Surveys in Mathematics and its Applications
ISSN 1842-6298 (electronic), 1843-7265 (print)
Volume 5 (2010), 297 – 310
HIGHER *-DERIVATIONS BETWEEN UNITAL
C∗ -ALGEBRAS
M. Eshaghi Gordji, R. Farokhzad Rostami and S. A. R. Hosseinioun
P
Abstract. Let A, B be two unital C ∗ −algebras. We prove that every sequence of mappings
from A into B, H = {h0 , h1 , ..., hm , ...}, which satisfy hm (3n uy) =
i+j=m
hi (3n u)hj (y) for each
m ∈ N0 , for all u ∈ U (A), all y ∈ A, and all n = 0, 1, 2, ..., is a higher derivation. Also, for
P
a unital C ∗ −algebra A of real rank zero, every sequence of continuous mappings from A into B,
H = {h0 , h1 , ..., hm , ...}, is a higher derivation when hm (3n uy) =
i+j=m
hi (3n u)hj (y) holds for
all u ∈ I1 (Asa ), all y ∈ A, all n = 0, 1, 2, ... and for each m ∈ N0 . Furthermore, by using the fixed
points methods, we investigate the Hyers–Ulam–Rassias stability of higher ∗−derivations between
unital C ∗ −algebras.
1
Introduction
The stability of functional equations was first introduced by S. M. Ulam [27] in 1940.
More precisely, he proposed the following problem: Given a group G1 , a metric group
(G2 , d) and a positive number , does there exist a δ > 0 such that if a function
f : G1 −→ G2 satisfies the inequality d(f (xy), f (x)f (y)) < δ for all x, y ∈ G1,
then there exists a homomorphism T : G1 → G2 such that d(f (x), T (x)) < for
all x ∈ G1 ? As mentioned above, when this problem has a solution, we say that
the homomorphisms from G1 to G2 are stable. In 1941, D. H. Hyers [10] gave a
partial solution of U lam, s problem for the case of approximate additive mappings
under the assumption that G1 and G2 are Banach spaces. In 1978, Th. M. Rassias
[24] generalized the theorem of Hyers by considering the stability problem with
unbounded Cauchy differences. This phenomenon of stability that was introduced
by Th. M. Rassias [24] is called the Hyers–Ulam–Rassias stability. According to
Th. M. Rassias theorem:
Theorem 1. Let f : E −→ E 0 be a mapping from a norm vector space E into a
Banach space E 0 subject to the inequality
kf (x + y) − f (x) − f (y)k ≤ (kxkp + kykp )
2010 Mathematics Subject Classification: 39B52; 39B82; 46B99; 17A40.
Keywords: Alternative fixed point; Hyers–Ulam–Rassias stability; Higher *-derivation.
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M. Eshaghi Gordji, R. Farokhzad Rostami and S. A. R. Hosseinioun
for all x, y ∈ E, where and p are constants with > 0 and p < 1. Then there exists
a unique additive mapping T : E −→ E 0 such that
kf (x) − T (x)k ≤
2
kxkp
2 − 2p
for all x ∈ E. If p < 0 then inequality (1.3) holds for all x, y 6= 0, and (1.4) for
x 6= 0. Also, if the function t 7→ f (tx) from R into E 0 is continuous for each fixed
x ∈ E, then T is linear.
During the last decades several stability problems of functional equations have
been investigated by many mathematicians. A large list of references concerning the
stability of functional equations can be found in [9, 12, 15].
D.G. Bourgin is the first mathematician dealing with the stability of ring homomorphisms. The topic of approximate ring homomorphisms was studied by a number
of mathematicians, see [1, 2, 3, 11, 17, 18, 20, 22, 25] and references therein.
Jun and Lee [14] proved the following: Let X and Y be Banach
Denote by
P∞ spaces.
−n
n
φ : X −{0}×Y −{0} → [0, ∞) a function such that φ̃(x, y) = n=0 3 φ(3 x, 3n y) <
∞ 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)k ≤ (φ̃(x, −x) + φ̃(−x, 3x))
3
for all x ∈ X − {0}.
Recently, C. Park and W. Park [21] applied the Jun and Lee’s result to the
Jensen’s equation in Banach modules over a C ∗ −algebra(See also [13]). Throughout
this paper, let A be a unital C ∗ −algebra with unit e, and B a unital C ∗ −algebra.
Let U (A) be the set of unitary elements in A, Asa := {x ∈ A|x = x∗ }, and
I1 (Asa ) = {v ∈ Asa |kvk = 1, v ∈ Inv(A)}.
A linear mapping d : A → A is said to be a derivation if d(xy) = d(x)y + xd(y)
holds for all x, y ∈ A.
Let N be the set of natural numbers. For m ∈ N ∪ {0} = N0 , a sequence
H = {h0 , h1 , ..., hm } (resp. H = {h0 , h1 , ..., hn , ...}) of linear mappings from A into
B is called a higher derivation of rank m (resp. infinite rank) from A into B if
X
hn (xy) =
hi (x)hj (y)
i+j=n
holds for each n ∈ {0, 1, ..., m} (resp. n ∈ N0 ) and all x, y ∈ A. The higher derivation H from A into B is said to be onto if h0 : A → B is onto. The higher derivation
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299
H on A is called be strong if h0 is an identity mapping on A. Of course, a higher
derivation of rank 0 from A into B (resp. a strong higher derivation of rank 1 on A)
is a homomorphism (resp. a derivation). So a higher derivation is a generalization
of both a homomorphism and a derivation.
In this paper, we prove that every sequence of mappings from A into B, H =
n
{h
,
0
P h1 , ..., hnm , ...} is a higher derivation when for each m ∈ N0 , hm (3 uy) =
i+j=m hi (3 u)hj (y) for all u ∈ U (A), all y ∈ A, and all n = 0, 1, 2, ..., and that
for a unital C ∗ −algebra A of real rank zero (see [4]), every sequence of continuous
mappings from A into B, HP
= {h0 , h1 , ..., hm , ...} is a higher derivation when for
n
n
each m ∈ N0 , hm (3 uy) =
i+j=m hi (3 u) hj (y) for all for all u ∈ I1 (Asa ), all
y ∈ A, and all n = 0, 1, 2, .... Furthermore, we investigate the Hyers–Ulam–Rassias
stability of higher ∗−derivations between unital C ∗ −algebras by using the fixed pint
methods.
Note that a unital C ∗ −algebra is of real rank zero, if the set of invertible selfadjoint elements is dense in the set of self–adjoint elements (see [4]). We denote the
algebric center of algebra A by Z(A).
2
Higher *-derivations on unital C∗ -algebras
By a following similar way as in [19], we obtain the next theorem.
Theorem 2. Suppose that F = {f0 , f1 , ..., fm , ...} is a sequence of mappings from
A into B such that fm (0) = 0 for each m ∈ N0 := N ∪ {0},
fm (3n uy) =
X
fi (3n u)fj (y)
(2.1)
i+j=m
for all u ∈ U (A), all y ∈ A, all n = 0, 1, 2, ... and for each
∈ N. If there exists a
Pm
∞
2
function φ : (A−{0}) ×A → [0, ∞) such that φ̃(x, y, z) = n=0 3−n φ(3n x, 3n y, 3n z) <
∞ for all x, y ∈ A − {0} and all z ∈ A and for each m ∈ N0 ,
k2fm (
µx + µy
) − µfm (x) − µfm (y) + fm (u∗ ) − fm (u)∗ k ≤ φ(x, y, u),
2
(2.2)
n
for all µ ∈ T and all x, y ∈ A, u ∈ (U (A) ∪ {0}). If limn fm3(3n e) ∈ U (B) ∩ Z(B),
then the sequence F = {f0 , f1 , ..., fm , ...} is a higher ∗−derivation.
Proof. Put u = 0, µ = 1 in (2.2), it follows from Theorem 1 of [14] that there exists
a unique additive mapping hm : A → B such that
1
kfm (x) − hm (x)k ≤ (φ̃(x, −x, 0) + φ̃(−x, 3x, 0))
3
(2.3)
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M. Eshaghi Gordji, R. Farokhzad Rostami and S. A. R. Hosseinioun
for all x ∈ A − {0} and for each m ∈ N0 . These mappings are given by
hm (x) = lim
n
fm (3n x)
3n
for all x ∈ A and for each m ∈ N0 . By the same reasoning as the proof of Theorem
1 of [19], hm is C−linear and ∗−preserving for each m ∈ N0 . It follows from (2.1)
and (2.2) that
X fi (3n u)fj (y)
fm (3n uy)
=
lim
n
n
3n
3n
i+j=m
X
hi (u)fj (y)
hm (uy) = lim
=
(2.4)
i+j=m
for all u ∈ U (A), all y ∈ A and for each m ∈ N0 . Since hm is additive, then by (2.4),
we have
X
3n hm (uy) = hm (u(3n y)) =
hi (u)fj (3n y)
i+j=m
for all u ∈ U (A), all y ∈ A and for each m ∈ N0 . Hence,
hm (uy) = lim
n
X
hi (u)
i+j=m
X
fj (3n y)
=
hi (u)hj (y)
n
3
(2.5)
i+j=m
for all u ∈ U (A), all y ∈ A and for each m ∈ N0 . By the assumption, we have
hm (e) = lim
n
fm (3n e)
∈ U (B) ∩ Z(B)
3n
and for each m ∈ N0 , hence, it follows by (2.4) and (2.5) that
X
i+j=m
hi (e)hj (y) = hm (ey) =
X
hi (e)fj (y)
i+j=m
for all y ∈ A and for each m ∈ N0 . Since hm (e) is invertible, then by induction
hm (y) = fm (y) for all y ∈ A and for each m ∈ N0 . We have to show that F =
{f0 , f1 , ..., fm , ...} is higher derivation. To this end, let x ∈ A. By Theorem
4.1.7 of
Pn
[16], x is a finite linear combination of unitary elements, i.e., x = j=1 cj uj (cj ∈
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301
C, uj ∈ U (A)), it follows from (2.5) that
n
n
X
X
fm (xy) = hm (xy) = hm (
ck uk y) =
ck hm (uk y)
k=1
=
n
X
ck (
i+j=m
n
X
k=1
=
X
hi (
i+j=m
=
X
X
k=1
hi (uk )hj (y))
ck uk )hj (y)
k=1
hi (x)hj (y)
i+j=m
=
X
fi (x)fj (y)
i+j=m
for all y ∈ A. And this completes the proof of theorem.
Corollary 3. Let p ∈ (0, 1), θ ∈ [0, ∞) be real numbers. Suppose that
F = {f0 , f1 , ..., fm , ...}
is a sequence of mappings from A into B such that fm (0) = 0 for each m ∈ N0 ,
X
fm (3n uy) =
fi (3n u)fj (y)
i+j=m
for all u ∈ U (A), all y ∈ A, all n = 0, 1, 2, ... and for each m ∈ N0 . Suppose that
k2fm (
µx + µy
) − µfm (x) − µfm (y) + fm (z ∗ ) − fm (z)∗ k ≤ θ(kxkp + kykp + kzkp )
2
n
for all µ ∈ T and all x, y, z ∈ A and for each m ∈ N0 . If limn fm3(3n e) ∈ U (B) ∩Z(B),
then the sequence F = {f0 , f1 , ..., fm , ...} is a higher ∗−derivation.
Proof. Setting φ(x, y, z) := θ(kxkp + kykp + kzkp ) all x, y, z ∈ A. Then by Theorem
1 we get the desired result.
Theorem 4. Let A be a C ∗ −algebra of real rank zero. Suppose that
F = {f0 , f1 , ..., fm , ...}
is a sequence of mappings from A into B such that fm (0) = 0 for each m ∈ N0 ,
X
fm (3n uy) =
fi (3n u)fj (y)
(2.6)
i+j=m
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302
M. Eshaghi Gordji, R. Farokhzad Rostami and S. A. R. Hosseinioun
for all u ∈ I1 (Asa ), all y ∈ A, all n = 0, 1, 2, ... and for each m ∈ N0 . Suppose that
there exists a function φ : (A−{0})2 ×A → [0, ∞) satisfying (2.2) and φ̃(x, y, z) < ∞
n
for all x, y ∈ A−{0} and all z ∈ A. If limn fm3(3n e) ∈ U (B)∩Z(B), then the sequence
F = {f0 , f1 , ..., fm , ...} is a higher ∗−derivation.
Proof. By the same reasoning as the proof of Theorem 1, there exist a unique involutive C−linear mappings hm : A → B satisfying (2.3) for each m ∈ N0 . It follows
from (2.6) that
hm (uy) = lim
n
X
X fi (3n u)fj (y)
fm (3n uy)
= lim
=
hi (u)fj (y)
n
n
n
3
3
(2.7)
i+j=m
i+j=m
for all u ∈ I1 (Asa ), and all y ∈ A and for each m ∈ N0 . By additivity of hm and
(2.7), we obtain that
3n hm (uy) = hm (u(3n y)) =
X
hi (u)fj (3n y)
i+j=m
for all u ∈ I1 (Asa ) and all y ∈ A and for each m ∈ N0 . Hence,
hm (uy) = lim
n
X
hi (u)
X
fj (3n y)
=
hi (u)hj (y)
3n
(2.8)
i+j=m
i+j=m
for all u ∈ I1 (Asa ) and all y ∈ A and for each m ∈ N0 . By the assumption, we have
hm (e) = lim
n
fm (3n e)
∈ U (B) ∩ Z(B)
3n
and for each m ∈ N0 . Similar to the proof of Theorem 1, it follows from (2.7) and
(2.8) that hm = fm on A for each m ∈ N0 . So hm is continuous for each m ∈ N0 .
On the other hand A is real rank zero. On can easily show that I1 (Asa ) is dense in
{x ∈ Asa : kxk = 1}. Let v ∈ {x ∈ Asa : kxk = 1}. Then there exists a sequence
{zn } in I1 (Asa ) such that limn zn = v. Since hm is continuous for each m ∈ N0 , it
follows from (2.8) that
hm (vy) = hm (lim(zn y)) = lim hm (zn y) = lim
n
=
X
i+j=m
n
hi (lim zn )hj (y) =
n
n
X
X
hi (zn )hj (y)
i+j=m
hi (v)hj (y)
(2.9)
i+j=m
for all y ∈ A and for each m ∈ N0 . Now, let x ∈ A. Then we have x = x1 + ix2 ,
∗
∗
where x1 := x+x
and x2 = x−x
2
2i are self–adjoint.
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First consider x1 = 0, x2 6= 0. Since hm is C−linear for each m ∈ N0 , it follows
from (2.9) that
x2
fm (xy) = hm (xy) = hm (ix2 y) = hm (ikx2 k
y)
kx2 k
X
x2
x2
= ikx2 khm (
y) = ikx2 k
hi (
)hj (y)
kx2 k
kx2 k
i+j=m
X
X
x2
=
hi (ikx2 k
)hj (y) =
hi (ix2 )hj (y)
kx2 k
i+j=m
i+j=m
X
X
=
hi (x)hj (y) =
fi (x)fj (y)
i+j=m
i+j=m
for all y ∈ A and for each m ∈ N0 .
If x2 = 0, x1 6= 0, then by (2.9), we have
x1
fm (xy) = hm (xy) = hm (x1 y) = hm (kx1 k
y)
kx1 k
X
x1
x1
= kx1 khm (
y) = kx1 k
)hj (y)
hi (
kx1 k
kx1 k
i+j=m
X
X
x1
=
)hj (y) =
hi (kx1 k
hi (x1 )hj (y)
kx1 k
i+j=m
i+j=m
X
X
=
hi (x)hj (y) =
fi (x)fj (y)
i+j=m
i+j=m
for all y ∈ A and for each m ∈ N0 .
Finally, consider the case that x1 6= 0, x2 6= 0. Then it follows from (2.9) that
f (xy) = hm (xy) = hm (x1 y + (ix2 )y)
x2
x1
= hm (kx1 k
y) + hm (ikx2 k
y)
kx1 k
kx2 k
x1
x2
= kx1 khm (
y) + ikx2 khm (
y)
kx1 k
kx2 k
X
X
x1
x2
)hj (y) + ikx2 k
)hj (y)
= kx1 k
hi (
hi (
kx1 k
kx2 k
i+j=m
i+j=m
X x1
x2
=
hi (kx1 k
)hj (y) + hi (ikx2 k
)hj (y)
kx1 k
kx2 k
i+j=m
X
X
X
=
[hi (x1 ) + hi (ix2 )]hj (y) =
hi (x)hj (y) =
fi (x)fj (y)
i+j=m
i+j=m
i+j=m
P
for all y ∈ A and for each m ∈ N0 . Hence, fm (xy) = i+j=m fi (x)fj (y) for all
x, y ∈ A, for each m ∈ N0 , and F = {f0 , f1 , ..., fm , ...} is higher ∗−derivation.
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304
M. Eshaghi Gordji, R. Farokhzad Rostami and S. A. R. Hosseinioun
Corollary 5. Let A be a C ∗ −algebra of rank zero. Let p ∈ (0, 1), θ ∈ [0, ∞) be real
numbers. Suppose that F = {f0 , f1 , ..., fm , ...} is a sequence of mappings from A
into B such that fm (0) = 0 for each m ∈ N0 ,
fm (3n uy) =
X
fi (3n u)fj (y)
i+j=m
for all u ∈ I1 (Asa ), all y ∈ A, all n = 0, 1, 2, ... and for each m ∈ N0 . Suppose that
k2fm (
µx + µy
) − µfm (x) − µfm (y) + fm (z ∗ ) − fm (z)∗ k ≤ θ(kxkp + kykp + kzkp )
2
n
for all µ ∈ T and all x, y, z ∈ A and for each m ∈ N0 . If limn fm3(3n e) ∈ U (B) ∩ Z(B)
for each m ∈ N0 , then the sequence F = {f0 , f1 , ..., fm , ...} is a higher ∗−derivation.
Proof. Setting φ(x, y, z) := θ(kxkp + kykp + kzkp ) all x, y, z ∈ A. Then by Theorem
4 we get the desired result.
3
Stability of higher *-derivations: a fixed point approach
We investigate the generalized Hyers–Ulam–Rassias stability of higher ∗-derivations
on unital C ∗ -algebras by using the alternative fixed point.
Recently, Cădariu and Radu applied the fixed point method to the investigation
of the functional equations. (see also [6, 7, 8, 22, 23, 26]). Before proceeding to the
main result of this section, we will state the following theorem.
Theorem 6. (The alternative of fixed point [5]). Suppose that we are given a
complete generalized metric space (Ω, d) and a strictly contractive mapping T : Ω →
Ω with Lipschitz constant L. Then for each given x ∈ Ω,
either
d(T m x, T m+1 x) = ∞ for all m ≥ 0,
or there exists a natural number m0 such that
d(T m x, T m+1 x) < ∞ for all m ≥ m0 ;
the sequence {T m x} is convergent to a fixed point y ∗ of T ;
y ∗ is the unique fixed point of T in the set Λ = {y ∈ Ω : d(T m0 x, y) < ∞};
d(y, y ∗ ) ≤
1
1−L d(y, T y)
for all y ∈ Λ.
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Theorem 7. Suppose that F = {f0 , f1 , ..., fm , ...} is a sequence of mappings from
A into B such that for each m ∈ N0 , fm (0) = 0 for which there exists a function
φ : A5 → [0, ∞) satisfying
kfm (
µx + µy + µz
µx − 2µy + µz
µx + µy − 2µz
) + fm (
) + fm (
)−
3
3
X 3
−µfm (x) + fm (uv) −
fi (u)fj (v) + fm (w∗ ) − fm (w)∗ k ≤
i+j=m
≤ φ(x, y, z, u, v, w),
(3.1)
for all µ ∈ T, and all x, y, z, u, v ∈ A, w ∈ U (A) ∪ {0} and for each m ∈ N0 .
If there exists an L < 1 such that φ(x, y, z, u, v, w) ≤ 3Lφ( x3 , y3 , z3 , u3 , v3 , w3 ) for all
x, y, z, u, v, w ∈ A, then there exists a unique higher ∗−derivation
H = {h0 , h1 , ..., hm , ...}
such that
kfm (x) − hm (x)k ≤
L
φ(x, 0, 0, 0, 0, 0)
1−L
(3.2)
for all x ∈ A and for each m ∈ N0 .
Proof. It follows from φ(x, y, z, u, v, w) ≤ 3Lφ( x3 , y3 , z3 , u3 , v3 , w3 ) that
limj 3−j φ(3j x, 3j y, 3j z, 3j u, 3j v, 3j w) = 0
(3.3)
for all x, y, z, u, v, w ∈ A.
Put y = z = w = u = 0 in (3.1) to obtain
x
k3fm ( ) − fm (x)k ≤ φ(x, 0, 0, 0, 0, 0)
3
(3.4)
for all x ∈ A and for each m ∈ N0 . Hence,
1
1
k fm (3x) − fm (x)k ≤ φ(3x, 0, 0, 0, 0, 0) ≤ Lφ(x, 0, 0, 0, 0, 0)
3
3
(3.5)
for all x ∈ A and for each m ∈ N0 .
Consider the set X := {gm | gm : A → B, m ∈ N0 } and introduce the generalized
metric on X:
d(h, g) := inf {C ∈ R+ : kg(x) − h(x)k ≤ Cφ(x, 0, 0, 0, 0, 0)∀x ∈ A}.
It is easy to show that (X, d) is complete. Now we define the linear mapping J :
X → X by
1
J(h)(x) = h(3x)
3
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306
M. Eshaghi Gordji, R. Farokhzad Rostami and S. A. R. Hosseinioun
for all x ∈ A. By Theorem 3.1 of [5],
d(J(g), J(h)) ≤ Ld(g, h)
for all g, h ∈ X.
It follows from (2.5) that
d(fm , J(fm )) ≤ L.
By Theorem 6, J has a unique fixed point in the set X1 := {h ∈ X : d(fm , h) < ∞}.
Let hm be the fixed point of J. hm is the unique mapping with
hm (3x) = 3hm (x)
for all x ∈ A and for each m ∈ N0 satisfying there exists C ∈ (0, ∞) such that
khm (x) − fm (x)k ≤ Cφ(x, 0, 0, 0, 0, 0)
for all x ∈ A and for each m ∈ N0 . On the other hand we have limn d(J n (fm ), hm ) =
0. It follows that
1
limn n fm (3n x) = hm (x)
(3.6)
3
1
d(fm , J(fm )),
for all x ∈ A and for each m ∈ N0 . It follows from d(fm , hm ) ≤ 1−L
that
L
d(fm , hm ) ≤
.
1−L
This implies the inequality (3.2). It follows from (3.1), (3.3) and (3.6) that
x+y+z
x − 2y + z
x + y − 2z
) + hm (
) + hm (
) − hm (x)k
3
3
3
1
= limn n kfm (3n−1 (x + y + z)) + fm (3n−1 (x − 2y + z))+
3
+fm (3n−1 (x + y − 2z)) − fm (3n x)k
1
≤ limn n φ(3n x, 3n y, , 3n z, 0, 0, 0) = 0
3
k3hm (
for all x, y, z ∈ A and for each m ∈ N0 . So
hm (
x+y+z
x − 2y + z
x + y − 2z
) + hm (
) + hm (
) = hm (x)
3
3
3
for all x, y, z ∈ A and for each m ∈ N0 . Put w = x+y+z
, t = x−2y+z
and s = x+y−2z
3
3
3
in above equation, we get hm (w + t + s) = hm (w) + hm (t) + hm (s) for all w, t, s ∈ A
and for each m ∈ N0 . Hence, hm for each m ∈ N0 is Cauchy additive. By putting
y = z = x, v = w = 0 in (2.1), we have
kµfm (
3µx
) − µfm (x)k ≤ φ(x, x, , x, 0, 0, 0)
3
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Higher *-derivations between unital C∗ -algebras
307
for all x ∈ A and for each m ∈ N0 . It follows that
khm (µx) − µhm (x)k =
1
1
= limm m kfm (µ3m x) − µfm (3m x)k ≤ limm m φ(3m x, 3m x, 3m x, 0, 0, 0) = 0
3
3
for all µ ∈ T, and all x ∈ A. One can show that the mapping h : A → B is C−linear.
By putting x = y = z = u = v = 0 in (2.1) it follows that
khm (w∗ ) − (hm (w))∗ k
1
1
= limm k m fm ((3m w)∗ ) − m (fm (3m w))∗ k
3
3
1
m
≤ limm m φ(0, 0, 0, 0, 0, 3 w)
3
=0
for all w ∈ U (A) and for each m ∈ N0 . By the same reasoning as the proof of
Theorem 1, we can show that hm : A → B is ∗−preserving for each m ∈ N0 .
Since hm is C−linear, by putting x = y = z = w = 0 in (2.1) it follows that
khm (uv) −
X
hi (u)hj (v)k = limm k
1 X
1
m
f
(9
(uv))
−
fi (3m u)fj (3m v)k
m
9m
9m
i+j=m
i+j=m
1
φ(0, 0, 0, 3m u, 3m v, 0)
9m
1
≤ limm m φ(0, 0, 0, 3m u, 3m v, 0)
3
= 0
≤ limm
for all u, v ∈ A and for each m ∈ N0 . Thus H = {h0 , h1 , ..., hm , ...} is higher
∗−derivation satisfying (3.2), as desired.
We prove the following Hyers–Ulam–Rassias stability problem for higher ∗-derivations
on unital C ∗ −algebras:
Corollary 8. Let p ∈ (0, 1), θ ∈ [0, ∞) be real numbers. Suppose that
F = {f0 , f1 , ..., fm , ...}
is a sequence of mappings from A into B such that for each m ∈ N0 , fm (0) = 0 and
µx + µy + µz
µx − 2µy + µz
µx + µy − 2µz
) + fm (
) + fm (
) − µfm (x) + fm (uv)
3
3
3
X
−
fi (u)fj (v)+fm (w∗ )−fm (w)∗ k ≤ θ(kxkp +kykp +kzkp +kukp +kvkp +kwkp ),
kfm (
i+j=m
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Surveys in Mathematics and its Applications 5 (2010), 297 – 310
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308
M. Eshaghi Gordji, R. Farokhzad Rostami and S. A. R. Hosseinioun
for all µ ∈ T and all x, y, z, u, v ∈ A, w ∈ U (A) ∪ {0}. Then there exists a unique
higher ∗−derivation H = {h0 , h1 , ..., hm , ...} such that
kfm (x) − hm (x)k ≤
3p θ
kxkp
3 − 3p
for all x ∈ A and for each m ∈ N0 .
Proof. Settingφ(x, y, z, u, v, w) := θ(kxkp + kykp + kzkp + kukp + kvkp + kwkp ) all
x, y, z, u, v, w ∈ A. Then by L = 3p−1 in Theorem 7, one can prove the result.
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M. Eshaghi Gordji
R. Farokhzad Rostami
Department of Mathematics, Semnan University,
Department of Mathematics,
P.O. Box 35195-363, Semnan, Iran.
Shahid Beheshti University,
e-mail: madjid.eshaghi@gmail.com
Tehran, Iran.
S. A. R. Hosseinioun
Department of Mathematics,
Shahid Beheshti University,
Tehran, Iran.
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Surveys in Mathematics and its Applications 5 (2010), 297 – 310
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