• Journal of applied mathematics & informatics
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• v.31 no.3_4
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• pp.559-564
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• 2013

In this paper we investigate necessary conditions for the mirror algebra $(M(X),{\bigoplus},(0,0))$ to be a $d$-algebra (having the condition (D5), resp.) when (X, *, 0) is a d-algebra (having the condition (D5), resp.). Moreover, we obtain the necessary conditions for M(X) of a $d^*$-algebra X to be a $d^*$-algebra.

• Journal of the Korean Mathematical Society
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• v.45 no.1
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• pp.249-271
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• 2008

We prove the Hyers-Ulam stability of linear d-isometries in linear d-normed Banach modules over a unital $C^*-algebra$ and of linear isometries in Banach modules over a unital $C^*-algebra$. The main purpose of this paper is to investigate d-isometric $C^*-algebra$ isomor-phisms between linear d-normed $C^*-algebras$ and isometric $C^*-algebra$ isomorphisms between $C^*-algebras$, and d-isometric Poisson $C^*-algebra$ isomorphisms between linear d-normed Poisson $C^*-algebras$ and isometric Poisson $C^*-algebra$ isomorphisms between Poisson $C^*-algebras$. We moreover prove the Hyers-Ulam stability of their d-isometric homomorphisms and of their isometric homomorphisms.

• Journal of the Chungcheong Mathematical Society
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• v.26 no.1
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• pp.91-103
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• 2013

Let (D,B) be an admissible pair. Then recall that $B\;{\times}^L_HD^{{\rightarrow}{\pi}_D}_{{\leftarrow}i_D}\;D$ are bialgebra maps satisfying ${\pi}_D{\circ}i_D=I$. We have solved a converse in case D is a Hopf algebra. Let D be a Hopf algebra with antipode $S_D$ and be a left H-comodule algebra and a left H-module coalgebra over a field $k$. Let A be a bialgebra over $k$. Suppose $A^{{\rightarrow}{\pi}}_{{\leftarrow}i}D$ are bialgebra maps satisfying ${\pi}{\circ}i=I_D$. Set ${\Pi}=I_D*(i{\circ}s_D{\circ}{\pi}),B=\Pi(A)$ and $j:B{\rightarrow}A$ be the inclusion. Suppose that ${\Pi}$ is an algebra map. We show that (D,B) is an admissible pair and $B^{\leftarrow{\Pi}}_{\rightarrow{j}}A^{\rightarrow{\pi}}_{\leftarrow{i}}D$ is an admissible mapping system and that the generalized biproduct bialgebra $B{\times}^L_HD$ is isomorphic to A as bialgebras.

• Journal of the Chungcheong Mathematical Society
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• v.10 no.1
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• pp.141-146
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• 1997

In this paper we prove that the corona theorem for the algebra $H^{\infty}(D){\cap}D(D)$. That is, we prove that $\mathcal{M}{\setminus}{\overline{D}}$ is an empty set where $\mathcal{M}$ is the maximal ideal space of the given algebra.

• Korean Journal of Mathematics
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• v.22 no.1
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• pp.91-121
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• 2014

Let X and Y be vector spaces. It is shown that a mapping $f:X{\rightarrow}Y$ satisfies the functional equation ${\ddag}$ $$2df(\frac{x_1+{\sum}_{j=2}^{2d}(-1)^jx_j}{2d})-2df(\frac{x_1+{\sum}_{j=2}^{2d}(-1)^{j+1}x_j}{2d})=2\sum_{j=2}^{2d}(-1)^jf(x_j)$$ if and only if the mapping $f:X{\rightarrow}Y$ is additive, and prove the Cauchy-Rassias stability of the functional equation (${\ddag}$) in Banach modules over a unital $C^*$-algebra, and in Poisson Banach modules over a unital Poisson $C^*$-algebra. Let $\mathcal{A}$ and $\mathcal{B}$ be unital $C^*$-algebras, Poisson $C^*$-algebras, Poisson $JC^*$-algebras or Lie $JC^*$-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 $h(d^nuy)=h(d^nu)h(y)$ or $h(d^nu{\circ}y)=h(d^nu){\circ}h(y)$ 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 $C^*$-algebras, Poisson $C^*$-algebras, Poisson $JC^*$-algebras or Lie $JC^*$-algebras, and of Lie $JC^*$-algebra derivations in Lie $JC^*$-algebras.

• Journal of the Chungcheong Mathematical Society
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• v.23 no.2
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• pp.323-333
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• 2010

The biproduct bialgebra has been generalized to generalized biproduct bialgebra $B{\times}^L_H\;D$ in [5]. Let (D, B) be an admissible pair and let D be a bialgebra. We show that if generalized biproduct bialgebra $B{\times}^L_H\;D$ is a Hopf algebra with antipode s, then D is a Hopf algebra and the identity $id_B$ has an inverse in the convolution algebra $Hom_k$(B, B). We show that if D is a Hopf algebra with antipode $s_D$ and $s_B$ in $Hom_k$(B, B) is an inverse of $id_B$ then $B{\times}^L_H\;D$ is a Hopf algebra with antipode s described by $s(b{\times}^L_H\;d)={\Sigma}(1_B{\times}^L_H\;s_D(b_{-1}{\cdot}d))(s_B(b_0){\times}^L_H\;1_D)$. We show that the mapping system $B{\leftrightarrows}^{{\Pi}_B}_{j_B}\;B{\times}^L_H\;D{\rightleftarrows}^{{\pi}_D}_{i_D}\;D$ (where $j_B$ and $i_D$ are the canonical inclusions, ${\Pi}_B$ and ${\pi}_D$ are the canonical coalgebra projections) characterizes $B{\times}^L_H\;D$. These generalize the corresponding results in [6].

• The Pure and Applied Mathematics
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• v.30 no.3
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• pp.237-248
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• 2023

The aim of this paper is to study the concept of s-topological d-algebras which is a d-algebra supplied with a certain type of topology that makes the binary operation defined on it d-topologically continuous. This concept is a generalization of the concept of topological d-algebra. We obtain several properties of s-topological d-algebras.

• Journal of the Korean Mathematical Society
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• v.56 no.5
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• pp.1333-1354
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• 2019

In authors' paper in 2007, it was shown that the BSE-extension of $C^1_0(R)$, the algebra of continuously differentiable functions f on the real number space R such that f and df /dx vanish at infinity, is the Lipschitz algebra $Lip_1(R)$. This paper extends this result to the case of $C^n_0(R^d)$ and $C^{n-1,1}_b(R^d)$, where n and d represent arbitrary natural numbers. Here $C^n_0(R^d)$ is the space of all n-times continuously differentiable functions f on $R^d$ whose k-times derivatives are vanishing at infinity for k = 0, ${\cdots}$, n, and $C^{n-1,1}_b(R^d)$ is the space of all (n - 1)-times continuously differentiable functions on $R^d$ whose k-times derivatives are bounded for k = 0, ${\cdots}$, n - 1, and (n - 1)-times derivatives are Lipschitz. As a byproduct of our investigation we obtain an important result that $C^{n-1,1}_b(R^d)$ has a predual.

• Bulletin of the Korean Mathematical Society
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• v.37 no.2
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• pp.265-272
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• 2000

Let D be a derivation on an Banach algebra A. Suppose that [[D(x), x], D(x)] lies in the nil radical of A for all $x{\;}{\in}{\;}A$. Then D(A) is contained in the Jacobson radical of A.

• Journal of the Korean Mathematical Society
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• v.34 no.2
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• pp.279-284
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• 1997

Let L/F be a finite separable extension of Henselian valued fields with same residue fields $\overline{L} = \overline{F}$. Let D be an inertially split division algebra over L, and let $^cD$ be the underlying division algebra of the corestriction $cor_{L/F} (D)$ of D. We show that the index $ind(^cD) of ^cD$ divides $[Z(\overline{D}) : Z(\overline {^cD})] \cdot ind(D), where Z(\overline{D})$ is the center of the residue division ring $\overline{D}$.