Journal of Lie Theory
Volume 15 (2005) 393–414
c 2005 Heldermann Verlag
Homomorphisms between Lie JC ∗ -Algebras and
Cauchy–Rassias Stability of Lie JC ∗ -Algebra Derivations
Chun-Gil Park ∗
Communicated by A. Valette
Abstract.
It is shown that every almost linear mapping h: A → B of
a unital Lie JC ∗ -algebra A to a unital Lie JC ∗ -algebra B is a Lie JC ∗ algebra homomorphism when h(2n u ◦ y) = h(2n u) ◦ h(y) , h(3n u ◦ y) =
h(3n u)◦h(y) or h(q n u◦y) = h(q n u)◦h(y) for all y ∈ A , all unitary elements
u ∈ A and n = 0, 1, 2, · · · , and that every almost linear almost multiplicative
mapping h: A → B is a Lie JC ∗ -algebra homomorphism when h(2x) =
2h(x) , h(3x) = 3h(x) or h(qx)qh(x) for all x ∈ A . Here the numbers 2, 3, q
depend on the functional equations given in the almost linear mappings or
in the almost linear almost multiplicative mappings. Moreover, we prove the
Cauchy–Rassias stability of Lie JC ∗ -algebra homomorphisms in Lie JC ∗ algebras, and of Lie JC ∗ -algebra derivations in Lie JC ∗ -algebras.
Mathematics Subject Classification: 17B40, 39B52, 46L05, 17A36.
Keywords and Phrases: Lie JC ∗ -algebra homomorphism. Lie JC ∗ -algebra
derivation, stability, linear functional equation.
1. Introduction
The original motivation to introduce the class of nonassociative algebras known
as Jordan algebras came from quantum mechanics (see [18]). Let L(H) be the
real vector space of all bounded self-adjoint linear operators on H , interpreted
as the (bounded) observables of the system. In 1932, Jordan observed that L(H)
.
is a (nonassociative) algebra via the anticommutator product x ◦ y := xy+yx
2
A commutative algebra X with product x ◦ y is called a Jordan algebra. A
unital Jordan C ∗ -subalgebra of a C ∗ -algebra, endowed with the anticommutator
product, is called a JC ∗ -algebra.
A unital C ∗ -algebra C , endowed with the Lie product [x, y] = xy−yx
2
on C , is called a Lie C ∗ -algebra. A unital C ∗ -algebra C , endowed with the Lie
product [·, ·] and the anticommutator product ◦, is called a Lie JC ∗ -algebra if
(C, ◦) is a JC ∗ -algebra and (C, [·, ·]) is a Lie C ∗ -algebra (see [5], [6]).
∗
Supported by grant No. R05-2003-000-10006-0 from the Basic Research Program of
the Korea Science & Engineering Foundation.
c Heldermann Verlag
ISSN 0949–5932 / $2.50 394
Park
Let X and Y be Banach spaces with norms || · || and k · k , respectively.
Consider f : X → Y to be a mapping such that f (tx) is continuous in t ∈ R for
each fixed x ∈ X . Assume that there exist constants θ ≥ 0 and p ∈ [0, 1) such
that
kf (x + y) − f (x) − f (y)k ≤ θ(||x||p + ||y||p )
for all x, y ∈ X . Rassias [11] showed that there exists a unique R -linear mapping
T : X → Y such that
kf (x) − T (x)k ≤
2θ
||x||p
p
2−2
for all x ∈ X . Găvruta [1] generalized the Rassias’ result: Let G be an abelian
group and Y a Banach space. Denote by ϕ : G × G → [0, ∞) a function such
that
∞
X
ϕ(x,
e y) =
2−j ϕ(2j x, 2j y) < ∞
j=0
for all x, y ∈ G . Suppose that f : G → Y is a mapping satisfying
kf (x + y) − f (x) − f (y)k ≤ ϕ(x, y)
for all x, y ∈ G . Then there exists a unique additive mapping T : G → Y such
that
1
e x)
kf (x) − T (x)k ≤ ϕ(x,
2
for all x ∈ G . C. Park [7] applied the Găvruta’s result to linear functional
equations in Banach modules over a C ∗ -algebra.
Jun and Lee [2] proved the following: Denote by ϕ : X \ {0} × X \ {0} →
[0, ∞) a function such that
ϕ(x,
e y) =
∞
X
3−j ϕ(3j x, 3j y) < ∞
j=0
for all x, y ∈ X \ {0} . Suppose that f : X → Y is a mapping satisfying
k2f (
x+y
) − f (x) − f (y)k ≤ ϕ(x, y)
2
for all x, y ∈ X \ {0} . Then there exists a unique additive mapping T : X → Y
such that
1
kf (x) − f (0) − T (x)| ≤ (ϕ(x,
e −x) + ϕ(−x,
e
3x))
3
for all x ∈ X \ {0}. C. Park and W. Park [9] applied the Jun and Lee’s result to
the Jensen’s equation in Banach modules over a C ∗ -algebra.
l
Recently, Trif [17] proved the following: Let q := l(d−1)
and r := − d−l
.
d−l
d
Denote by ϕ : X → [0, ∞) a function such that
ϕ(x
e 1 , · · · , xd ) =
∞
X
j=0
q −j ϕ(q j x1 , · · · , q j xd ) < ∞
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for all x1 , · · · , xd ∈ X . Suppose that f : X → Y is a mapping satisfying
d
X
x1 + · · · + xd
) + d−2 Cl−1
f (xj )
kd d−2 Cl−2 f (
d
j=1
X
xj + · · · + xjl
−l
f( 1
)k ≤ ϕ(x1 , · · · , xd )
l
1≤j1 <···<jl ≤d
for all x1 , · · · , xd ∈ X . Then there exists a unique additive mapping T : X → Y
such that
1
kf (x) − f (0) − T (x)k ≤
ϕ(qx,
e
rx, · · · , rx)
| {z }
l · d−1 Cl−1
d−1 times
for all x ∈ X . And C. Park [8] applied the Trif’s result to the Trif functional
equation in Banach modules over a C ∗ -algebra. Several authors have investigated
functional equations (see [10]–[16]).
l
Throughout this paper, let q = l(d−1)
d−l and r = − d−l for positive integers
l, d with 2 ≤ l ≤ d − 1 . Let A be a unital Lie JC ∗ -algebra with norm || · || ,
unit e and unitary group U (A) = {u ∈ A | uu∗ = u∗ u = e} , and B a unital Lie
JC ∗ -algebra with norm k · k and unit e0 .
Using the stability methods of linear functional equations, we prove that
every almost linear mapping h : A → B is a Lie JC ∗ -algebra homomorphism
when h(2n u ◦ y) = h(2n u) ◦ h(y) , h(3n u ◦ y) = h(3n u) ◦ h(y) or h(q n u ◦ y) =
h(q n u) ◦ h(y) for all y ∈ A , all u ∈ U (A) and n = 0, 1, 2, · · ·, and that every
almost linear almost multiplicative mapping h : A → B is a Lie JC ∗ -algebra
homomorphism when h(2x) = 2h(x) , h(3x) = 3h(x) or h(qx) = qh(x) for all
x ∈ A. We moreover prove the Cauchy–Rassias stability of Lie JC ∗ -algebra
homomorphisms in unital Lie JC ∗ -algebras, and of Lie JC ∗ -algebra derivations
in unital Lie JC ∗ -algebras.
2. Homomorphisms between Lie JC ∗ -algebras
Definition 2.1. A C -linear mapping H : A → B is called a Lie JC ∗ -algebra
homomorphism if H : A → B satisfies
H(x ◦ y) = H(x) ◦ H(y),
H([x, y]) = [H(x), H(y)],
H(x∗ ) = H(x)∗
for all x, y ∈ A.
Remark 2.1. A C -linear mapping H : A → B is a C ∗ -algebra homomorphism
if and only if the mapping H : A → B is a Lie JC ∗ -algebra homomorphism.
Assume that H is a Lie JC ∗ -algebra homomorphism. Then
H(xy) = H([x, y] + x ◦ y) = H([x, y]) + H(x ◦ y)
= [H(x), H(y)] + H(x) ◦ H(y) = H(x)H(y)
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for all x, y ∈ A. So H is a C ∗ -algebra homomorphism.
Assume that H is a C ∗ -algebra homomorphism. Then
H(x)H(y) − H(y)H(x)
xy − yx
)=
= [H(x), H(y)],
2
2
H(x)H(y) + H(y)H(x)
xy + yx
H(x ◦ y) = H(
)=
= H(x) ◦ H(y)
2
2
H([x, y] = H(
for all x, y ∈ A. So H is a Lie JC ∗ -algebra homomorphism.
We are going to investigate Lie JC ∗ -algebra homomorphisms between
∗
Lie JC -algebras associated with the Cauchy functional equation.
Theorem 2.1.
Let h : A → B be a mapping satisfying h(0) = 0 and
n
n
h(2 u ◦ y) = h(2 u) ◦ h(y) for all y ∈ A, all u ∈ U (A) and n = 0, 1, 2, · · ·,
for which there exists a function ϕ : A4 → [0, ∞) such that
ϕ(x,
e y, z, w) :=
∞
X
2−j ϕ(2j x, 2j y, 2j z, 2j w) < ∞,
(2.i)
j=0
kh(µx + µy + [z, w]) − µh(x) − µh(y) − [h(z), h(w)]k
≤ ϕ(x, y, z, w),
n ∗
n ∗
kh(2 u ) − h(2 u) k ≤ ϕ(2n u, 2n u, 0, 0)
(2.ii)
(2.iii)
for all µ ∈ T1 := {λ ∈ C | |λ| = 1}, all u ∈ U (A), all x, y, z, w ∈ A and
n
n = 0, 1, 2, · · ·. Assume that (2.iv) limn→∞ h(22n e) = e0 . Then the mapping
h : A → B is a Lie JC ∗ -algebra homomorphism.
Proof.
Put z = w = 0 and µ = 1 ∈ T1 in (2.ii). It follows from Găvruta’s
Theorem [1] that there exists a unique additive mapping H : A → B such that
kh(x) − H(x)k ≤
1
ϕ(x,
e x, 0, 0)
2
(2.0)
for all x ∈ A. The additive mapping H : A → B is given by
1
h(2n x)
n→∞ 2n
H(x) = lim
(2.1)
for all x ∈ A.
By the assumption, for each µ ∈ T1 ,
kh(2n µx) − 2µh(2n−1 x)k ≤ ϕ(2n−1 x, 2n−1 x, 0, 0)
for all x ∈ A. And one can show that
kµh(2n x) − 2µh(2n−1 x)k ≤ |µ| · kh(2n x) − 2h(2n−1 x)k ≤ ϕ(2n−1 x, 2n−1 x, 0, 0)
for all µ ∈ T1 and all x ∈ A. So
kh(2n µx) − µh(2n x)k ≤kh(2n µx) − 2µh(2n−1 x)k + k2µh(2n−1 x) − µh(2n x)k
≤ϕ(2n−1 x, 2n−1 x, 0, 0) + ϕ(2n−1 x, 2n−1 x, 0, 0)
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for all µ ∈ T1 and all x ∈ A. Thus 2−n kh(2n µx) − µh(2n x)k → 0 as n → ∞
for all µ ∈ T1 and all x ∈ A. Hence
h(2n µx)
µh(2n x)
=
lim
= µH(x)
n→∞
n→∞
2n
2n
H(µx) = lim
(2.2)
for all µ ∈ T1 and all x ∈ A.
Now let λ ∈ C (λ 6= 0) and M an integer greater than 4|λ|. Then
λ
|M
| < 14 < 1 − 23 = 13 . By [3, Theorem 1], there exist three elements µ1 , µ2 , µ3 ∈
λ
T1 such that 3 M
= µ1 + µ2 + µ3 . And H(x) = H(3 · 31 x) = 3H( 31 x) for all
x ∈ A . So H( 31 x) = 13 H(x) for all x ∈ A . Thus by (2.2)
M
λ
1
λ
M
λ
· 3 x) = M · H( · 3 x) =
H(3 x)
3
M
3 M
3
M
M
M
H(µ1 x + µ2 x + µ3 x) =
(H(µ1 x) + H(µ2 x) + H(µ3 x))
=
3
3
M
M
λ
(µ1 + µ2 + µ3 )H(x) =
· 3 H(x)
=
3
3
M
= λH(x)
H(λx) = H(
for all x ∈ A. Hence
H(ζx + ηy) = H(ζx) + H(ηy) = ζH(x) + ηH(y)
for all ζ, η ∈ C(ζ, η 6= 0) and all x, y ∈ A. And H(0x) = 0 = 0H(x) for all
x ∈ A . So the unique additive mapping H : A → B is a C -linear mapping.
Since h(2n u ◦ y) = h(2n u) ◦ h(y) for all y ∈ A, all u ∈ U (A) and
n = 0, 1, 2, · · ·,
1
1
h(2n u ◦ y) = lim n h(2n u) ◦ h(y) = H(u) ◦ h(y)
n
n→∞ 2
n→∞ 2
H(u ◦ y) = lim
(2.3)
for all y ∈ A and all u ∈ U (A) . By the additivity of H and (2.3),
2n H(u ◦ y) = H(2n u ◦ y) = H(u ◦ (2n y)) = H(u) ◦ h(2n y)
for all y ∈ A and all u ∈ U (A) . Hence
H(u ◦ y) =
1
1
H(u) ◦ h(2n y) = H(u) ◦ n h(2n y)
n
2
2
(2.4)
for all y ∈ A and all u ∈ U (A) . Taking the limit in (2.4) as n → ∞, we obtain
H(u ◦ y) = H(u) ◦ H(y)
(2.5)
for all y ∈ A and all u ∈ U (A) . Since H is C -linear and each x ∈ A is
a finite
Pm linear combination of unitary elements (see [4, Theorem 4.1.7]), i.e.,
x = j=1 λj uj (λj ∈ C, uj ∈ U (A)) ,
m
m
m
X
X
X
λj uj ◦ y) =
λj H(uj ◦ y) =
λj H(uj ) ◦ H(y)
H(x ◦ y) = H(
j=1
m
X
= H(
j=1
j=1
λj uj ) ◦ H(y) = H(x) ◦ H(y)
j=1
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for all x, y ∈ A.
By (2.iv), (2.3) and (2.5),
H(y) = H(e ◦ y) = H(e) ◦ h(y) = e0 ◦ h(y) = h(y)
for all y ∈ A. So
H(y) = h(y)
for all y ∈ A.
It follows from (2.1) that
h(22n x)
n→∞
22n
H(x) = lim
(2.6)
for all x ∈ A. Let x = y = 0 in (2.ii). Then we get
kh([z, w]) − [h(z), h(w)]k ≤ ϕ(0, 0, z, w)
for all z, w ∈ A. So
1
1
kh([2n z, 2n w]) − [h(2n z), h(2n w)]k ≤ 2n ϕ(0, 0, 2n z, 2n w)
2n
2
2
1
≤ n ϕ(0, 0, 2n z, 2n w)
2
(2.7)
for all z, w ∈ A. By (2.i), (2.6), and (2.7),
h(22n [z, w])
h([2n z, 2n w])
=
lim
n→∞
n→∞
22n
22n
1
h(2n z) h(2n w)
= lim 2n [h(2n z), h(2n w)] = lim [
,
]
n→∞ 2
n→∞
2n
2n
= [H(z), H(w)]
H([z, w]) = lim
for all z, w ∈ A.
By (2.i) and (2.iii), we get
h(2n u∗ )
h(2n u)∗
h(2n u) ∗
=
lim
=
(
lim
)
n→∞
n→∞
n→∞
2n
2n
2n
= H(u)∗
H(u∗ ) = lim
for all u ∈ U (A) . Since H : A → B is C -linear
Pm and each x ∈ A is a finite linear
combination of unitary elements, i.e., x = j=1 λj uj (λj ∈ C, uj ∈ U (A)) ,
m
m
m
X
X
X
H(x∗ ) = H(
λj u∗j ) =
λj H(u∗j ) =
λj H(uj )∗
j=1
m
X
j=1
j=1
m
X
λj H(uj ))∗ = H(
=(
j=1
λj uj )∗ = H(x)∗
j=1
for all x ∈ A.
Therefore, the mapping h : A → B is a Lie JC ∗ -algebra homomorphism,
as desired.
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399
Corollary 2.2.
Let h : A → B be a mapping satisfying h(0) = 0 and
n
n
h(2 u ◦ y) = h(2 u) ◦ h(y) for all y ∈ A, all u ∈ U (A) and n = 0, 1, 2, · · ·, for
which there exist constants θ ≥ 0 and p ∈ [0, 1) such that
kh(µx + µy + [z, w])−µh(x) − µh(y) − [h(z), h(w)]k
≤ θ(||x||p + ||y||p + ||z||p + ||w||p ),
kh(2n u∗ ) − h(2n u)∗ k ≤ 2 · 2np θ
for all µ ∈ T1 , all u ∈ U (A), all x, y, z, w ∈ A and n = 0, 1, 2, · · ·. Assume
n
that limn→∞ h(22n e) = e0 . Then the mapping h : A → B is a Lie JC ∗ -algebra
homomorphism.
Proof.
Define ϕ(x, y, z, w) = θ(||x||p + ||y||p + ||z||p + ||w||p ) , and apply
Theorem 2.1.
Theorem 2.3.
Let h : A → B be a mapping satisfying h(0) = 0 and
h(2n u ◦ y) = h(2n u) ◦ h(y) for all y ∈ A, all u ∈ U (A) and n = 0, 1, 2, · · ·,
for which there exists a function ϕ : A4 → [0, ∞) satisfying (2.i), (2.iii) and
(2.iv) such that
kh(µx + µy + [z, w]) − µh(x) − µh(y) − [h(z), h(w)]k ≤ ϕ(x, y, z, w)
(2.v)
for µ = 1, i, and all x, y, z, w ∈ A. If h(tx) is continuous in t ∈ R for each
fixed x ∈ A, then the mapping h : A → B is a Lie JC ∗ -algebra homomorphism.
Proof.
Put z = w = 0 and µ = 1 in (2.v). By the same reasoning as in
the proof of Theorem 2.1, there exists a unique additive mapping H : A → B
satisfying (2.0). The additive mapping H : A → B is given by
1
h(2n x)
n→∞ 2n
H(x) = lim
for all x ∈ A. By the same reasoning as in the proof of [11, Theorem], the
additive mapping H : A → B is R -linear.
Put y = z = w = 0 and µ = i in (2.v). By the same method as in the
proof of Theorem 2.1, one can obtain that
h(2n ix)
ih(2n x)
=
lim
= iH(x)
n→∞
n→∞
2n
2n
H(ix) = lim
for all x ∈ A. For each element λ ∈ C , λ = s + it, where s, t ∈ R . So
H(λx) = H(sx + itx) = sH(x) + tH(ix) = sH(x) + itH(x) = (s + it)H(x)
= λH(x)
for all λ ∈ C and all x ∈ A. So
H(ζx + ηy) = H(ζx) + H(ηy) = ζH(x) + ηH(y)
for all ζ, η ∈ C , and all x, y ∈ A . Hence the additive mapping H : A → B is
C -linear.
The rest of the proof is the same as in the proof of Theorem 2.1.
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Theorem 2.4. Let h : A → B be a mapping satisfying h(2x) = 2h(x) for all
x ∈ A for which there exists a function ϕ : A4 → [0, ∞) satisfying (2.i), (2.ii),
(2.iii) and (2.iv) such that
kh(2n u ◦ y) − h(2n u) ◦ h(y)k ≤ ϕ(u, y, 0, 0)
(2.vi)
for all y ∈ A, all u ∈ U (A) and n = 0, 1, 2, · · ·. Then the mapping h : A → B
is a Lie JC ∗ -algebra homomorphism.
Proof.
By the same reasoning as in the proof of Theorem 2.1, there exists a
unique C -linear mapping H : A → B satisfying (2.0).
By (2.vi) and the assumption that h(2x) = 2h(x) for all x ∈ A,
1
kh(2m 2n u ◦ 2m y) − h(2m 2n u) ◦ h(2m y)k
4m
1
1
≤ m ϕ(2m u, 2m y, 0, 0) ≤ m ϕ(2m u, 2m y, 0, 0),
4
2
kh(2n u ◦ y) − h(2n u) ◦ h(y)k =
which tends to zero as m → ∞ by (2.i). So
h(2n u ◦ y) = h(2n u) ◦ h(y)
for all y ∈ A, all u ∈ U (A) and n = 0, 1, 2, · · ·. But by (2.1),
1
h(2n x) = h(x)
n→∞ 2n
H(x) = lim
for all x ∈ A.
The rest of the proof is the same as in the proof of Theorem 2.1.
Now we are going to investigate Lie JC ∗ -algebra homomorphisms between Lie JC ∗ -algebras associated with the Jensen functional equation.
Theorem 2.5.
Let h : A → B be a mapping satisfying h(0) = 0 and
n
n
h(3 u ◦ y) = h(3 u) ◦ h(y) for all y ∈ A, all u ∈ U (A) and n = 0, 1, 2, · · ·,
for which there exists a function ϕ : (A \ {0})4 → [0, ∞) such that
ϕ(x,
e y, z, w) :=
∞
X
3−j ϕ(3j x, 3j y, 3j z, 3j w) < ∞,
(2.vii)
j=0
k2h(
µx + µy + [z, w]
) − µh(x) − µh(y)−[h(z), h(w)]k
2
≤ ϕ(x, y, z, w),
n ∗
n ∗
kh(3 u ) − h(3 u) k ≤ϕ(3n u, 3n u, 0, 0)
(2.viii)
(2.ix)
for all µ ∈ T1 , all u ∈ U (A), all x, y, z, w ∈ A and n = 0, 1, 2, · · ·. Assume
n
that limn→∞ h(33n e) = e0 . Then the mapping h : A → B is a Lie JC ∗ -algebra
homomorphism.
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401
Proof.
Put z = w = 0 and µ = 1 ∈ T1 in (2.viii). It follows from Jun
and Lee’s Theorem [2, Theorem 1] that there exists a unique additive mapping
H : A → B such that
kh(x) − H(x)k ≤
1
(ϕ(x,
e −x, 0, 0) + ϕ(−x,
e
3x, 0, 0))
3
for all x ∈ A \ {0}. The additive mapping H : A → B is given by
1
h(3n x)
n→∞ 3n
H(x) = lim
for all x ∈ A.
By the assumption, for each µ ∈ T1 ,
k2h(3n µx) − µh(2 · 3n−1 x) − µh(4 · 3n−1 x)k ≤ ϕ(2 · 3n−1 x, 4 · 3n−1 x, 0, 0)
for all x ∈ A \ {0} . And one can show that
kµh(2 · 3n−1 x) + µh(4 · 3n−1 x) − 2µh(3n x)k
≤ |µ| · kh(2 · 3n−1 x) + h(4 · 3n−1 x) − 2h(3n x)k
≤ ϕ(2 · 3n−1 x, 4 · 3n−1 x, 0, 0)
for all µ ∈ T1 and all x ∈ A \ {0}. So
1
1
kh(3n µx) − µh(3n x)k =kh(3n µx) − µh(2 · 3n−1 x) − µh(4 · 3n−1 x)
2
2
1
1
+ µh(2 · 3n−1 x) + µh(4 · 3n−1 x) − µh(3n x)k
2
2
1
≤ k2h(3n µx) − µh(2 · 3n−1 x) − µh(4 · 3n−1 x)k
2
1
+ kµh(2 · 3n−1 x) + µh(4 · 3n−1 x) − 2µh(3n x)k
2
2
≤ ϕ(2 · 3n−1 x, 4 · 3n−1 x, 0, 0)
2
for all µ ∈ T1 and all x ∈ A \ {0} . Thus 3−n kh(3n µx) − µh(3n x)k → 0 as
n → ∞ for all µ ∈ T1 and all x ∈ A \ {0}. Hence
h(3n µx)
µh(3n x)
=
lim
= µH(x)
n→∞
n→∞
3n
3n
H(µx) = lim
for all µ ∈ T1 and all x ∈ A \ {0}.
By the same reasoning as in the proof of Theorem 2.1, the unique additive
mapping H : A → B is a C -linear mapping.
By a similar method to the proof of Theorem 2.1, one can show that the
mapping h : A → B is a Lie JC ∗ -algebra homomorphism.
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Corollary 2.6.
Let h : A → B be a mapping satisfying h(0) = 0 and
n
n
h(3 u ◦ y) = h(3 u) ◦ h(y) for all y ∈ A, all u ∈ U (A) and n = 0, 1, 2, · · ·, for
which there exist constants θ ≥ 0 and p ∈ [0, 1) such that
µx + µy + [z, w]
k2h(
)−µh(x) − µh(y) − [h(z), h(w)]k
2
≤ θ(||x||p + ||y||p + ||z||p + ||w||p ),
kh(3n u∗ ) − h(3n u)∗ k ≤ 2 · 3np θ
for all µ ∈ T1 , all u ∈ U (A), all x, y, z, w ∈ A \ {0} and n = 0, 1, 2, · · ·. Assume
n
that limn→∞ h(33n e) = e0 . Then the mapping h : A → B is a Lie JC ∗ -algebra
homomorphism.
Proof.
Define ϕ(x, y, z, w) = θ(||x||p + ||y||p + ||z||p + ||w||p ) , and apply
Theorem 2.5.
One can obtain similar results to Theorems 2.3 and 2.4 for the Jensen
functional equation.
Finally, we are going to investigate Lie JC ∗ -algebra homomorphisms
between Lie JC ∗ -algebras associated with the Trif functional equation.
Theorem 2.7.
Let h : A → B be a mapping satisfying h(0) = 0 and
h(q n u ◦ y) = h(q n u) ◦ h(y) for all y ∈ A , all u ∈ U (A) and n = 0, 1, 2, · · ·,
for which there exists a function ϕ : Ad+2 → [0, ∞) such that
ϕ(x
e 1 , · · · , xd , z, w) :=
∞
X
q −j ϕ(q j x1 , · · · ,q j xd , q j z, q j w) < ∞, (2.x)
j=0
kd
d
X
[z, w]
µx1 + · · · + µxd
+
)+d−2 Cl−1
µh(xj )
d
d d−2 Cl−2
j=1
X
xj + · · · + xjl
) − [h(z), h(w)]k
−l
µh( 1
l
d−2 Cl−2 h(
(2.xi)
1≤j1 <···<jl ≤d
≤ϕ(x1 , · · · , xd , z, w),
kh(q n u∗ ) − h(q n u)∗ k ≤ϕ(q n u, · · · , q n u, 0, 0) (2.xii)
|
{z
}
d times
1
for all µ ∈ T , all u ∈ U (A), all x1 , · · · , xd , z, w ∈ A and n = 0, 1, 2, · · ·.
n
Assume that limn→∞ h(qqn e) = e0 . Then the mapping h : A → B is a Lie JC ∗ algebra homomorphism.
Proof.
Put z = w = 0 and µ = 1 ∈ T1 in (2.xi). It follows from Trif’s
Theorem [17, Theorem 3.1] that there exists a unique additive mapping H :
A → B such that
1
kh(x) − H(x)k ≤
ϕ(qx,
e
rx, · · · , rx, 0, 0)
| {z }
l · d−1 Cl−1
d−1 times
for all x ∈ A. The additive mapping H : A → B is given by
1
H(x) = lim n h(q n x)
n→∞ q
for all x ∈ A.
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Put x1 = · · · = xd = x and z = w = 0 in (2.xi). For each µ ∈ T1 ,
kd
d−2 Cl−2 (h(µx)
− µh(x))k ≤ ϕ(x, · · · , x, 0, 0)
| {z }
d times
for all x ∈ A. So
q −n kd
d−2 Cl−2 (h(µq
n
x) − µh(q n x))k ≤ q −n ϕ(q n x, · · · , q n x, 0, 0)
{z
}
|
d times
for all x ∈ A. By (2.x),
q −n kd
d−2 Cl−2 (h(µq
n
x) − µh(q n x))k → 0
as n → ∞ for all µ ∈ T1 and all x ∈ A. Thus
q −n kh(µq n x) − µh(q n x)k → 0
as n → ∞ for all µ ∈ T1 and all x ∈ A. Hence
h(q n µx)
µh(q n x)
H(µx) = lim
= lim
= µH(x)
n→∞
n→∞
qn
qn
for all µ ∈ T1 and all x ∈ A.
By the same reasoning as in the proof of Theorem 2.1, the unique additive
mapping H : A → B is a C -linear mapping.
By a similar method to the proof of Theorem 2.1, one can show that the
mapping h : A → B is a Lie JC ∗ -algebra homomorphism.
Corollary 2.8.
Let h : A → B be a mapping satisfying h(0) = 0 and
n
n
h(q u ◦ y) = h(q u) ◦ h(y) for all y ∈ A, all u ∈ U (A) and n = 0, 1, 2, · · ·, for
which there exist constants θ ≥ 0 and p ∈ [0, 1) such that
d
X
µx1 + · · · + µxd
[z, w]
kd d−2 Cl−2 h(
+
) + d−2 Cl−1
µh(xj )
d
d d−2 Cl−2
j=1
X
xj1 + · · · + xjl
−l
µh(
) − [h(z), h(w)]k
l
1≤j1 <···<jl ≤d
d
X
≤ θ(
||xj ||p + ||z||p + ||w||p ),
j=1
n ∗
n
∗
kh(q u ) − h(q u) k ≤ dq np θ
for all µ ∈ T1 , all u ∈ U (A), all x1 , · · · , xd , z, w ∈ A and n = 0, 1, 2, · · ·.
n
Assume that limn→∞ h(qqn e) = e0 . Then the mapping h : A → B is a Lie JC ∗ algebra homomorphism.
Pd
Proof. Define ϕ(x1 , · · · , xd , z, w) = θ( j=1 ||xj ||p + ||z||p + ||w||p ) , and apply
Theorem 2.7.
One can obtain similar results to Theorems 2.3 and 2.4 for the Trif
functional equation.
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3. Stability of Lie JC ∗ -algebra homomorphisms in Lie JC ∗ -algebras
We are going to show the Cauchy–Rassias stability of Lie JC ∗ -algebra homomorphisms in Lie JC ∗ -algebras associated with the Cauchy functional equation.
Theorem 3.1.
Let h : A → B be a mapping with h(0) = 0 for which there
exists a function ϕ : A6 → [0, ∞) such that
ϕ(x,
e y, z, w, a, b) :=
∞
X
2−j ϕ(2j x, 2j y, 2j z, 2j w, 2j a, 2j b) < ∞,
(3.i)
j=0
kh(µx + µy + [z, w] + a ◦ b) − µh(x) − µh(y) − [h(z), h(w)] − h(a) ◦ h(b)k
≤ ϕ(x, y, z, w, a, b),
(3.ii)
n ∗
n ∗
n
n
kh(2 u ) − h(2 u) k ≤ ϕ(2 u, 2 u, 0, 0, 0, 0) (3.iii)
for all µ ∈ T1 , all u ∈ U (A), n = 0, 1, 2, · · ·, and all x, y, z, w, a, b ∈ A . Then
there exists a unique Lie JC ∗ -algebra homomorphism H : A → B such that
kh(x) − H(x)k ≤
1
ϕ(x,
e x, 0, 0, 0, 0)
2
(3.iv)
for all x ∈ A.
Proof.
Put z = w = a = b = 0 and µ = 1 ∈ T1 in (3.ii). It follows from
Găvruta’s Theorem [1] that there exists a unique additive mapping H : A → B
satisfying (3.iv). The additive mapping H : A → B is given by
1
h(2n x)
n→∞ 2n
H(x) = lim
for all x ∈ A.
The rest of the proof is similar to the proof of Theorem 2.1.
Corollary 3.2.
Let h : A → B be a mapping with h(0) = 0 for which there
exist constants θ ≥ 0 and p ∈ [0, 1) such that
kh(µx + µy + [z, w] + a ◦ b) − µh(x) − µh(y) − [h(z), h(w)] − h(a) ◦ h(b)k
≤ θ(||x||p + ||y||p + ||z||p + ||w||p + ||a||p + ||b||p ),
kh(2n u∗ ) − h(2n u)∗ k ≤ 2 · 2np θ
for all µ ∈ T1 , all u ∈ U (A), n = 0, 1, 2, · · ·, and all x, y, z, w, a, b ∈ A . Then
there exists a unique Lie JC ∗ -algebra homomorphism H : A → B such that
kh(x) − H(x)k ≤
2θ
||x||p
2 − 2p
for all x ∈ A.
Proof. Define ϕ(x, y, z, w, a, b) = θ(||x||p +||y||p +||z||p +||w||p +||a||p +||b||p ) ,
and apply Theorem 3.1.
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Theorem 3.3.
Let h : A → B be a mapping with h(0) = 0 for which there
exists a function ϕ : A6 → [0, ∞) satisfying (3.i) and (3.iii) such that
kh(µx + µy + [z, w] + a ◦ b) − µh(x) − µh(y)−[h(z), h(w)] − h(a) ◦ h(b)k
≤ ϕ(x, y, z, w, a, b)
for µ = 1, i, and all x, y, z, w, a, b ∈ A. If h(tx) is continuous in t ∈ R for
each fixed x ∈ A, then there exists a unique Lie JC ∗ -algebra homomorphism
H : A → B satisfying (3.iv).
Proof.
The proof is similar to the proof of Theorem 2.3.
We are going to show the Cauchy–Rassias stability of Lie JC ∗ -algebra
homomorphisms in Lie JC ∗ -algebras associated with the Jensen functional equation.
Theorem 3.4.
Let h : A → B be a mapping with h(0) = 0 for which there
exists a function ϕ : (A \ {0})6 → [0, ∞) such that
ϕ(x,
e y, z, w, a, b) =
∞
X
3−j ϕ(3j x, 3j y, 3j z, 3j w, 3j a, 3j b) < ∞, (3.v)
j=0
k2h(
µx + µy + [z, w] + a ◦ b
) − µh(x) − µh(y) − [h(z), h(w)] − h(a) ◦ h(b)k
2
≤ ϕ(x, y, z, w, a, b),
(3.vi)
n ∗
n ∗
n
n
kh(3 u ) − h(3 u) k ≤ϕ(3 u, 3 u, 0, 0, 0, 0)
(3.vii)
for all µ ∈ T1 , all u ∈ U (A), n = 0, 1, 2, · · ·, and all x, y, z, w, a, b ∈ A \ {0}.
Then there exists a unique Lie JC ∗ -algebra homomorphism H : A → B such
that
kh(x) − H(x)k ≤
1
(ϕ(x,
e −x, 0, 0, 0, 0) + ϕ(−x,
e
3x, 0, 0, 0, 0))
3
(3.viii)
for all x ∈ A \ {0} .
Proof. Put z = w = a = b = 0 and µ = 1 ∈ T1 in (3.vi). It follows from Jun
and Lee’s Theorem [2, Theorem 1] that there exists a unique additive mapping
H : A → B satisfying (3.viii). The additive mapping H : A → B is given by
1
h(3n x)
n→∞ 3n
H(x) = lim
for all x ∈ A.
The rest of the proof is similar to the proof of Theorem 2.5.
Corollary 3.5.
Let h : A → B be a mapping with h(0) = 0 for which there
exist constants θ ≥ 0 and p ∈ [0, 1) such that
k2h(
µx + µy + [z, w] + a ◦ b
) − µh(x) − µh(y) − [h(z), h(w)] − h(a) ◦ h(b)k
2
≤ θ(||x||p + ||y||p + ||z||p + ||w||p + ||a||p + ||b||p ),
kh(3n u∗ ) − h(3n u)∗ k ≤ 2 · 3np θ
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Park
for all µ ∈ T1 , all u ∈ U (A), n = 0, 1, 2, · · ·, and all x, y, z, w, a, b ∈ A \ {0}.
Then there exists a unique Lie JC ∗ -algebra homomorphism H : A → B such
that
3 + 3p
kh(x) − H(x)k ≤
θ||x||p
p
3−3
for all x ∈ A \ {0} .
Proof. Define ϕ(x, y, z, w, a, b) = θ(||x||p +||y||p +||z||p +||w||p +||a||p +||b||p ) ,
and apply Theorem 3.4.
One can obtain a similar result to Theorem 3.3 for the Jensen functional
equation.
Now we are going to show the Cauchy–Rassias stability of Lie JC ∗ algebra homomorphisms in Lie JC ∗ -algebras associated with the Trif functional
equation.
Theorem 3.6.
Let h : A → B be a mapping with h(0) = 0 for which there
exists a function ϕ : Ad+4 → [0, ∞) such that
ϕ(x
e 1 , · · · , xd , z, w, a, b) :=
∞
X
q −j ϕ(q j x1 , · · · ,q j xd , q j z, q j w, q j a, q j b)
j=0
< ∞,
kd
(3.ix)
d
X
µx1 + · · · + µxd
[z, w] + a ◦ b
+
) + d−2 Cl−1
µh(xj )
d
d d−2 Cl−2
j=1
X
xj1 + · · · + xjl
) − [h(z),h(w)] − h(a) ◦ h(b)k (3.x)
µh(
l
d−2 Cl−2 h(
−l
1≤j1 <···<jl ≤d
≤ ϕ(x1 , · · · , xd , z, w, a, b),
kh(q u ) − h(q u) k ≤ ϕ(q u, · · · , q n u,0, 0, 0, 0)
|
{z
}
n ∗
n
∗
n
(3.xi)
d times
for all µ ∈ T1 , all u ∈ U (A), n = 0, 1, 2, · · ·, and all x1 , · · · , xd , z, w, a, b ∈ A.
Then there exists a unique Lie JC ∗ -algebra homomorphism H : A → B such
that
1
(3.xii)
kh(x) − H(x)k ≤
ϕ(qx,
e
rx, · · · , rx, 0, 0, 0, 0)
| {z }
l · d−1 Cl−1
d−1 times
for all x ∈ A.
Proof.
Put z = w = a = b = 0 and µ = 1 ∈ T1 in (3.x). It follows from
Trif’s Theorem [17, Theorem 3.1] that there exists a unique additive mapping
H : A → B satisfying (3.xii). The additive mapping H : A → B is given by
H(x) = lim
n→∞
1
h(q n x)
n
q
for all x ∈ A.
The rest of the proof is similar to the proof of Theorem 2.7.
Park
407
Corollary 3.7.
Let h : A → B be a mapping with h(0) = 0 for which there
exist constants θ ≥ 0 and p ∈ [0, 1) such that
d
X
µx1 + · · · + µxd
[z, w] + a ◦ b
kd d−2 Cl−2 h(
+
) + d−2 Cl−1
µh(xj )
d
d d−2 Cl−2
j=1
X
xj + · · · + xjl
−l
µh( 1
)−[h(z), h(w)] − h(a) ◦ h(b)k
l
1≤j1 <···<jl ≤d
d
X
≤ θ(
||xj ||p +||z||p + ||w||p + ||a||p + ||b||p ),
j=1
n ∗
kh(q u ) − h(q n u)∗ k ≤ dq np θ
for all µ ∈ T1 , all u ∈ U (A), n = 0, 1, 2, · · ·, and all x1 , · · · , xd , z, w, a, b ∈ A.
Then there exists a unique Lie JC ∗ -algebra homomorphism H : A → B such
that
q 1−p (q p + (d − 1)rp )θ
||x||p
kh(x) − H(x)k ≤
l d−1 Cl−1 (q 1−p − 1)
for all x ∈ A.
Pd
Proof. Define ϕ(x1 , · · · , xd , z, w, a, b) = θ( j=1 ||xj ||p +||z||p +||w||p +||a||p +
||b||p ), and apply Theorem 3.6.
One can obtain a similar result to Theorem 3.3 for the Trif functional
equation.
4. Stability of Lie JC ∗ -algebra derivations in Lie JC ∗ -algebras
Definition 4.1. A C -linear mapping D : A → A is called a Lie JC ∗ -algebra
derivation if D : A → A satisfies
D(x ◦ y) = (Dx) ◦ y + x ◦ (Dy),
D([x, y]) = [Dx, y] + [x, Dy],
D(x∗ ) = D(x)∗
for all x, y ∈ A.
Remark 4.1. A C -linear mapping D : A → A is a C ∗ -algebra derivation if
and only if the mapping D : A → A is a Lie JC ∗ -algebra derivation.
Assume that D is a Lie JC ∗ -algebra derivation. Then
D(xy) = D([x, y] + x ◦ y) = D([x, y]) + D(x ◦ y)
= [Dx, y] + [x, Dy] + (Dx) ◦ y + x ◦ (Dy) = (Dx)y + x(Dy)
for all x, y ∈ A. So D is a C ∗ -algebra derivation.
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Park
Assume that D is a C ∗ -algebra derivation. Then
(Dx)y + x(Dy) − (Dy)x − y(Dx)
xy − yx
)=
2
2
= [Dx, y] + [x, Dy],
xy + yx
(Dx)y + x(Dy) + (Dy)x + y(Dx)
D(x ◦ y) = D(
)=
2
2
= (Dx) ◦ y + x ◦ (Dy)
D([x, y]) = D(
for all x, y ∈ A. So H is a Lie JC ∗ -algebra derivation.
We are going to show the Cauchy–Rassias stability of Lie JC ∗ -algebra
derivations in Lie JC ∗ -algebras associated with the Cauchy functional equation.
Theorem 4.1.
Let h : A → A be a mapping with h(0) = 0 for which there
exists a function ϕ : A6 → [0, ∞) satisfying (3.i) and (3.iii) such that
kh(µx + µy + [z, w] + a ◦ b) − µh(x) − µh(y) − [h(z), w] − [z, h(w)]
− h(a) ◦ b − a ◦ h(b)k ≤ ϕ(x, y, z, w, a, b) (4.i)
for all µ ∈ T1 and all x, y, z, w, a, b ∈ A. Then there exists a unique Lie JC ∗ algebra derivation D : A → A such that
kh(x) − D(x)k ≤
1
ϕ(x,
e x, 0, 0, 0, 0)
2
(4.ii)
for all x ∈ A.
Proof. Put z = w = a = b = 0 in (4.i). By the same reasoning as in the proof
of Theorem 2.1, there exists a unique C -linear involutive mapping D : A → A
satisfying (4.ii). The C -linear mapping D : A → A is given by
1
h(2n x)
n→∞ 2n
(4.1)
h(22n x)
n→∞
22n
(4.2)
D(x) = lim
for all x ∈ A.
It follows from (4.1) that
D(x) = lim
for all x ∈ A. Let x = y = a = b = 0 in (4.i). Then we get
kh([z, w]) − [h(z), w] − [z, h(w)]k ≤ ϕ(0, 0, z, w, 0, 0)
for all z, w ∈ A. Since
1
1
n
n
ϕ(0,
0,
2
z,
2
w,
0,
0)
≤
ϕ(0, 0, 2n z, 2n w, 0, 0),
22n
2n
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Park
1
1
kh([2n z, 2n w]) − [h(2n z), 2n w] − [2n z, h(2n w)]k ≤ 2n ϕ(0, 0, 2n z, 2n w, 0, 0)
2n
2
2
1
≤ n ϕ(0, 0, 2n z, 2n w, 0, 0) (4.3)
2
for all z, w ∈ A. By (3.i), (4.2), and (4.3),
h(22n [z, w])
h([2n z, 2n w])
=
lim
n→∞
n→∞
22n
22n
n
n
n
2 z h(2n w)
h(2 z) 2 w
])
,
]
+
[
,
= lim ([
n→∞
2n
2n
2n
2n
= [D(z), w] + [z, D(w)]
D([z, w]) = lim
for all z, w ∈ A.
Similarly, one can obtain that
h(22n a ◦ b)
h((2n a) ◦ (2n b))
=
lim
2n
n→∞
n→∞
22n
2 n
n
n
h(2 a)
2 b
2 a
h(2n b)
= lim (
)◦( n )+( n ◦(
)
n→∞
2n
2
2
2n
D(a ◦ b) = lim
= (Da) ◦ b + a ◦ (Db)
for all a, b ∈ A. Hence the C -linear mapping D : A → A is a Lie JC ∗ -algebra
derivation satisfying (4.ii), as desired.
Corollary 4.2.
Let h : A → A be a mapping with h(0) = 0 for which there
exist constants θ ≥ 0 and p ∈ [0, 1) such that
kh(µx + µy + [z, w] + a ◦ b) − µh(x) − µh(y) − [h(z), w] − [z, h(w)]
− h(a) ◦ b − a ◦ h(B)k
≤ θ(kxkp + kykp + kzkp + kwkp + kakp + kbkp ),
kh(2n u∗ ) − h(2n u)∗ k ≤ 2 · 2np θ
for all µ ∈ T1 , all u ∈ U (A), n = 0, 1, 2, · · ·, and all x, y, z, w, a, b ∈ A . Then
there exists a unique Lie JC ∗ -algebra derivation D : A → A such that
kh(x) − D(x)k ≤
2θ
kxkp
p
2−2
for all x ∈ A.
Proof. Define ϕ(x, y, z, w, a, b) = θ(kxkp + kykp + kzkp + kwkp + kakp + kbkp ) ,
and apply Theorem 4.1.
Theorem 4.3.
Let h : A → A be a mapping with h(0) = 0 for which there
exists a function ϕ : A6 → [0, ∞) satisfying (3.i) and (3.iii) such that
kh(µx + µy + [z, w] + a ◦ b) − µh(x) − µh(y) − [h(z), w] − [z, h(w)]
− h(a) ◦ b − a ◦ h(b)k ≤ ϕ(x, y, z, w, a, b)
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Park
for µ = 1, i, and all x, y, z, w, a, b ∈ A. If h(tx) is continuous in t ∈ R for each
fixed x ∈ A, then there exists a unique Lie JC ∗ -algebra derivation D : A → A
satisfying (4.ii).
Proof.
By the same reasoning as in the proof of Theorem 2.3, there exists a
unique C -linear mapping D : A → A satisfying (4.ii).
The rest of the proof is the same as in the proofs of Theorems 2.1, 3.1
and 4.1.
We are going to show the Cauchy–Rassias stability of Lie JC ∗ -algebra
derivations in Lie JC ∗ -algebras associated with the Jensen functional equation.
Theorem 4.4.
Let h : A → A be a mapping with h(0) = 0 for which there
exists a function ϕ : (A \ {0})6 → [0, ∞) satisfying (3.v) and (3.vii) such that
k2h(
µx + µy + [z, w] + a ◦ b
) − µh(x) − µh(y) − [h(z), w] − [z, h(w)]
2
− h(a) ◦ b − a ◦ h(b)k ≤ ϕ(x, y, z, w, a, b)(4.iii)
for all µ ∈ T1 and all x, y, z, w, a, b ∈ A \ {0} . Then there exists a unique Lie
JC ∗ -algebra derivation D : A → A such that
kh(x) − D(x)k ≤
1
(ϕ(x,
e −x, 0, 0, 0, 0) + ϕ(−x,
e
3x, 0, 0, 0, 0))
3
(4.iv)
for all x ∈ A \ {0} .
Proof.
Put z = w = a = b = 0 in (4.iii). By the same reasoning as in
the proof of Theorem 2.5, there exists a unique C -linear involutive mapping
D : A → A satisfying (4.iv). The C -linear mapping D : A → A is given by
1
h(3n x)
n→∞ 3n
(4.4)
h(32n x)
n→∞
32n
(4.5)
D(x) = lim
for all x ∈ A.
It follows from (4.4) that
D(x) = lim
for all x ∈ A. Let x = y = a = b = 0 in (4.iii). Then we get
k2h(
[z, w]
) − [h(z), w] − [z, h(w)]k ≤ ϕ(0, 0, z, w, 0, 0)
2
for all z, w ∈ A. Since
1
1
n
n
ϕ(0,
0,
3
z,
3
w,
0,
0)
≤
ϕ(0, 0, 3n z, 3n w, 0, 0),
32n
3n
1
1
1
k2h( [3n z, 3n w]) − [h(3n z), 3n w] − [3n z, h(3n w)]k ≤ 2n ϕ(0, 0, 3n z, 3n w, 0, 0)
2n
3
2
3
1
≤ n ϕ(0, 0,3n z, 3n w, 0, 0)
(4.6)
3
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for all z, w ∈ A. By (3.v), (4.5), and (4.6),
2n
2h( 32 [z, w])
2h( 12 [3n z, 3n w])
[z, w]
2D(
) = lim
= lim
n→∞
n→∞
2
32n
32n
n
n
n
n
3 z h(3 w)
h(3 z) 3 w
= lim ([
, n ]+[ n ,
])
n
n→∞
3
3
3
3n
= [D(z), w] + [z, D(w)]
for all z, w ∈ A. But since D is C -linear,
D([z, w]) = 2D(
[z, w]
) = [D(z), w] + [z, D(w)]
2
for all z, w ∈ A.
Similarly, one can obtain that
2n
2h( 32 a ◦ b)
2h( 12 (3n a) ◦ (3n b))
a◦b
2D(
) = lim
= lim
2n
n→∞
n→∞
2
32n
3n
n
n
h(3 a)
3 b
3 a
h(3n b)
= lim (
)◦( n )+( n ◦(
)
n→∞
3n
3
3
3n
= (Da) ◦ b + a ◦ (Db)
for all a, b ∈ A. So
a◦b
) = (Da) ◦ b + a ◦ (Db)
2
for all a, b ∈ A. Hence the C -linear mapping D : A → A is a Lie JC ∗ -algebra
derivation satisfying (4.iv), as desired.
D(a ◦ b) = 2D(
Corollary 4.5.
Let h : A → A be a mapping with h(0) = 0 for which there
exist constants θ ≥ 0 and p ∈ [0, 1) such that
k2h(
µx + µy + [z, w] + a ◦ b
) − µh(x) − µh(y) − [h(z), w] − [z, h(w)]
2
−h(a) ◦ b − a ◦ h(b)k ≤ θ(kxkp + kykp + kzkp + kwkp + kakp + kbkp ),
kh(3n u∗ ) − h(3n u)∗ k ≤ 2 · 3np θ
for all µ ∈ T1 , all u ∈ U (A), n = 0, 1, 2, · · ·, and all x, y, z, w, a, b ∈ A \ {0} .
Then there exists a unique Lie JC ∗ -algebra derivation D : A → A such that
kh(x) − D(x)k ≤
3 + 3p
θkxkp
3 − 3p
for all x ∈ A \ {0} .
Proof. Define ϕ(x, y, z, w, a, b) = θ(kxkp + kykp + kzkp + kwkp + kakp + kbkp ) ,
and apply Theorem 4.4.
One can obtain a similar result to Theorem 4.3 for the Jensen functional
equation.
Finally, we are going to show the Cauchy–Rassias stability of Lie JC ∗ algebra derivations in Lie JC ∗ -algebras associated with the Trif functional equation.
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Theorem 4.6.
Let h : A → A be a mapping with h(0) = 0 for which there
exists a function ϕ : Ad+4 → [0, ∞) satisfying (3.ix) and (3.xi) such that
d
X
µx1 + · · · + µxd
[z, w] + a ◦ b
kd d−2 Cl−2 h(
+
) + d−2 Cl−1
µh(xj )
d
d d−2 Cl−2
j=1
X
xj1 + · · · + xjl
µh(
) − [h(z), w] − [z, h(w)]
−l
l
(4.v)
1≤j1 <···<jl ≤d
−h(a) ◦ b − a ◦ h(b)k ≤ ϕ(x1 , · · · , xd , z, w, a, b)
for all µ ∈ T1 and all x1 , · · · , xd , z, w, a, b ∈ A. Then there exists a unique Lie
JC ∗ -algebra derivation D : A → A such that
kh(x) − D(x)k ≤
1
ϕ(qx,
e
rx, · · · , rx, 0, 0, 0, 0)
| {z }
l · d−1 Cl−1
(4.vi)
d−1 times
for all x ∈ A.
Proof. Put z = w = a = b = 0 in (4.v). By the same reasoning as in the proof
of Theorem 2.7, there exists a unique C -linear involutive mapping D : A → A
satisfying (4.vi). The C -linear mapping D : A → A is given by
1
h(q n x)
n→∞ q n
(4.7)
h(q 2n x)
n→∞
q 2n
(4.8)
D(x) = lim
for all x ∈ A.
It follows from (4.7) that
D(x) = lim
for all x ∈ A. Let x1 = · · · = xd = a = b = 0 in (4.v). Then we get
kd
d−2 Cl−2 h(
[z, w]
) − [h(z), w] − [z, h(w)]k ≤ ϕ(0, · · · , 0, z, w, 0, 0)
| {z }
d d−2 Cl−2
d times
for all z, w ∈ A. Since
1
q 2n
1
ϕ(0, · · · , 0, q n z, q n w, 0, 0) ≤ n ϕ(0, · · · , 0, q n z, q n w, 0, 0),
| {z }
| {z }
q
d times
1
q 2n
d times
1
[q n z, q n w]) − [h(q n z), q n w] − [q n z, h(q n w)]k
dd−2 Cl−2
1
1
≤ 2n ϕ(0, · · · , 0, q n z, q n w, 0, 0) ≤ n ϕ(0, · · · , 0, q n z, q n w, 0, 0) (4.9)
| {z }
| {z }
q
q
kdd−2 Cl−2 h(
d times
d times
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Park
for all z, w ∈ A. By (3.ix), (4.8), and (4.9),
2n
q
dd−2 Cl−2 h( dd−2
[z, w]
Cl−2 [z, w])
dd−2 Cl−2 D(
) = lim
n→∞
dd−2 Cl−2
q 2n
dd−2 Cl−2 h( dd−21Cl−2 [q n z, q n w])
h(q n z) q n w
q n z h(q n w)
= lim
=
lim
([
,
]
+
[
,
])
n→∞
n→∞
q 2n
qn
qn
qn
qn
= [D(z), w] + [z, D(w)]for all z, w ∈ A.
But since D is C -linear, vglue-8pt
D([z, w]) = d d−2 Cl−2 D([z, w]
d d−2 Cl−2 ) = [D(z), w] + [z, D(w)] for all z, w ∈ A.
Similarly, one can obtain that D(a ◦ b) = (Da) ◦ b + a ◦ (Db) for all a, b ∈ A.
Hence the C -linear mapping D: A → A is a Lie JC ∗ -algebra derivation satisfying
(4.vi), as desired.
Corollary 4.7.
Let h : A → A be a mapping with h(0) = 0 for which there
exist constants θ ≥ 0 and p ∈ [0, 1) such that
d
X
µx1 + · · · + µxd
[z, w] + a ◦ b
kd d−2 Cl−2 h(
+
) + d−2 Cl−1
µh(xj )
d
d d−2 Cl−2
j=1
X
xj + · · · + xjl
−l
µh( 1
) − [h(z), w] − [z, h(w)] − h(a) ◦ b
l
1≤j1 <···<jl ≤d
d
X
−a ◦ h(b)k ≤ θ(
kxj kp + kzkp + kwkp + kakp + kbkp ),
j=1
n ∗
kh(q u ) − h(q n u)∗ k ≤ dq np θ
for all µ ∈ T1 , all u ∈ U (A), n = 0, 1, · · ·, and all x1 , · · · , xd , z, w, a, b ∈ A .
Then there exists a unique Lie JC ∗ -algebra derivation D : A → A such that
kh(x) − D(x)k ≤
q 1−p (q p + (d − 1)rp )θ
kxkp
l d−1 Cl−1 (q 1−p − 1)
for all x ∈ A.
Pd
Proof. Define ϕ(x1 , · · · , xd , z, w, a, b) = θ( j=1 kxj kp + kzkp + kwkp + kakp +
kbkp ), and apply Theorem 4.6.
One can obtain a similar result to Theorem 4.3 for the Trif functional
equation.
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Chun-Gil Park
Department of Mathematics
Chungnam National University
Daejeon 305–764
South Korea
cgpark@cnu.ac.kr
Received March 13, 2004
and in final form December 7, 2004