HYERS-ULAM STABILITY OF TERNARY (σ,τ,ξ)-DERIVATIONS ON C*-TERNARY ALGEBRAS: REVISITED

Abstract

In [1], the definition of C*-Lie ternary (σ,τ,ξ)-derivation is not well-defined and so the results of [1, Section 4] on C*-Lie ternary (σ,τ,ξ)-derivations should be corrected.

1. HYERS-ULAM STABILITY OF C*-LIE TERNARY (σ, τ, ξ)-DERIVATIONS

A C*-ternary algebra is a complex Banach space A, equipped with a ternary product (x, y, z) → [xyz] 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 [xy[zwv]] = [x[wzy]v] = [[xyz]wv], and satisfies ‖[xyz]‖ ≤ ‖x‖ · ‖y‖ · ‖z‖ and ‖[xxx]‖ = ‖x‖3.

Definition 1.1 ([1]). Let A be a C*-ternary algebra and let σ, τ, ξ : A → A be ℂ-linear mappings. A ℂ-linear mapping L : A → A is called a C*-Lie ternary (σ, τ, ξ)-derivation if

for all x, y, z ∈ A, where [xyz](σ, τ, ξ) = xτ(y)ξ(z) − σ(z)τ(y)x.

The x- and z-variables of the left side are ℂ-linear and the y-variable of the left side is conjugate ℂ-linear. But the x-variable of the right side is not ℂ-linear and the y-variable of the right side is not conjugate ℂ-linear. Furthermore, the y-variable of the right side in the definition of [xyz] is ℂ-linear. But the y-variable of the left side is conjugate ℂ-linear. Thus we correct the definition of C*-Lie ternary (σ, τ, ξ)-derivation as follows.

Definition 1.2. Let A be a C*-ternary algebra and let σ, τ, ξ : A → A be ℂ-linear mappings. A ℂ-linear mapping L : A → A is called a C*-Lie ternary (σ, τ, ξ)-derivation if

for all x, y, z ∈ A, where [xyz](σ, τ, ξ) = xτ(y)*ξ(z) − σ(z)τ(y)*x.

Throughout this paper, assume that A is a C*-ternary with norm ‖ · ‖, and that σ, τ, ξ : A → A are ℂ-linear mappings. Let q be a positive rational number.

We prove the Hyers-Ulam stability of C*-Lie ternary (σ, τ, ξ)-derivations on C*-ternary algebras, associated with the Euler-Lagrange type additive mapping.

Theorem 1.3. Let n ∈ ℕ: Assume that r > 3 if nq > 1 and that 0 < r < 1 if nq < 1. Let θ be a positive real number, and let f : A → A be an odd mapping for which there exist mappings g, h, k : A → A with g(0) = h(0) = k(0) = 0 satisfying (2.1), (2.3)–(2.5) of [1] and

for all x, y, z ∈ A. Then there exist unique ℂ-linear mappings σ, τ, ξ : A → A and a unique C*-Lie ternary (σ, τ, ξ)-derivation L : A → A satisfying (2.6)-(2.8) of [1] and

for all x ∈ A.

Proof. By the same reasoning as in the proof of [1, Theorem 2.1], one can show that there exist unique ℂ-linear mappings σ, τ, ξ : A → A and a unique ℂ-linear mapping L : A → A satisfying (2.6)–(2.8) of [1] and (1.2). The mapping L : A → A is defined by

for all x ∈ A.

It follows from (1.1) that

for all x, y, z ∈ A. So

for all x, y, z ∈ A.

The rest of the proof is similar to the proof of [1, Theorem 2.1].   ☐

Theorem 1.4. Let n ∈ ℕ: Assume that 0 < r < 1 if nq > 1 and that r > 3 if nq < 1. Let θ be a positive real number, and let f : A → A be an odd mapping for which there exist mappings g, h, k : A → A with g(0) = h(0) = k(0) = 0 satisfying (2.1), (2.3)−(2.5) of [1] and (1.1). Then there exist unique ℂ-linear mappings σ, τ, ξ : A → A and a unique C*-Lie ternary (σ, τ, ξ)-derivation L : A → A satisfying (2.12)−(2.14) of [1] and

for all x ∈ A.

Proof. By the same reasoning as in the proof of [1, Theorem 2.2], there exist unique ℂ-linear mappings σ, τ, ξ : A → A and a unique ℂ-linear mapping L : A → A satisfying (2.1), (2.3)−(2.5) of [1] and (1.3). The mapping L : A → A is defined by

for all x ∈ A.

The rest of the proof is similar to the proofs of Theorem 1.3 and [1, Theorem 2.1].   ☐

Theorem 1.5. Let n ∈ ℕ: Assume that r > 1 if nq > 1 and that 0 < nr < 1 if nq < 1. Let θ be a positive real number, and let f : A → A be an odd mapping for which there exist mappings g, h, k : A → A with g(0) = h(0) = k(0) = 0 satisfying (2.3)−(2.5), (2.17) of [1] and

for all x, y, z ∈ A. Then there exist unique ℂ-linear mappings σ, τ, ξ : A → A and a unique C*-Lie ternary (σ, τ, ξ)-derivation L : A → A satisfying (2.6)−(2.8) of [1] and

for all x ∈ A.

Proof. By the same reasoning as in the proof of [1, Theorem 2.3], there exist unique ℂ-linear mappings σ, τ, ξ : A → A and a unique ℂ-linear mapping L : A → A satisfying (2.6)−(2.8) of [1] and (1.5). The mapping L : A → A is defined by

for all x ∈ A.

It follows from (1.4) that

for all x ∈ A. Hence

for all x, y, z ∈ A and the proof of the theorem is complete.   ☐

Theorem 1.6. Let n ∈ ℕ: Assume that r > 1 if nq < 1 and that 0 < nr < 1 if nq > 1. Let θ be a positive real number, and let f : A → A be an odd mapping for which there exist mappings g, h, k : A → A with g(0) = h(0) = k(0) = 0 satisfying (2.3)−(2.5), (2.17) of [1] and (1.4). Then there exist unique ℂ-linear mappings σ, τ, ξ : A → A to A and a unique C*-ternary (σ, τ, ξ)-derivation L : A → A satisfying (2.12)−(2.14) of [1] and

for all x ∈ A.

Proof. By the same reasoning as in the proof of [1, Theorem 2.4], there exist unique ℂ-linear mappings σ, τ, ξ : A → A and a unique ℂ-linear mapping L : A → A satisfying (2.12)−(2.14) of [1] and (1.6). The mapping L : A → A is defined by

for all x ∈ A.

It follows from (1.4) that

for all x, y, z ∈ A. So

for all x ∈ A and the proof of the theorem is complete.   ☐