we gain
f2x, 2z f0, 0 − 2fx, z 2fx, −z ≤ 2ε xr zr
3.4
for all x, z ∈ X. Putting x z 0 in 3.4, we get f0, 0 0. Putting x z 0 in 3.1, we get
f y, −w f −y, w 2f y, w ≤ ε yr wr
3.5
for all y, w ∈ X. Replacing y by x and w by z in the above inequality, we have
fx, −z f−x, z 2fx, z ≤ εxr zr 3.6
for all x, z ∈ X. Setting y −x and w z in 3.1, we obtain
f2x, 2z − 2fx, z 2f−x, z ≤ 2ε xr zr
3.7
for all x, z ∈ X. By 3.4 and 3.6, we gain
f2x, 2z − 4fx, z fx, −z − f−x, z ≤ 3ε xr zr
3.8
for all x, z ∈ X. By 3.4 and 3.7, we get
fx, −z − f−x, z ≤ 2ε xr zr
3.9
for all x, z ∈ X. By 3.4, 3.6, and 3.7, we have
f2x, 2z − 4fx, z ≤ 3ε xr zr
3.10
for all x, z ∈ X. Replacing x by 2j x and z by 2j z and dividing 4j1 , we obtain that
rj 1 j
f 2 x, 2j z − 1 f 2j1 x, 2j1 z ≤ 3ε · 2 xr zr
4j
j1
j1
4
4
3.11
for all x, z ∈ X and all j 0, 1, 2, . . .. For given integers l, m 0 ≤ l < m, we obtain that
m−1
1 l
3ε · 2rj f 2 x, 2l z − 1 f2m x, 2m z ≤
xr zr
4l
m
j1
4
4
jl
3.12
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Discrete Dynamics in Nature and Society
for all x, z ∈ X. By 3.12, the sequence {1/4j f2j x, 2j y} is a Cauchy sequence for all x, y ∈
X. Since Y is complete, the sequence {1/4j f2j x, 2j y} converges for all x, y ∈ X. Define
F : X × X → Y by Fx, y : limj → ∞ 1/4j f2j x, 2j y for all x, y ∈ X. By 3.1, we have
1 j
f 2 x y , 2j z − w 1 f 2j x − y , 2j z w − 2 f 2j x, 2j z 2 f 2j y, 2j w 4j
j
j
j
4
4
4
≤ε
r
2rj xr y zr wr
j
4
3.13
for all x, y, z, w ∈ X and all j 0, 1, 2, . . .. Letting j → ∞ in the above inequality, we see that
F satisfies 1.3. Setting l 0 and taking m → ∞ in 3.12, one can obtain inequality 3.2. If
G : X × X → Y is another mapping satisfying 1.3 and 3.2, by Theorem 2.1, we obtain that
F x, y − G x, y 1 F 2n x, 2n y − G 2n x, 2n y n
4
1 1 ≤ n F 2n x, 2n y − f 2n x, 2n y n f 2n x, 2n y − G 2n x, 2n y 4
4
≤
r 6ε · 2nr−2 xr y −→ 0
4 − 2r
as n −→ ∞
3.14
for all x, y ∈ X. Hence the mapping F is the unique bi-additive mapping satisfying 1.3, as
desired.
The proof of the case r ∈ {0} ∪ 2, ∞ is similar to that of the case r ∈ 0, 2.
From now on, let A be a unital C -algebra with a norm | · |, and let A M and A N be left
Banach A-modules with norms || · || and · , respectively. Put A1 : {a ∈ A | |a| 1}.
A bi-additive mapping F : A M × A M → A N satisfying 1.3 is called A-quadratic if
Fax, ay a2 Fx, y for all a ∈ A and all x, y ∈ A M.
Theorem 3.2. Let f :
AM
× AM →
AN
be a mapping such that
f ax ay, az − aw f ax − ay, az aw − 2a2 fx, z 2a2 f y, w 4ε,
≤
ε xr yr zr wr ,
r 0,
2
0 < r /
3.15
for all a ∈ A1 and all x, y, z, w ∈ A M. If ftx, ty is continuous in t ∈ R for each fixed x, y ∈ A M,
then there exists a unique bi-additive A-quadratic mapping F : A M × A M → A N satisfying 1.3
and inequality 3.2.
Proof. Consider the case r ∈ 0, 2. By Lemma 3.1, it follows from the inequality of the
statement for a 1 that there exists a unique bi-additive mapping F : A M × A M → A N
satisfying 1.3 and inequality 3.2. Let x0 , y0 ∈ A M be fixed. And let L : A N → R be any
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real continuous linear functional, that is, L is an arbitrary real functional element of the dual
space of A N restricted to the scalar field R. For n ∈ N, consider the functions ψn : R → R
defined by ψn t : 1/4n Lf2n tx0 , 2n ty0 for all t ∈ R. By the assumption that ftx, ty is
continuous in t ∈ R for each fixed x, y ∈ A M, the function ψn is continuous for all n ∈ N. Note
that ψn t 1/4n Lf2n tx0 , 2n ty0 L1/4n f2n tx0 , 2n ty0 for all n ∈ N and all t ∈ R.
By the proof of Lemma 3.1, the sequence ψn t is a Cauchy sequence for all t ∈ R. Define a
function ψ : R → R by ψt : limn → ∞ ψn t for all t ∈ R. Note that ψt LFtx0 , ty0 for
all t ∈ R. Since F is bi-additive, we get
ψs t ψs − t L F s tx0 , s ty0 L F s − tx0 , s − ty0
L F s tx0 , s ty0 F s − tx0 , s − ty0
L F sx0 tx0 , sy0 ty0 F sx0 − tx0 , sy0 − ty0
L 2F sx0 , sy0 2F tx0 , ty0
2L F sx0 , sy0 2L F tx0 , ty0 2ψs 2ψt
3.16
for all s, t ∈ R. Since ψ is the pointwise limit of continuous functions, it is a Borel function.
Thus the function ψ as a measurable quadratic function is continuous see 26 so has the
form ψt t2 ψ1 for all t ∈ R. Hence we have
L F tx0 , ty0 ψt t2 ψ1 t2 L F x0 , y0 L t2 F x0 , y0
for all t ∈ R. Since L is any continuous linear functional, the bi-additive mapping F :
→ A N satisfies Ftx0 , ty0 t2 Fx0 , y0 for all t ∈ R. Therefore we obtain
AM
F tx, ty t2 F x, y
3.17
AM
×
3.18
for all t ∈ R and all x, y ∈ A M. Let j be an arbitrary positive integer. Replacing x and z by 2j x
and 2j z, respectively, and letting y w 0 in inequality 3.15, we gain
f 2j ax, 2j az − a2 f 2j x, 2j z a2 f0, 0 ≤ 2rj−1 ε xr zr
3.19
for all a ∈ A1 and all x, z ∈ A M. Note that there is a constant K > 0 such that the condition
av ≤ K|a|v
3.20
for each a ∈ A and each v ∈ A N see 27, Definition 12. For all a ∈ A1 and all x, y ∈ A M,
we get
1
K|a|2 r
r
j
j
2
j
j
r−2j−1
f0, 0 −→ 0
2
ax,
2
ay
−
a
f
2
x,
2
y
≤
2
ε
x
z
f
j
j
4
4
3.21
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Discrete Dynamics in Nature and Society
as j → ∞. Hence we have
1 1 F ax, ay lim j f 2j ax, 2j ay a2 lim j f 2j x, 2j y a2 F x, y
j →∞4
j →∞4
for all a ∈ A1 and all x, y ∈
obtain
A M.
3.22
Since Fax, ay a2 Fx, y for each a ∈ A1 , by 3.18, we
a
a
a
a
x, y a2 F x, y
F ax, ay F |a| x, |a| y |a|2 F
|a|
|a|
|a| |a|
3.23
for all nonzero a ∈ A and all x, y ∈ A M. By 3.18, we get F0x, 0y 02 Fx, y for all x, y ∈
A M. Therefore the bi-additive mapping F is the unique A-quadratic mapping satisfying the
inequality 3.2.
The proof of the case r ∈ {0} ∪ 2, ∞ is similar to that of the case r ∈ 0, 2.
We obtain the Hyers-Ulam stability of 1.3 as a corollary of Theorem 3.2.
Corollary 3.3. Let E be a complex normed space and f : E × E → C a function such that
f λx λy, λz − λw f λx − λy, λz λw − 2λ2 fx, z 2λ2 f y, w ≤ ε
3.24
for all λ ∈ T : {λ ∈ C : |λ| 1} and all x, y, z, w ∈ E. If ftx, ty is continuous in t ∈ R for each
fixed x, y ∈ E, then there exists a unique bi-additive C-quadratic mapping F : E × E → C satisfying
1.3 such that fx, y − Fx, y ≤ ε/2 f0, 0 for all x, y ∈ E.
Put Ain : {a ∈ A | a is invertible in A}, Asa : {a ∈ A | a a}, A : {a ∈ Asa |
Spa ⊂ 0, ∞}, and A1 : A1 ∩ A .
A unital C -algebra A is said to have real rank 0 see 28 if the invertible self-adjoint
elements are dense in Asa .
For any element a ∈ A, a a1 ia2 , where a1 : a a /2 and a2 : a − a /2i are
self-adjoint elements, furthermore, a a1 − a−1 ia2 − ia−2 , where a1 , a−1 , a2 , and a−2 are positive
elements see 27, Lemma 38.8.
Theorem 3.4. Let A be of real rank 0, and let f :
AM
×
AM
→
AN
be a mapping such that
f ax ay, bz − bw f ax − ay, bz bw − 2abfx, z 2ab y, w 4ε,
r 0,
r
≤
r
r
r
ε x y z w , 0 < r /
2
3.25
for all a, b ∈ A1 ∩ Ain ∪ {i} and all x, y, z, w ∈ A M. For each fixed x, y ∈ A M, let the sequence
{1/4j f2j ax, 2j by} converge uniformly on A1 × A1 . If fax, by is continuous in a, b ∈ A1 ∪
R2 for each fixed x, y ∈ A M, then there exists a unique bi-additive A-quadratic mapping F : A M ×
A M → A N satisfying 1.3 and inequality 3.2 such that Fax, by abFx, y for all a, b ∈
A1 ∪ {i} and all x, y ∈ A M.
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Proof. Consider the case r ∈ 0, 2. By Lemma 3.1, there exists a unique bi-additive mapping
F : A M × A M → A N satisfying 1.3 and inequality 3.2 on A M × A M. Let x0 , y0 ∈ A M
be fixed. And let L be an arbitrary real functional element of the dual space of A N restricted
to the scalar field R. For n ∈ N, consider the functions ψn : R × R → R defined by ψn s, t :
1/4n Lf2n sx0 , 2n ty0 for all s, t ∈ R. By the assumption that fax, by is continuous in
a, b ∈ A1 ∪ R2 for each fixed x, y ∈ A M, the function ψn is continuous for all n ∈ N.
Note that ψn s, t 1/4n Lf2n sx0 , 2n ty0 L1/4n f2n sx0 , 2n ty0 for all n ∈ N and all
s, t ∈ R. By the proof of Lemma 3.1, the sequence ψn s, t is a Cauchy sequence for all s, t ∈ R.
Define a function ψ : R × R → R by ψs, t : limn → ∞ ψn s, t for all s, t ∈ R. Note that
ψs, t LFsx0 , ty0 for all s, t ∈ R. Since the mapping F is bi-additive, we have
ψs1 s2 , t1 − t2 ψs1 − s2 , t1 t2 L F s1 s2 x0 , t1 − t2 y0 L F s1 − s2 x0 , t1 t2 y0
L F s1 s2 x0 , t1 − t2 y0 F s1 − s2 x0 , t1 t2 y0
L F s1 x0 s2 x0 , t1 y0 − t2 y0 F s1 x0 − s2 x0 , t1 y0 t2 y0
L 2F s1 x0 , t1 y0 − 2F s2 x0 , t2 y0 2L F s1 x0 , t1 y0 − 2L F s2 x0 , t2 y0
3.26
2ψs1 , t1 − 2ψs2 , t2 for all s1 , s2 , t1 , t2 ∈ R. Since ψ is the pointwise limit of continuous functions, it is a Borel
function. By Theorem 2.2, we gain ψs, t stψ1, 1 for all s, t ∈ R. Hence we get
L F sx0 , ty0 ψs, t stψ1, 1 stL F x0 , y0 L stF x0 , y0
for all s, t ∈ R. Since L is any continuous linear functional, the bi-additive mapping F :
A M → A N satisfies Fsx0 , ty0 stFx0 , y0 for all s, t ∈ R. Therefore we obtain
F sx, ty stF x, y
3.27
AM ×
3.28
for all s, t ∈ R and all x, y ∈ A M. Let j be an arbitrary positive integer. Replacing x and z by
2j x and 2j z, respectively, and letting y w 0 in inequality 3.25, we get
f 2j ax, 2j bz − abf 2j x, 2j z abf0, 0 ≤ 2rj−1 ε xr zr
3.29
for all a, b ∈ A1 ∩ Ain ∪ {i} and all x, z ∈ A M. By inequality 3.20 and the above inequality,
for all a, b ∈ A1 ∩ Ain ∪ {i} and all x, z ∈ A M, we have
1
j
j
j
j
2
ax,
2
bz
−
abf
2
x,
2
z
f
4j
r−2j−1
≤2
r
ε x z
r
K|a||b| f0, 0 −→ 0 as j −→ ∞.
j
4
3.30
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Discrete Dynamics in Nature and Society
Hence we obtain that
1 1 F ax, by lim j f 2j ax, 2j by ab lim j f 2j x, 2j y abF x, y
j →∞4
j →∞4
3.31
for all a, b ∈ A1 ∩ Ain ∪ {i} and all x, y ∈ A M. Let c, d ∈ A1 \ Ain . Since Ain ∩ Asa is dense
in Asa , there exists two sequences {cj } and {dj } in Ain ∩ Asa such that cj → c and dj → d
as j → ∞. Put pj : 1/|cj |cj and qj : 1/|dj |dj for all j ∈ N. Then pj → c and qj → d as
j → ∞. Set aj : pj pj and bj : qj qj for all j ∈ N. Then aj → c and bj → d as j → ∞
and aj , bj ∈ A1 ∩ Ain . Since {1/4j f2j ax, 2j by} is uniformly converges on A1 × A1 for each
x, y ∈ A M and fax, by is continuous in a, b ∈ A1 for each x, y ∈ A M, we see that Fax, by
is also continuous in a, b ∈ A1 for each x, y ∈ A M. In fact, we gain
lim
1 j
j
f
2
ax,
2
by
a,b → c,d j → ∞ 4j
1 j
1 j
j
j
lim
f
2
ax,
2
by
lim
f
2
cx,
2
dy
F cx, dy
lim
j
j
j → ∞ a,b → c,d 4
j →∞4
3.32
F ax, by a,b → c,d
for all x, y ∈
A M.
lim
lim
Thus we get
lim F aj x, bj y F lim aj x, lim bj y F cx, dy
j →∞
for all x, y ∈
A M.
j →∞
j →∞
3.33
By equality 3.31, we have
F aj x, bj y − cdF x, y aj bj F x, y − cdF x, y −→ cdF x, y − cdF x, y 0
as j → ∞ for all x, y ∈
A M.
3.34
By equality 3.33 and the above convergence, we see that
F cx, dy − cdF x, y ≤ F cx, dy − F aj x, bj y F aj x, bj y − cdF x, y −→ 0
3.35
as j → ∞ for all x, y ∈
A M.
By equality 3.31 and the above convergence, we obtain
F ax, by abF x, y
3.36
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for all a, b ∈ A1 ∪ {i} and all x, y ∈
A M.
Since the mapping F is bi-additive, we see that
F ax, ay F a1 x − a−1 x ia2 x − ia−2 x, a1 y − a−1 y ia2 y − ia−2 y
F a1 x, a1 y − F a1 x, a−1 y F a1 x, ia2 y − F a1 x, ia−2 y
− F a−1 x, a1 y F a−1 x, a−1 y − F a−1 x, ia2 y F a−1 x, ia−2 y
F ia2 x, a1 y − F ia2 x, a−1 y F ia2 x, ia2 y − F ia2 x, ia−2 y
− F ia−2 x, a1 y F ia−2 x, a−1 y − F ia−2 x, ia2 y F ia−2 x, ia−2 y
for all a ∈ A and all x, y ∈
A M.
3.37
By 3.28 and equality 3.36, we have
q
p
p
q
x, q
y p qF
x,
y pqF x, y
F px, qy F p
p
q
p q
11
3.38
for all p, q ∈ {a1 , a−1 , a2 , a−2 } and all x, y ∈A M. Note that a1 a−1 a−1 a1 a2 a−2 a−2 a2 0.
Hence we obtain that
2 2 F ax, ay a2 F x, y ia1 a2 F x, y − ia1 a−2 F x, y a−1 F x, y
− ia−1 a2 F x, y ia−1 a−2 F x, y ia2 a1 F x, y − ia2 a−1 F x, y
2 2 − a2 F x, y − ia−2 a1 F x, y ia−2 a−1 F x, y − a−2 F x, y
2
2
a1 ia1 a2 − ia1 a−2 a−1 − ia−1 a2 ia−1 a−2
3.39
2
2 ia2 a1 − ia2 a−1 − a2 − ia−2 a1 ia−2 a−1 − a−2 F x, y
2 a1 − a−1 ia2 − ia−2 F x, y a2 F x, y
for all a ∈ A and all x, y ∈ A M.
The proof of the case r ∈ {0} ∪ 2, ∞ is similar to that of the case r ∈ 0, 2.
Acknowledgment
This research was supported by Basic Science Research Program through the National
Research Foundation of Korea NRF funded by the Ministry of Education, Science and
Technology Grant no. 2012003499.
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