• Bulletin of the Korean Mathematical Society
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• v.55 no.4
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• pp.1109-1124
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• 2018

For a metric space (X, d) and ${\alpha}$ > 0. We study the structure and properties of vector-valued Lipschitz algebra $Lip{\alpha}(X,E)$ and $lip{\alpha}(X,E)$ of order ${\alpha}$. We investigate the approximate and Character amenability of vector-valued Lipschitz algebras.

• Honam Mathematical Journal
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• v.20 no.1
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• pp.31-43
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• 1998

Exact definitions of four kinds of points shall be defined associated to Lie algebras L over an algebraically closed field F of prime characteristic p > 0. Next, rough bound of dimensions for L-irreducible modules associated to subregular points shall be established by taking advantage of Premet's result.

• Korean Journal of Mathematics
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• v.29 no.2
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• pp.271-291
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• 2021

Hom-Lie-Yamaguti color algebras are defined and their representation and cohomology theory is considered. The (2, 3)-cocycles of a given Hom-Lie-Yamaguti color algebra T are shown to be very useful in a study of its deformations. In particular, it is shown that any (2, 3)-cocycle of T gives rise to a Hom-Lie-Yamaguti color structure on T⊕V , where V is a T-module, and that a one-parameter infinitesimal deformation of T is equivalent to that a (2, 3)-cocycle of T (with coefficients in the adjoint representation) defines a Hom-Lie-Yamaguti color algebra of deformation type.

• East Asian mathematical journal
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• v.11 no.2
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• pp.199-205
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• 1995

• Journal of the Chungcheong Mathematical Society
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• v.17 no.1
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• pp.85-101
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• 2004

Let A be a Hopf algebra with a linear form ${\sigma}:k{\rightarrow}A{\otimes}A$, which is convolution invertible, such that ${\sigma}_{21}({\Delta}{\otimes}id){\tau}({\sigma}(1))={\sigma}_{32}(id{\otimes}{\Delta}){\tau}({\sigma}(1))$. We define Hopf algebras, ($A_{\sigma}$, m, u, ${\Delta}_{\sigma}$, ${\varepsilon}$, $S_{\sigma}$). If B and C are opposite skew copaired Hopf algebras and $A=B{\otimes}_kC$ then we find Hopf algebras, ($A_{[{\sigma}]}$, $m_B{\otimes}m_C$, $u_B{\otimes}u_C$, ${\Delta}_{[{\sigma}]}$, ${\varepsilon}B{\otimes}{\varepsilon}_C$, $S_{[{\sigma}]}$). Let H be a finite dimensional commutative Hopf algebra with dual basis $\{h_i\}$ and $\{h_i^*\}$, and let $A=H^{op}{\otimes}H^*$. We show that if we define ${\sigma}:k{\rightarrow}H^{op}{\otimes}H^*$ by ${\sigma}(1)={\sum}h_i{\otimes}h_i^*$ then ($A_{[{\sigma}]}$, $m_A$, $u_A$, ${\Delta}_{[{\sigma}]}$, ${\varepsilon}_A$, $S_{[{\sigma}]}$) is the dual space of Drinfeld double, $D(H)^*$, as Hopf algebra.

• Journal of the Korean Mathematical Society
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• v.43 no.2
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• pp.323-356
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• 2006

Let X and Y be vector spaces. It is shown that a mapping $f\;:\;X{\rightarrow}Y$ satisfies the functional equation $$mn_{mn-2}C_{k-2}f(\frac {x_1+...+x_{mn}} {mn})$$ $(\ddagger)\;+mn_{mn-2}C_{k-1}\;\sum\limits_{i=1}^n\;f(\frac {x_{mi-m+1}+...+x_{mi}} {m}) =k\;{\sum\limits_{1{\leq}i_1<... if and only if the mapping $f : X{\rightarrow}Y$ is additive, and we prove the Cauchy-Rassias stability of the functional equation $(\ddagger)$ in Banach modules over a unital $C^*-algebra$. Let A and B be unital $C^*-algebra$ or Lie $JC^*-algebra$. As an application, we show that every almost homomorphism h : $A{\rightarrow}B$ of A into B is a homomorphism when $h(2^d{\mu}y) = h(2^d{\mu})h(y)\;or\;h(2^d{\mu}\;o\;y)=h(2^d{\mu})\;o\;h(y)$ for all unitaries ${\mu}{\in}A,\;all\;y{\in}A$, and d = 0,1,2,..., and that every almost linear almost multiplicative mapping $h:\;A{\rightarrow}B$ is a homomorphism when h(2x)=2h(x) for all $x{\in}A$. Moreover, we prove the Cauchy-Rassias stability of homomorphisms in $C^*-algebras$ or in Lie $JC^*-algebras$, and of Lie $JC^*-algebra$ derivations in Lie $JC^*-algebras$.

• Bulletin of the Korean Mathematical Society
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• v.38 no.4
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• pp.709-718
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• 2001

In this paper we show that if D and G are continuous linear Jordan derivations on a Banach algebra A satisfying [D(x), x]x - x[G(x),x] $\epsilon$ rad(A)for all $\epsilon$ A, then both D and G map A into rad(A).

• Communications of the Korean Mathematical Society
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• v.31 no.2
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• pp.229-235
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• 2016

In this paper we study the regularity of inside (or outside) (${\theta},{\phi}$)-derivations in BCI-algebras X and prove that let $d_{({\theta},{\phi})}:X{\rightarrow}X$ be an inside (${\theta},{\phi}$)-derivation of X. If there exists a ${\alpha}{\in}X$ such that $d_{({\theta},{\phi})}(x){\ast}{\theta}(a)=0$, then $d_{({\theta},{\phi})}$ is regular for all $x{\in}X$. It is also shown that if X is a BCK-algebra, then every inside (or outside) (${\theta},{\phi}$)-derivation of X is regular. Furthermore the concepts of ${\theta}$-ideal, ${\phi}$-ideal and invariant inside (or outside) (${\theta},{\phi}$)-derivations of X are introduced and their related properties are investigated. Finally we obtain the following result: If $d_{({\theta},{\phi})}:X{\rightarrow}X$ is an outside (${\theta},{\phi}$)-derivation of X, then $d_{({\theta},{\phi})}$ is regular if and only if every ${\theta}$-ideal of X is $d_{({\theta},{\phi})}$-invariant.

• Journal of applied mathematics & informatics
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• v.26 no.5_6
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• pp.1221-1232
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• 2008

The notions of vague d-subalgebras, vague BCK-ideals, vague d-ideals, vague $d^#$-ideals and vague $d^*$-ideals are introduced, and their properties are investigated. Relations between vague d-subalgebras, vague BCK-ideals, vague d-ideals, vague $d^#$-ideals and vague $d^*$-ideals are established.

• Communications of the Korean Mathematical Society
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• v.12 no.2
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• pp.251-256
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• 1997

We show that a strong system of derivations ${D_0, D_1,\cdots,D_m}$ on a commutative Banach algebra A is contained in the radical of A if it satisfies one of the following conditions for separating spaces; (1) $\partial(D_i) \subseteq rad(A) and \partial(D_i) \subseteq K D_i(rad(A))$ for all i, where $K D_i(rad(A)) = {x \in rad(A))$ : for each $m \geq 1, D^m_i(x) \in rad(A)}$. (2) $(D^m_i) \subseteq rad(A)$ for all i and m. (3) $\bar{x\partial(D_i)} = \partial(D_i)$ for all i and all nonzero x in rad(A).