Section of Mathematics and Informatics, Pedagogical Department, National and Kapodistrian University of Athens, 4, Agamemnonos Street, Aghia Paraskevi, 15342 Athens, Greece uoa.gr

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M. B. Savadkouhi,

M. B. Savadkouhi

Department of Mathematics, Islamic Azad University, Mahdishahr Branch, Mahdishahr, Semnan, Iran iau.ac.ir

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M. Eshaghi Gordji,

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M. Eshaghi Gordji

[email protected]

Department of Mathematics, Semnan University, Semnan 35195-363, Iran semnan.ac.ir

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A. Ebadian,

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A. Ebadian

[email protected]

Department of Mathematics, Payame Noor University, Tabriz Branch, Tabriz, Iran pnu.ac.ir

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N. Ghobadipour,

N. Ghobadipour

Department of Mathematics, Urmia University, Urmia, Iran urmia.ac.ir

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J. M. Rassias,

J. M. Rassias

Section of Mathematics and Informatics, Pedagogical Department, National and Kapodistrian University of Athens, 4, Agamemnonos Street, Aghia Paraskevi, 15342 Athens, Greece uoa.gr

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M. B. Savadkouhi,

M. B. Savadkouhi

Department of Mathematics, Islamic Azad University, Mahdishahr Branch, Mahdishahr, Semnan, Iran iau.ac.ir

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First published: 25 March 2012

https://doi.org/10.1155/2012/984160

Citations: 2

Academic Editor: Josip E. Pecaric

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Abstract

We investigate the stability and superstability of ternary homomorphisms between C^*-ternary algebras and derivations on C^*-ternary algebras, associated with the following functional equation f((x₂ − x₁)/3) + f((x₁ − 3x₃)/3) + f((3x₁ + 3x₃ − x₂)/3) = f(x₁).

1. Introduction

A C^*-ternary algebra is a complex Banach space A, equipped with a ternary product (x, y, z)↣[x, y, z] of A³ into A, which is ℂ-linear in the outer variables, conjugate ℂ-linear in the middle variable, and associative in the sense that [x, y, [z, w, v]] = [x, [w, z, y], v] = [[x, y, z], w, v], and satisfies ∥[x, y, z]∥≤∥x∥·∥y∥·∥z∥ and ∥[x, x, x]∥ = ∥x∥³. If a C^*-ternary algebra (A, [·, ·, ·]) has an identity, that is, an element e ∈ A such that x = [x, e, e] = [e, e, x] for all x ∈ A, then it is routine to verify that A, endowed with xoy : = [x, e, y] and x^* : = [e, x, e], is a unital C^*-algebra. Conversely, if (A, o) is a unital C^*-algebra, then [x, y, z]: = xoy^*oz makes A into a C^*-ternary algebra. A ℂ-linear mapping H : A → B is called a C^*-ternary algebra homomorphism if

()

for all x, y, z ∈ A. A ℂ-linear mapping δ : A → A is called a C^*-ternary algebra derivation if

()

for all x, y, z ∈ A.

Ternary structures and their generalization the so-called n-ary structures raise certain hopes in view of their applications in physics (see [1–8]).

We say a functional equation ζ is stable if any function g satisfying the equation ζ approximately is near to true solution of ζ. Moreover, ζ is superstable if every approximately solution of ζ is an exact solution of it.

The study of stability problems originated from a famous talk given by Ulam [9] in 1940: “Under what condition does there exist a homomorphism near an approximate homomorphism?” In the next year 1941, Hyers [10] answered affirmatively the question of Ulam for additive mappings between Banach spaces.

A generalized version of the theorem of Hyers for approximately additive maps was given by Rassias [11] in 1978 as follows.

Theorem 1.1. Let f : E₁ → E₂ be a mapping from a normed vector space E₁ into a Banach space E₂ subject to the inequality:

()

for all x, y ∈ E₁, where ϵ and p are constants with ϵ > 0 and p < 1. Then, there exists a unique additive mapping T : E₁ → E₂ such that

()

for all x ∈ E₁.

The stability phenomenon that was introduced and proved by Rassias is called Hyers-Ulam-Rassias stability. And then the stability problems of several functional equations have been extensively investigated by a number of authors, and there are many interesting results concerning this problem (see [12–27]).

Throughout this paper, we assume that A is a C^*-ternary algebra with norm ∥·∥_A and that B is a C^*-ternary algebra with norm ∥·∥_B. Moreover, we assume that n₀ ∈ ℕ is a positive integer and suppose that .

2. Superstability

In this section, first we investigate homomorphisms between C^*-ternary algebras. We need the following Lemma in the main results of the paper.

Lemma 2.1. Let f : A → B be a mapping such that

()

for all x₁, x₂, x₃ ∈ A. Then f is additive.

Proof. Letting x₁ = x₂ = x₃ = 0 in (2.1), we get

()

So f(0) = 0. Letting x₁ = x₂ = 0 in (2.1), we get

()

for all x₃ ∈ A. Hence f(−x₃) = −f(x₃) for all x₃ ∈ A. Letting x₁ = 0 and x₂ = 6x₃ in (2.1), we get

()

for all x₃ ∈ A. Hence

()

for all x₃ ∈ A. Letting x₁ = 0 and x₂ = 9x₃ in (2.1), we get

()

for all x₃ ∈ A. Hence

()

for all x₃ ∈ A. Letting x₁ = 0 in (2.1), we get

()

for all x₂, x₃ ∈ A. So

()

for all x₂, x₃ ∈ A. Let t₁ = x₃ − (x₂/3) and t₂ = x₂/3 in (2.9). Then

()

for all t₁, t₂ ∈ A, this means that f is additive.

Now, we prove the first result in superstability as follows.

Theorem 2.2. Let p ≠ 1 and θ be nonnegative real numbers, and let f : A → B be a mapping such that

()

for all

and all x₁, x₂, x₃ ∈ A. Then, the mapping f : A → B is a C^*-ternary algebra homomorphism.

Proof. Assume p > 1.

Let μ = 1 in (2.11). By Lemma 2.1, the mapping f : A → B is additive. Letting x₁ = x₂ = 0 in (2.11), we get

()

for all x₃ ∈ A and μ ∈ 𝕋¹. So

()

for all x₃ ∈ A and all μ ∈ 𝕋¹. Hence f(μx₃) = μf(x₃) for all x₃ ∈ A and all

. By same reasoning as proof of Theorem 2.2 of [28], the mapping f : A → B is ℂ-linear. It follows from (2.12) that

()

for all x₁, x₂, x₃ ∈ A. Thus,

()

for all x₁, x₂, x₃ ∈ A. Hence, the mapping f : A → B is a C^*-ternary algebra homomorphism. Similarly, one obtains the result for the case p < 1.

Now, we establish the superstability of derivations on C^*-ternary algebras as follows.

Theorem 2.3. Let p ≠ 1 and θ be nonnegative real numbers, and let f : A → A be a mapping satisfying (2.11) such that

()

for all x₁, x₂, x₃ ∈ A. Then the mapping f : A → A is a C^*-ternary derivation.

Proof. Assume p > 1.

By the Theorem 2.2, the mapping f : A → A is ℂ-linear. It follows from (2.17) that

()

for all x₁, x₂, x₃ ∈ A. So

()

for all x₁, x₂, x₃ ∈ A. Thus, the mapping f : A → A is a C^*-ternary derivation. Similarly, one obtains the result for the case p < 1.

3. Stability

First we prove the generalized Hyers-Ulam-Rassias stability of homomorphisms in C^*-ternary algebras.

Theorem 3.1. Let p > 1 and θ be nonnegative real numbers, and let f : A → B be a mapping such that

()

for all

, and all x₁, x₂, x₃ ∈ A. Then there exists a unique C^*-ternary homomorphism H : A → B such that

()

for all x₁ ∈ A.

Proof. Let us assume μ = 1, x₂ = 2x₁ and x₃ = 0 in (3.1). Then we get

()

for all x₁ ∈ A. So by induction, we have

()

for all x₁ ∈ A. Hence

()

for all nonnegative integers m and n with n ≥ m, and all x₁ ∈ A. It follows that the sequence {3ⁿf(x₁/3ⁿ)} is a Cauchy sequence for all x₁ ∈ A. Since B is complete, the sequence {3ⁿf(x₁/3ⁿ)} converges. Thus, one can define the mapping H : A → B by

()

for all x₁ ∈ A. Moreover, letting m = 0 and passing the limit n → ∞ in (3.6), we get (3.3). It follows from (3.1) that

()

for all

, and all x₁, x₂, x₃ ∈ A. So

()

for all

, and all x₁, x₂, x₃ ∈ A. By the same reasoning as proof of Theorem 2.2 of [28], the mapping H : A → B is ℂ-linear.

Now, let H^′ : A → B be another additive mapping satisfying (3.3). Then, we have

()

which tends to zero as n → ∞ for all x₁ ∈ A. So we can conclude that H(x₁) = H^′(x₁) for all x₁ ∈ A. This proves the uniqueness of H.

It follows from (3.2) that

()

for all x₁, x₂, x₃ ∈ A.

Thus, the mapping H : A → B is a unique C^*-ternary homomorphism satisfying (3.3).

Theorem 3.2. Let p < 1 and θ be nonnegative real numbers, and let f : A → B be a mapping satisfying (3.1) and (3.2). Then, there exists a unique C^*-ternary homomorphism H : A → B such that

()

for all x₁ ∈ A.

Proof. The proof is similar to the proof of Theorem 3.1.

Now, we prove the generalized Hyers-Ulam-Rassias stability of derivations on C^*-ternary algebras.

Theorem 3.3. Let p > 1 and θ be nonnegative real numbers, and let f : A → A be a mapping such that

()

for all

, and all x₁, x₂, x₃ ∈ A. Then, there exists a unique C^*-ternary derivation D : A → A such that

()

for all x₁ ∈ A.

Proof. By the same reasoning as in the proof of the Theorem 3.1, there exists a unique ℂ-linear mapping D : A → A satisfying (3.15). The mapping D : A → A is defined by

()

for all x₁ ∈ A. It follows from (3.14) that

()

for all x₁, x₂, x₃ ∈ A. So

()

for all x₁, x₂, x₃ ∈ A.

Thus, the mapping D : A → A is a unique C^*-ternary derivation satisfying (3.15).

Theorem 3.4. Let p < 1 and θ be nonnegative real numbers, and let f : A → A be a mapping satisfying (3.13)and (3.14). Then, there exists a unique C^*-ternary derivation D : A → A such that

()

for all x₁ ∈ A.

Proof. The proof is similar to the proof of Theorems 3.1 and 3.3.

4. Conclusions

In this paper, we have analyzed some detail C^*-ternary algebras and derivations on C^*-ternary algebras, associated with the following functional equation:

()

A detailed study of how we can have the generalized Hyers-Ulam-Rassias stability of homomorphisms and derivations on C^*-ternary algebras is given.

Acknowledgment

This paper was partially supported by Research Program of Payame Noor University of Tabriz.

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Approximately Ternary Homomorphisms and Derivations on C^∗-Ternary Algebras

Abstract

1. Introduction

2. Superstability

3. Stability

4. Conclusions

Acknowledgment

References

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Approximately Ternary Homomorphisms and Derivations on C∗-Ternary Algebras

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Approximately Ternary Homomorphisms on C*-Ternary Algebras

Abstract

1. Introduction

2. Superstability

3. Stability

4. Conclusions

Acknowledgment

References

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Approximately Ternary Homomorphisms and Derivations on C^∗-Ternary Algebras

Approximately Ternary Homomorphisms on C^*-Ternary Algebras