• Korean Journal of Mathematics
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• v.18 no.1
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• pp.1-10
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• 2010

It is shown that for an almost algebra homomorphism between Banach algebras, there exists a unique algebra homomorphism near the almost algebra homomorphism. Moreover, we prove that for an almost algebra ${\ast}$-homomorphism between $C^{\ast}$-algebras, there exists a unique algebra ${\ast}$-homomorphism near the almost algebra ${\ast}$-homomorphism, and that for an almost algebra ${\ast}$-homomorphism between $JB^{\ast}$-algebras, there exists a unique algebra ${\ast}$-homomorphism near the almost algebra ${\ast}$-homomorphism.

• Journal of the Chungcheong Mathematical Society
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• v.24 no.2
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• pp.287-301
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• 2011

Let X and Y be vector spaces. It is shown that a mapping $f\;:\;X{\rightarrow}Y$ satisfies the functional equation $$(\ddag)\hspace{50}dk\;f\left(\frac{\sum_{j=1}^{dk}x_j}{dk}\right)=\displaystyle\sum_{j=1}^{dk}f(x_j)$$ if and only if the mapping $f$ : X ${\rightarrow}$ Y is Cauchy additive, and prove the Cauchy-Rassias stability of the functional equation ($\ddag$) in Banach modules over a unital $C^{\ast}$-algebra. Let $\mathcal{A}$ and $\mathcal{B}$ be unital $C^{\ast}$-algebras. As an application, we show that every almost homomorphism $h\;:\;\mathcal{A}{\rightarrow}\mathcal{B}$ of $\mathcal{A}$ into $\mathcal{B}$ is a homomorphism when $h((k-1)^nuy)=h((k-1)^nu)h(y)$ for all unitaries $u{\in}\mathcal{A}$, all $y{\in}\mathcal{A}$, and $n$ = 0,1,2,$\cdots$. Moreover, we prove the Cauchy-Rassias stability of homomorphisms in $C^{\ast}$-algebras.

• Honam Mathematical Journal
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• v.42 no.4
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• pp.757-768
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• 2020

In this paper, the Hyers-Ulam-Rassias stability of mixed n-Jordan homomorphisms on Banach algebras and the superstability of mixed n-Jordan ∗-homomorphism between C∗-algebras are investigated.

• The Pure and Applied Mathematics
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• v.7 no.1
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• pp.61-69
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• 2000

Let($X^{\ast},\tau^{\ast}$) be the space with one point Lindeloffication topology of space (X,$\tau$). This paper offers the definition of the space with one point Lin-deloffication topology of a topological space and proves that the retraction regu-lar closed function f: $K^{\ast}(X^{\ast}$) defined f($A^{\ast})=A^{\ast}$ if p $\in A^{\ast}$ or ($f(A^{\ast})=A^{\ast}-{p}$ if $p \in A^{\ast}$ is a homomorphism. There are two examples in this paper to show that the retraction regular closed function f is neither a surjection nor an injection.

• Bulletin of the Korean Mathematical Society
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• v.56 no.1
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• pp.57-71
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• 2019

Let R be an associative ring with identity, M a t.u.p. monoid with only one unit and ${\omega}:M{\rightarrow}End(R)$ a monoid homomorphism. Let R be a reversible, M-compatible ring and ${\alpha}=a_1g_1+{\cdots}+a_ng_n$ a non-zero element in skew monoid ring $R{\ast}M$. It is proved that if there exists a non-zero element ${\beta}=b_1h_1+{\cdots}+b_mh_m$ in $R{\ast}M$ with ${\alpha}{\beta}=c$ is a constant, then there exist $1{\leq}i_0{\leq}n$, $1{\leq}j_0{\leq}m$ such that $g_{i_0}=e=h_{j_0}$ and $a_{i_0}b_{j_0}=c$ and there exist elements a, $0{\neq}r$ in R with ${\alpha}r=ca$. As a consequence, it is proved that ${\alpha}{\in}R*M$ is unit if and only if there exists $1{\leq}i_0{\leq}n$ such that $g_{i_0}=e$, $a_{i_0}$ is unit and aj is nilpotent for each $j{\neq}i_0$, where R is a reversible or right duo ring. Furthermore, we determine the relation between clean and nil clean elements of R and those elements in skew monoid ring $R{\ast}M$, where R is a reversible or right duo ring.