• 충청수학회지
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• 제24권2호
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• pp.147-162
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• 2011

In this paper, we investigate the Ulam-Hyers stability of $C^{\star}$-ternary algebra 3-homomorphisms for the functional equation $$f(x_1+x_2,y_1+y_2,z_1+z_2)=\;\displaystyle\sum_{1{\leq}i,j,k{\leq}2}\;f(x_i,y_j,z_k)$$ in $C^{\star}$-ternary algebras.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제22권4호
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• pp.383-387
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• 2015

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.

• Kyungpook Mathematical Journal
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• 제55권4호
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• pp.893-907
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• 2015

The Hyers-Ulam-Rassias stability of the k-th partial ternary quadratic derivations is investigated in non-Archimedean Banach ternary algebras and non-Archimedean $C^*$-ternary algebras by using the fixed point theorem.

• 대한수학회논문집
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• 제38권1호
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• pp.123-135
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• 2023

It is proved that there is a one to one correspondence between representations of C^*-ternary ring M and C^*-algebra 𝒜(M). We discuss primitive and modular ideals of a C^*-ternary ring and prove that a closed ideal I is primitive or modular if and only if so is the ideal 𝒜(I) of 𝒜(M). We also show that a closed ideal in M is primitive if and only if it is the kernel of some irreducible representation of M. Lastly, we obtain approximate identity characterization of strongly quasi-central C^*-ternary ring and the ideal structure of the TRO V ⊗^tmin B for a C^*-algebra B.

• Korean Journal of Mathematics
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• 제17권1호
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• pp.83-90
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• 2009

In this paper, we investigate isomorphisms between $C^*$-ternary algebras and derivations on $C^*$-ternary algebras associated with the Cauchy-Jensen functional equation $$2f(\frac{x+y}{2}+z)=f(x)+f(y)+2f(z)$$, which was introduced and investigated by Baak in [2].

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제24권3호
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• pp.171-178
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• 2017

In this paper, we define $C^*-ternary$ quadratic 3-Jordan homomorphisms associated with the quadratic mapping f(x + y) + f(x - y) = 2f(x) + 2f(y), and prove the Hyers-Ulam stability of $C^*-ternary$ quadratic 3-Jordan homomorphisms.

• 대한수학회보
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• 제47권1호
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• pp.195-209
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• 2010

In this paper, we prove the generalized Hyers-Ulam stability of bi-homomorphisms in $C^*$-ternary algebras and of bi-derivations on $C^*$-ternary algebras for the following bi-additive functional equation f(x + y, z - w) + f(x - y, z + w) = 2f(x, z) - 2f(y, w). This is applied to investigate bi-isomorphisms between $C^*$-ternary algebras.

• Kyungpook Mathematical Journal
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• 제53권2호
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• pp.233-245
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• 2013

Let $\mathcal{A}$ be a Banach ternary algebra over a scalar field R or C and $\mathcal{X}$ be a ternary Banach $\mathcal{A}$-module. A quartic mapping $D\;:\;(\mathcal{A},[\;]_{\mathcal{A}}){\rightarrow}(\mathcal{X},[\;]_{\mathcal{X}})$ is called a $4^{th}$- order ternary derivation if $D([x,y,z])=[D(x),y^4,z^4]+[x^4,D(y),z^4]+[x^4,y^4,D(z)]$ for all $x,y,z{\in}\mathcal{A}$. In this paper, we prove Ulam stability generalizations of $4^{th}$- order ternary derivations associated to the following JMRassias quartic functional equation on fr$\acute{e}$che algebras: $$f(kx+y)+f(kx-y)=k^2[f(x+y)+f(x-y)]+2k^2(k^2-1)f(x)-2(k^2-1)f(y)$$.

• Korean Journal of Mathematics
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• 제21권2호
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• pp.161-170
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• 2013

Bae and W. Park [3] proved the Hyers-Ulam stability of bi-homomorphisms and bi-derivations in $C^*$-ternary algebras. It is easy to show that the definitions of bi-homomorphisms and bi-derivations, given in [3], are meaningless. So we correct the definitions of bi-homomorphisms and bi-derivations. Under the conditions in the main theorems, we can show that the related mappings must be zero. In this paper, we correct the statements and the proofs of the results, and prove the corrected theorems.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제23권3호
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• pp.237-246
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• 2016

Ebadian et al. proved the Hyers-Ulam stability of bimultipliers and Jordan bimultipliers in C*-ternary algebras by using the fixed point method. Under the conditions in the main theorems for bimultipliers, we can show that the related mappings must be zero. Moreover, there are some mathematical errors in the statements and the proofs of the results. In this paper, we correct the statements and the proofs of the results, and prove the corrected theorems by using the direct method.