θ-centralizers on semiprime Banach *-algebras
We generalize the celebrated theorem of Johnson and prove that every left θ -centralizer on a semisimple Banach algebra with left approximate identity is continuous. We also investigate the generalized Hyers–Ulam–Rassias stability and the superstability of θ -centralizers on semiprime Banach *-algeb...
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irk-123456789-1653062020-02-14T01:26:25Z θ-centralizers on semiprime Banach *-algebras Nikoufar, I. Rassias, Th.M. Статті We generalize the celebrated theorem of Johnson and prove that every left θ -centralizer on a semisimple Banach algebra with left approximate identity is continuous. We also investigate the generalized Hyers–Ulam–Rassias stability and the superstability of θ -centralizers on semiprime Banach *-algebras. Шляхом узагальнення відомої теореми Джонсона доведено, що кожний лівий 0-централізатор на напівпростій банаховій алгебрі з лівою наближеною одиницею є неперервним. Також досліджено узагальнену стійкість Хайерса-Улама-Рассіаса та надстійкість θ-централізаторів на напівпростих *-алгебрах. 2014 Article θ-centralizers on semiprime Banach *-algebras / I. Nikoufar, Th.M. Rassias // Український математичний журнал. — 2014. — Т. 66, № 2. — С. 269–278. — Бібліогр.: 23 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/165306 517.9 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Nikoufar, I. Rassias, Th.M. θ-centralizers on semiprime Banach *-algebras Український математичний журнал |
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We generalize the celebrated theorem of Johnson and prove that every left θ -centralizer on a semisimple Banach algebra with left approximate identity is continuous. We also investigate the generalized Hyers–Ulam–Rassias stability and the superstability of θ -centralizers on semiprime Banach *-algebras. |
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Nikoufar, I. Rassias, Th.M. |
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Nikoufar, I. Rassias, Th.M. |
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Nikoufar, I. |
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θ-centralizers on semiprime Banach *-algebras |
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θ-centralizers on semiprime Banach *-algebras |
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θ-centralizers on semiprime Banach *-algebras |
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θ-centralizers on semiprime Banach *-algebras |
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θ-centralizers on semiprime Banach *-algebras |
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θ-centralizers on semiprime banach *-algebras |
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Інститут математики НАН України |
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2014 |
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θ-centralizers on semiprime Banach *-algebras / I. Nikoufar, Th.M. Rassias // Український математичний журнал. — 2014. — Т. 66, № 2. — С. 269–278. — Бібліогр.: 23 назв. — англ. |
| series |
Український математичний журнал |
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AT nikoufari thcentralizersonsemiprimebanachalgebras AT rassiasthm thcentralizersonsemiprimebanachalgebras |
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2025-07-14T18:18:34Z |
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2025-07-14T18:18:34Z |
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1837647386367754240 |
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UDC 517.9
I. Nikoufar (Payame Noor Univ., Iran),
Th. M. Rassias (Nat. Techn. Univ. Athens, Greece)
θ-CENTRALIZERS ON SEMIPRIME BANACH ∗-ALGEBRAS
θ-ЦЕНТРАЛIЗАТОРИ НА НАПIВПРОСТИХ БАНАХОВИХ ∗-АЛГЕБРАХ
By generalizing the celebrated theorem of Johnson, we prove that every left θ-centralizer on a semisimple Banach algebra
with left approximate identity is continuous. We also investigate the generalized Hyers – Ulam – Rassias stability and the
superstability of θ-centralizers on semiprime Banach ∗-algebras.
Шляхом узагальнення вiдомої теореми Джонсона доведено, що кожний лiвий θ-централiзатор на напiвпростiй
банаховiй алгебрi з лiвою наближеною одиницею є неперервним. Також дослiджено узагальнену стiйкiсть Хайерса –
Улама – Рассiаса та надстiйкiсть θ-централiзаторiв на напiвпростих ∗-алгебрах.
1. Introduction. The notion of centralizers has been generalized as θ-centralizer by Albas [1]. Let
A be a ∗-algebra and θ be an algebra automorphism of A. A mapping T : A −→ A is called a left
(right) θ-centralizer on A if T (xy) = T (x)θ(y)
(
T (xy) = θ(x)T (y)
)
holds for all x, y ∈ A. T is
called a θ-centralizer if it is a left as well as a right θ-centralizer. The concept of left and right θ-
centralizer covers the concept of left and right centralizer (in case θ = id, the identity automorphism
on A). The properties of θ-centralizers have been studied by Albas [1], Ali and Haetinger [2], Cortis
and Haetinger [7], Daif [8] and Ullah and Chaudhry [22].
A classical question in the theory of functional equations is the following: When is it true that
a function which approximately satisfies a functional equation ζ must be close to an exact solution
of ζ ? If the problem accepts a solution, we say that the equation ζ is stable. There are cases in
which each approximate solution is actually a true solution. In such cases, we call the equation ζ
superstable. The first stability problem concerning group homomorphisms was raised by Ulam [23]
in 1940. Ulam problem was partially solved by Hyers [12] for Banach spaces. Hyers’ theorem was
generalized by Aoki [3] for additive mappings and by Th. M. Rassias [21] for linear mappings by
considering an unbounded Cauchy difference. The paper of Th. M. Rassias [21] has provided a lot of
influence in the development of what is called the generalized Hyers – Ulam stability or the Hyers –
Ulam – Rassias stability of functional equations. A generalization of the Th. M. Rassias theorem was
obtained by Gavruta [11] in 1994 by replacing the unbounded Cauchy difference by a general control
function in the spirit of Th. M. Rassias’ approach. Badora [5] proved the generalized Hyers – Ulam
stability of ring homomorphisms, which generalizes the result of D. G. Bourgin. Miura [18] proved
the generalized Hyers – Ulam stability of Jordan homomorphisms. For more details about the stability
of functional equations see [9 – 14].
In Section 2, by generalizing the celebrated theorem of Johnson [17], we prove that every left
θ-centralizer on a semisimple Banach algebra with a left approximate identity is continuous. In
Section 3, we prove the superstability of θ-centralizers on semiprime Banach ∗-algebras and we
provide conditions for which a given mapping f is a left (right) θ-centralizer. In Section 4, we
investigate the generalized Hyers – Ulam stability of θ-centralizers on semiprime Banach ∗-algebras.
Throughout this paper, it is assumed that A is a semiprime Banach (complex) ∗-algebra.
c© I. NIKOUFAR, TH. M. RASSIAS, 2014
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 2 269
270 I. NIKOUFAR, TH. M. RASSIAS
2. Automatic continuity of θ-centralizers. In this section, we show that every left (right) θ-
centralizer is homogenous. Also, we apply a classical theorem of B. E. Johnson to prove that every
left θ-centralizer on a semisimple Banach algebra with a left approximate identity is continuous.
Following [6], a Banach algebra B is said to have a left approximate identity (in Cohen’s sense), if
there exists a constant C, such that given ε > 0, and xi ∈ B, 1 ≤ i ≤ m, there exists an e ∈ B,
satisfying
‖e‖ < C, ‖exi − xi‖ < ε.
Proposition 2.1. Let B be a semiprime algebra. If T : B −→ B is a left (right) θ-centralizer,
then T is homogenous.
Proof. Set a := T (µx)− µT (x) for every x ∈ B and every µ ∈ C. Let y ∈ B. Then there exists
a z ∈ B such that y = θ(z). Therefore,
aya =
(
T (µx)− µT (x)
)
θ(z)a =
(
T (µx)θ(z)− µT (x)θ(z)
)
a =
=
(
T (µxz)− T (x)θ(µz)
)
a =
(
T (µxz)− T (xµz)
)
a = 0.
From the semiprimeness of B it follows that a = 0. Thus, T is homogenous.
Proposition 2.1 is proved.
We now generalize the result of [17] for continuity of θ-centralizers on Banach algebras.
Theorem 2.1. Let B be a semisimple Banach algebra with a left approximate identity (in Cohen’s
sense). If T : B −→ B is a left θ-centralizer, then T is linear and continuous.
Proof. If x1, x2 ∈ B, then by Johnson’s Theorem (see [17]) one can find y1, y2, z ∈ B such that
x1 = zy1 and x2 = zy2. Thus,
T (x1 + x2) = T
(
z(y1 + y2)
)
= T (z)θ(y1 + y2) =
= T (z)θ(y1) + T (z)θ(y2) = T (zy1) + T (zy2) = T (x1) + T (x2).
Now, Proposition 2.1 implies T is linear.
If xm ∈ B and xm → 0, then by Johnson’s Theorem (see [17]) it follows that there exists a z ∈ B
and a sequence ym in B with ym → 0 such that xm = zym, m = 1, 2, . . . . Hence,
T (xm) = T (zym) = T (z)θ(ym).
But a classical theorem of B. E. Johnson (see [4]) yields θ(ym) → 0 as m → ∞. Therefore, T is
continuous.
Theorem 2.1 is proved.
3. Superstability. In this section, we prove the superstability of θ-centralizers on semiprime
Banach ∗-algebras. Note that throughout this section n > 4 is a fixed integer.
We first summarize the following corollaries from [22].
Corollary 3.1. If T : A −→ A is an additive mapping such that T (xx∗) = T (x)θ(x∗) holds for
all x ∈ A, then T is a left θ-centralizer.
Proof. The result follows from Theorem 2.2 of [22] and the fact that every complex ∗-algebra is
a 2-torsion free ring.
Corollary 3.2. If T : A −→ A is an additive mapping such that T (xx∗) = θ(x∗)T (x) holds for
all x ∈ A, then T is a right θ-centralizer.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 2
θ-CENTRALIZERS ON SEMIPRIME BANACH ∗-ALGEBRAS 271
Corollary 3.3. If T : A −→ A is an additive mapping such that T (xx∗) = T (x)θ(x∗) =
= θ(x∗)T (x) holds for all x ∈ A, then T is a θ-centralizer.
We now provide conditions which imply the superstability of θ-centralizers on semiprime Banach
∗-algebras.
Theorem 3.1. Let p 6= 2 and α be nonnegative real numbers and f : A −→ A be a mapping
such that ∥∥∥∥∥∥ 1
n− 2
n∑
i=1
f
(
− xi +
n∑
j=1,j 6=i
xj
)
−
n−1∑
i=1
f(xi)
∥∥∥∥∥∥ ≤ ∥∥f(xn)∥∥, (3.1)
∥∥f(aa∗)− f(a)θ(a∗)∥∥ ≤ α‖a‖p (3.2)
for all a, xi ∈ A, 1 ≤ i ≤ n. Then the mapping f : A −→ A is a linear left θ-centralizer. Moreover,
if A is a semisimple Banach ∗-algebra with a left approximate identity (in Cohen’s sense), then f is
continuous.
Proof. Letting x1 = . . . = xn = 0 and using n > 4 we conclude that f(0) = 0. Letting x1 = x
and x2 = . . . = xn = 0 we infer that f is odd for all x ∈ A. Setting x3 = . . . = xn = 0, we get
1
n− 2
(
f(−x1 + x2) + f(−x2 + x1)
)
+ f(x1 + x2) = f(x1) + f(x2)
for all x1, x2 ∈ A. From the oddness of f it follows that f is additive. Assume that p < 2. By using
the inequality (3.2), we have∥∥f(aa∗)− f(a)θ(a∗)∥∥ =
1
n2
∥∥∥f((na)(na)∗)− f(na)θ((na)∗)∥∥∥ ≤ 1
n2
αnp‖a‖p
for all a ∈ A. Thus, by letting n tend to ∞ in the last inequality, we obtain f(aa∗) = f(a)θ(a∗)
for all a ∈ A. Hence Corollary 3.1 implies f is a left θ-centralizer. The additivity of f together with
Proposition 2.1 yield f is linear. Moreover, the continuity of f follows from Theorem 2.1. Similarly,
one can obtain the result for the case p > 2.
Theorem 3.1 is proved.
Theorem 3.2. Let p 6= 2 and α be nonnegative real numbers and f : A −→ A be a mapping
satisfying the inequality (3.1) and∥∥f(aa∗)− θ(a∗)f(a)∥∥ ≤ α‖a‖p (3.3)
for all a ∈ A. Then the mapping f : A −→ A is a linear right θ-centralizer.
Proof. The proof is similar to the proof of Theorem 3.1 and the result follows from Corollary 3.2.
Theorem 3.3. Let p 6= 2 and α be nonnegative real numbers and f : A −→ A be a mapping
satisfying the inequality (3.1) and∥∥f(aa∗ + bb∗)− f(a)θ(a∗)− θ(b∗)f(b)
∥∥ ≤ α(‖a‖p + ‖b‖p) (3.4)
for all a, b ∈ A. Then the mapping f : A −→ A is a linear θ-centralizer. Moreover, if A is a semi-
simple Banach ∗-algebra with a left approximate identity (in Cohen’s sense), then f is continuous.
Proof. Setting b = 0 in (3.4) and applying Theorem 3.1, we conclude that f is a linear left
θ-centralizer. Letting a = 0 in (3.4) and using Theorem 3.2, we deduce that f is a right θ-centralizer.
Theorem 3.3 is proved.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 2
272 I. NIKOUFAR, TH. M. RASSIAS
4. Stability. In this section we prove the generalized Hyers – Ulam stability of θ-centralizers on
semiprime Banach ∗-algebras. Throughout this section n > 3 is a fixed integer.
The following lemma (see [19]) is needed in the rest of the paper.
Lemma 4.1. Let X and Y be linear spaces. A mapping f : X −→ Y satisfies
n∑
i=1
f
(
− xi +
n∑
j=1,j 6=i
xj
)
= (n− 2)
n∑
i=1
f(xi) (4.1)
for x1, . . . , xn ∈ X, if and only if f is additive.
Theorem 4.1. Let f : A −→ A be a mapping for which f(0) = 0 and there exists a control
function ϕ : An+1 −→ [0,∞) such that
ϕ̃(x) :=
∞∑
i=1
1
2i
ϕ
(
2i−1x, 2i−1x, 0, . . . , 0
)
<∞, (4.2)
lim
k→∞
1
2k
ϕ
(
2kx1, . . . , 2
kxn, 2
ka
)
= 0, (4.3)
∥∥∥∥∥
n∑
i=1
f
(
− xi +
n∑
j=1,j 6=i
xj
)
− (n− 2)
n∑
i=1
f(xi) + f(aa∗)− f(a)θ(a∗)
∥∥∥∥∥ ≤
≤ ϕ(x1, . . . , xn, a) (4.4)
for all a, x1, . . . , xn ∈ A. Then there exists a unique linear left θ-centralizer T : A −→ A such that∥∥T (x)− f(x)∥∥ ≤ 1
n− 2
ϕ̃(x) (4.5)
for all x ∈ A.
Proof. Setting x1 = x2 = x, a = x3 = . . . = xn = 0 in (4.4) and using f(0) = 0, we obtain∥∥∥∥12f(2x)− f(x)
∥∥∥∥ ≤ 1
2(n− 2)
ϕ(x, x, 0, . . . , 0) (4.6)
for all x ∈ A. Applying induction method on m, we have∥∥∥∥ 1
2m
f(2mx)− f(x)
∥∥∥∥ ≤ 1
n− 2
m∑
i=1
1
2i
ϕ
(
2i−1x, 2i−1x, 0, . . . , 0
)
(4.7)
for all x ∈ A. In order to show that the functions Tm(x) =
1
2m
f(2mx) form a convergent sequence,
we use the Cauchy convergence criterion. Replace x by 2lx and divide by 2l in (4.7), where l is an
arbitrary positive integer, to find that∥∥∥∥ 1
2m+l
f(2m+lx)− 1
2l
f(2lx)
∥∥∥∥ ≤ 1
n− 2
m+l∑
i=1+l
1
2i
ϕ
(
2i−1x, 2i−1x, 0, . . . , 0
)
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 2
θ-CENTRALIZERS ON SEMIPRIME BANACH ∗-ALGEBRAS 273
for all positive integers m ≥ l and all x ∈ A. Hence by the Cauchy criterion the limit T (x) :=
:= limm→∞ Tm(x) exists for each x ∈ A. By taking the limit as m → ∞ in (4.7) we see that the
inequality (4.5) holds for all x ∈ A. Setting a = 0 in (4.4), we get∥∥∥∥∥
n∑
i=1
f
(
− xi +
n∑
j=1,j 6=i
xj
)
− (n− 2)
n∑
i=1
f(xi)
∥∥∥∥∥ ≤ ϕ(x1, . . . , xn, 0)
for all xi ∈ A, 1 ≤ i ≤ n. Replacing xi by 2mxi, 1 ≤ i ≤ n and dividing both sides by 2m and
taking the limit as m → ∞ and using (4.3) we deduce that T satisfies (4.1). Thus, it follows from
Lemma 4.1 that T is additive. Setting x1 = . . . = xn = 0 in (4.4), we get∥∥f(aa∗)− f(a)θ(a∗)∥∥ ≤ ϕ(0, . . . , 0, a) (4.8)
for all a ∈ A. Replacing a by 2ma in (4.8) and dividing its both sides by 22m, we obtain∥∥∥∥ 1
22m
f(22maa∗)− 1
2m
f(2ma)θ(a∗)
∥∥∥∥ ≤ 1
22m
ϕ
(
0, . . . , 0, 2ma
)
for all a ∈ A. Taking the limit as m → ∞ and using (4.3), we conclude that T (aa∗) = T (a)θ(a∗).
So Corollary 3.1 implies T is a left θ-centralizer. Now, let T ′ : A −→ A be another additive mapping
satisfying (4.5). Consequently, we have∥∥T (x)− T ′(x)∥∥ =
1
2m
∥∥T (2mx)− T ′(2mx)∥∥ ≤
≤ 1
2m
(∥∥T (2mx)− f(2mx)∥∥+ ∥∥T ′(2mx)− f(2mx)∥∥) ≤ 2
2m(n− 2)
ϕ̃(2mx) =
=
2
n− 2
∞∑
i=m+1
1
2i
ϕ
(
2i−1x, 2i−1x, 0, . . . , 0
)
for all x ∈ A. The right-hand side tends to zero as m → ∞. This proves the uniqueness of T. The
linearity of T follows from Proposition 2.1.
Theorem 4.1 is proved.
Theorem 4.2. Let f : A −→ A be a mapping for which f(0) = 0 and there exists a control
function ϕ : An+1 −→ [0,∞) that satisfies (4.2), (4.3) and∥∥∥∥∥
n∑
i=1
f
(
− xi +
n∑
j=1,j 6=i
xj
)
− (n− 2)
n∑
i=1
f(xi) + f(aa∗)− θ(a∗)f(a)
∥∥∥∥∥ ≤
≤ ϕ(x1, . . . , xn, a) (4.9)
for all a, x1, . . . , xn ∈ A. Then there exists a unique linear right θ-centralizer T : A −→ A such
that, ∥∥T (x)− f(x)∥∥ ≤ 1
n− 2
ϕ̃(x) (4.10)
for all x ∈ A.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 2
274 I. NIKOUFAR, TH. M. RASSIAS
Proof. The proof is similar to the proof of Theorem 4.1.
Theorem 4.3. Let f : A −→ A be a mapping for which f(0) = 0 and there exists a control
function φ : An+2 −→ [0,∞) such that
φ̃(x) :=
∞∑
i=1
1
2i
φ
(
2i−1x, 2i−1x, 0, . . . , 0
)
<∞, (4.11)
lim
k→∞
1
2k
φ
(
2kx1, . . . , 2
kxn, 2
ka, 2kb
)
= 0, (4.12)
∥∥∥∥∥
n∑
i=1
f
(
− xi +
n∑
j=1,j 6=i
xj
)
− (n− 2)
n∑
i=1
f(xi)+
+f(aa∗ + bb∗)− f(a)θ(a∗)− θ(b∗)f(b)
∥∥∥∥∥ ≤ φ(x1, . . . , xn, a, b) (4.13)
for all a, b, x1, . . . , xn ∈ A. Then there exists a unique linear θ-centralizer T : A −→ A such that
∥∥T (x)− f(x)∥∥ ≤ 1
n− 2
φ̃(x) (4.14)
for all x ∈ A.
Proof. Setting b = 0 in (4.13), we obtain∥∥∥∥∥
n∑
i=1
f
(
− xi +
n∑
j=1,j 6=i
xj
)
− (n− 2)
n∑
i=1
f(xi) + f(aa∗)− f(a)θ(a∗)
∥∥∥∥∥ ≤ φ(x1, . . . , xn, a, 0)
for all a, x1, . . . , xn ∈ A. By taking ϕ(x1, . . . , xn, a) := φ(x1, . . . , xn, a, 0) for all a, x1, . . . , xn ∈ A
and applying the same method as in the proof of Theorem 4.1, we obtain the Cauchy sequence{
1
2m
f(2mx)
}
for all x ∈ A. Completeness of A gives a unique mapping T : A −→ A which is a
linear left θ-centralizer and ∥∥T (x)− f(x)∥∥ ≤ 1
n− 2
ϕ̃(x) =
1
n− 2
φ̃(x). (4.15)
Setting a = 0 in (4.13), we obtain∥∥∥∥∥
n∑
i=1
f
(
− xi +
n∑
j=1,j 6=i
xj
)
− (n− 2)
n∑
i=1
f(xi) + f(bb∗)− θ(b∗)f(b)
∥∥∥∥∥ ≤ φ(x1, . . . , xn, 0, b)
for all b, x1, . . . , xn ∈ A. By taking ϕ(x1, . . . , xn, b) := φ(x1, . . . , xn, 0, b) for all b, x1, . . . , xn ∈ A
and applying the same method as in the proof of Theorem 4.2, we obtain the above Cauchy sequence
which converges to the mapping T : A −→ A. Now, Theorem 4.2 implies the mapping T is a linear
right θ-centralizer and satisfies (4.15). Therefore, T is a unique linear θ-centralizer satisfying (4.14).
Theorem 4.3 is proved.
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θ-CENTRALIZERS ON SEMIPRIME BANACH ∗-ALGEBRAS 275
Corollary 4.1. Let α and rj , 1 ≤ j ≤ n+2, be nonnegative real numbers such that 0 < rj < 1.
Suppose that a mapping f : A −→ A with f(0) = 0 satisfies∥∥∥∥∥
n∑
i=1
f
(
− xi +
n∑
j=1,j 6=i
xj
)
− (n− 2)
n∑
i=1
f(xi) + f(xn+1x
∗
n+1 + xn+2x
∗
n+2)−
−f(xn+1)θ(x
∗
n+1)− θ(x∗n+2)f(xn+2)
∥∥∥∥∥ ≤ α
n+2∑
j=1
‖xj‖rj (4.16)
for all x1, . . . , xn+2 ∈ A. Then there exists a unique linear θ-centralizer T : A −→ A such that∥∥T (x)− f(x)∥∥ ≤ α
n− 2
(
‖x‖r1
2− 2r1
+
‖x‖r2
2− 2r2
)
for all x ∈ A.
Proof. It is an immediate consequence of Theorem 4.3 by taking
φ(x1, . . . , xn+2) := α
n+2∑
j=1
‖xj‖rj
for all x1, . . . , xn+2 ∈ A.
The following Corollary is Isac – Rassias type stability (see [15, 16]) for θ-centralizers on semiprime
Banach ∗-algebras.
Corollary 4.2. Let ψ : R+ ∪ {0} −→ R+ ∪ {0} be a function with ψ(0) = 0 such that
lim
t→∞
ψ(t)
t
= 0, ψ(ts) ≤ ψ(t)ψ(s)
for t, s ∈ R+, and ψ(t) < t for t > 1. Suppose that α is a nonnegative real number and f : A −→ A
is a mapping with f(0) = 0 satisfies∥∥∥∥∥
n∑
i=1
f
(
− xi +
n∑
j=1,j 6=i
xj
)
− (n− 2)
n∑
i=1
f(xi) + f(xn+1x
∗
n+1 + xn+2x
∗
n+2)−
−f(xn+1)θ(x
∗
n+1)− θ(x∗n+2)f(xn+2)
∥∥∥∥∥ ≤ α
n+2∑
j=1
ψ
(
‖xj‖
)
for all x1, . . . , xn+2 ∈ A. Then there exists a unique linear θ-centralizer T : A −→ A such that∥∥T (x)− f(x)∥∥ ≤ 2αψ(2)ψ(2−1)
(n− 2)(2− ψ(2))
ψ
(
‖x‖
)
for all x ∈ A.
Proof. The result follows from Theorem 4.3 by letting
φ(x1, . . . , xn+2) := α
n+2∑
j=1
ψ
(
‖xj‖
)
for all x1, . . . , xn+2 ∈ A.
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276 I. NIKOUFAR, TH. M. RASSIAS
Theorem 4.4. Let f : A −→ A be a mapping for which there exists a control function ϕ : An+1 −→
−→ [0,∞) that satisfies (4.4) and
ϕ̃(x) :=
∞∑
i=1
2iϕ
(
1
2i−1
x,
1
2i−1
x, 0, . . . , 0
)
<∞, (4.17)
lim
k→∞
4kϕ
(x1
2k
, . . . ,
xn
2k
,
a
2k
)
= 0 (4.18)
for all a, x1, . . . , xn ∈ A. Then there exists a unique linear left θ-centralizer T : A −→ A such that∥∥T (x)− f(x)∥∥ ≤ 1
n− 2
ϕ̃(x) (4.19)
for all x ∈ A.
Proof. Setting a = x1 = . . . = xn = 0 in (4.18) we conclude that ϕ(0, . . . , 0) = 0. Setting
a = x1 = . . . = xn = 0 in (4.4) and using n > 3 we see that f(0) = 0. Therefore by a similar
calculation as in the proof of Theorem 4.1 we can obtain (4.6). Now, replace x by
x
2
and multiply
both sides by 2 in (4.6), to get∥∥∥f(x)− 2f
(x
2
)∥∥∥ ≤ 1
n− 2
ϕ
(x
2
,
x
2
, 0, . . . , 0
)
for all x ∈ A. Using induction method on m, we have∥∥∥f(x)− 2mf
( x
2m
)∥∥∥ ≤ 1
n− 2
m∑
i=1
2i−1ϕ
( x
2i
,
x
2i
, 0, . . . , 0
)
(4.20)
for all x ∈ A. Replacing x by
x
2l
and multiplying by 2l in (4.20), where l is an arbitrary positive
integer, we get
∥∥∥2lf ( x
2l
)
− 2m+lf
( x
2m+l
)∥∥∥ ≤ 1
n− 2
m+l∑
i=1+l
2i−1ϕ
( x
2i
,
x
2i
, 0, . . . , 0
)
(4.21)
for all positive integers m ≥ l. Due to completeness of A the sequence
{
2mf
( x
2m
)}
converges
for all x ∈ A. Hence we can define the mapping T : A −→ A by T (x) := limn→∞ 2mf
( x
2m
)
. By
taking the limit as m→∞ in (4.20) we obtain the desired inequality (4.19). The rest of the proof is
similar to the proof of Theorem 4.1 and we omit it.
Theorem 4.5. Let f : A −→ A be a mapping for which there exists a control function
ϕ : An+1 −→ [0,∞) that satisfies (4.9), (4.17) and (4.18). Then there exists a unique linear right
θ-centralizer T : A −→ A that satisfies the inequality (4.19).
Theorem 4.6. Let f : A −→ A be a mapping for which there exists a control function
φ : An+2 −→ [0,∞) that satisfies (4.13) and
φ̃(x) :=
∞∑
i=1
2iφ
(
1
2i−1
x,
1
2i−1
x, 0, . . . , 0
)
<∞, (4.22)
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θ-CENTRALIZERS ON SEMIPRIME BANACH ∗-ALGEBRAS 277
lim
k→∞
4kφ
(
x1
2k
, . . . ,
xn
2k
,
a
2k
,
b
2k
)
= 0 (4.23)
for all a, b, x1, . . . , xn ∈ A. Then there exists a unique linear θ-centralizer T : A −→ A such that∥∥T (x)− f(x)∥∥ ≤ 1
n− 2
φ̃(x) (4.24)
for all x ∈ A.
Corollary 4.3. Let α and rj , 1 ≤ j ≤ n + 2, be nonnegative real numbers such that rj > 1.
Suppose that a mapping f : A −→ A satisfies (4.16). Then there exists a unique linear θ-centralizer
T : A −→ A such that∥∥T (x)− f(x)∥∥ ≤ 2α
n− 2
(
2r1
2r1 − 2
‖x‖r1 + 2r2
2r2 − 2
‖x‖r2
)
for all x ∈ A.
Proof. It is enough to define
φ(x1, . . . , xn+2) := α
n+2∑
j=1
‖xj‖rj
for all x1, . . . , xn+2 ∈ A and apply Theorem 4.6.
Remark 4.1. In Theorems 4.3, 4.4, and 4.6 and Corollaries 4.1, 4.2, and 4.3 if A is replaced
by a semisimple Banach ∗-algebra with a left approximate identity (in Cohen’s sense), then T is
continuous. Note that in this case the result follows from Theorem 2.1.
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Received 09.12.11
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