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HYERS-ULAM STABILITY OF TERNARY (σ,τ,ξ)-DERIVATIONS ON C*-TERNARY ALGEBRAS: REVISITED
HYERS-ULAM STABILITY OF TERNARY (σ,τ,ξ)-DERIVATIONS ON C*-TERNARY ALGEBRAS: REVISITED
Journal of the Korean Society of Mathematical Education Series B The Pure and Applied Mathematics. 2015. Nov, 22(4): 383-387
Copyright © 2015, Korean Society of Mathematical Education
  • Received : October 14, 2015
  • Accepted : October 24, 2015
  • Published : November 30, 2015
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About the Authors
EON WHA SHIM
DEPARTMENT OF MATHEMATICS, HANYANG UNIVERSITY, SEOUL 04763, KOREAEmail address:stardal@daum.net
MADJID ESHAGHI GORDJI
DEPARTMENT OF MATHEMATICS, SEMNAN UNIVERSITY, P. O. BOX 35195-363, SEMNAN, IRANEmail address:madjid.eshaghi@gmail.com
JUNG RYE LEE
DEPARTMENT OF MATHEMATICS, DAEJIN UNIVERSITY, KYEONGGI 11159, KOREAEmail address:jrlee@daejin.ac.kr

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.
Keywords
1. HYERS-ULAM STABILITY OFC*-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 A 3 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
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for all x , y , z A , where [ xyz ] (σ, τ, ξ) = ( 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
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for all x , y , z A , where [ xyz ] (σ, τ, ξ) = ( 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
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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
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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
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for all x A .
It follows from (1.1) that
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for all x , y , z A . So
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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
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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
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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
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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
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Lager Image
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
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for all x A .
It follows from (1.4) that
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for all x A . Hence
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Lager Image
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
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Lager Image
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
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Lager Image
for all x A .
It follows from (1.4) that
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Lager Image
for all x , y , z A . So
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Lager Image
for all x A and the proof of the theorem is complete.   ☐
References
Gordji M. Eshaghi , Farrokhzad R. , Hosseinioun S.A.R. (2012) Hyers-Ulam stability ofternary (σ, τ, ζ)-derivations on C*-ternary algebras J. Math. Phys. Anal. Geom. 8 (1) 3 - 20