• Journal of the Korean Mathematical Society
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• v.35 no.2
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• pp.331-340
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• 1998

We define the spherical non-commutative torus $L_{\omega}$/ as the crossed product obtained by an iteration of l crossed products by actions of, the first action on C( $S^{2n＋l}$). Assume the fibres are isomorphic to the tensor product of a completely irrational non-commutative torus $A_{p}$ with a matrix algebra $M_{m}$ ( ) (m > 1). We prove that $L_{\omega}$/ $M_{p}$ (C) is not isomorphic to C(Prim( $L_{\omega}$/)) $A_{p}$ $M_{mp}$ (C), and that the tensor product of $L_{\omega}$/ with a UHF-algebra $M_{p{\infty}}$ of type $p^{\infty}$ is isomorphic to C(Prim( $L_{\omega}$/)) $A_{p}$ $M_{m}$ (C) $M_{p{\infty}}$ if and only if the set of prime factors of m is a subset of the set of prime factors of p. Furthermore, it is shown that the tensor product of $L_{\omega}$/, with the C＊-algebra K(H) of compact operators on a separable Hilbert space H is not isomorphic to C(Prim( $L_{\omega}$/)) $A_{p}$ $M_{m}$ (C) K(H) if Prim( $L_{\omega}$/) is homeomorphic to $L^{k}$ (n)$\times$ $T^{l'}$ for k and l' non-negative integers (k > 1), where $L^{k}$ (n) is the lens space.$T^{l'}$ for k and l' non-negative integers (k > 1), where $L^{k}$ (n) is the lens space.e.

• 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.30 no.2
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• pp.193-199
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• 1993

A hereditary $C^{*}$-subalgebra B of a $C^{*}$-algebra A is said to be full if B is not contained in any proper closed two-sided ideal in A, so each hereditary $C^{*}$-subalgebra of a simple $C^{*}$-algebra is always full. It is well known that every $C^{*}$-algebra is strong Morita equivalent to its full hereditary $C^{*}$-subalgebra, but the strong Morita equivalence of a $C^{*}$-algebra A and its hereditary $C^{*}$-subalgebra B does not imply the fullness of B, ingeneral. We present the following lemma for our computational convenience in the course of the proof of the main theorem. Note that $L_{B}$, $L_{B}$$^{*}$ and $L_{B}$ $L_{B}$$^{*}$ are all .alpha.-invariant whenever B is .alpha.-invariant under the action .alpha. of G.a. of G.a. of G.a. of G.f G.

• Journal of the Korean Mathematical Society
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• v.56 no.1
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• pp.1-23
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• 2019

Let ${\mathcal{G}}$ be an ultragraph and let $C^*({\mathcal{G}})$ be the associated $C^*$-algebra introduced by Tomforde. For any gauge invariant ideal $I_{(H,B)}$ of $C^*({\mathcal{G}})$, we approach the quotient $C^*$-algebra $C^*({\mathcal{G}})/I_{(H,B)}$ by the $C^*$-algebra of finite graphs and prove versions of gauge invariant and Cuntz-Krieger uniqueness theorems for it. We then describe primitive gauge invariant ideals and determine purely infinite ultragraph $C^*$-algebras (in the sense of Kirchberg-Rørdam) via Fell bundles.

• Journal of Educational Research in Mathematics
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• v.17 no.3
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• pp.253-270
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• 2007

by A.C. Clairaut was written based on the historico-genetic principle such as his . In this paper, by analyzing his we can induce six principles that Clairaut adopted to teach algebra: necessity and curiosity as a motive of studying algebra, harmony of discovery and proof, complementarity of generalization and specialization, connection of knowledge to be learned with already known facts, semantic approaches to procedural knowledge of mathematics, reversible approach. These can be considered as strategies for teaching algebra accorded with beginner's mind. Some of them correspond with characteristics of , but the others are unique in the domain of algebra. And by comparing Clairaut's approaches with school algebra, we discuss about some mathematical subjects: setting equations in relation to problem situations, operations and signs of letters, rule of signs in multiplication, solving quadratic equations, and general relationship between roots and coefficients of equations.

• Journal of the Chungcheong Mathematical Society
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• v.13 no.1
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• pp.1-11
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• 2000

The non-commutative torus $A_{\omega}=C^*(\mathbb{Z}^n,{\omega})$ may be realized as the $C^*$-algebra of sections of a locally trivial $C^*$-algebra bundle over $\widehat{S_{\omega}}$ with fibres $C^*(\mathbb{Z}^n/S_{\omega},{\omega}_1)$ for some totally skew multiplier ${\omega}_1$ on $\mathbb{Z}^n/S_{\omega}$. It is shown that $A_{\omega}{\otimes}M_l(\mathbb{C})$ has the trivial bundle structure if and only if $\mathbb{Z}^n/S_{\omega}$ is torsion-free.

• Honam Mathematical Journal
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• v.7 no.1
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• pp.119-127
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• 1985

A study is made of a special type of $C^{\ast}$ -dynamical systems, consisting of a class $n^{\infty}$ uniformly hyperfinite $C^{\ast}$-algebra A, the torus group $G=T^{d}$ ($$1{\leq_-}d{\leq_-}n-1$$) and a natural product action of G on A by $^{*}-automorphisms$. We give some conditions for product states on the fixed point algebra $A^{G}$ of A by G to be factorial.

• Bulletin of the Korean Mathematical Society
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• v.40 no.1
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• pp.53-61
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• 2003

We prove the generalized Hyers-Ulam-Rassias stability of the invertible mapping in a Banach module over a unital Banach algebra, and prove the generalized Hyers-Ulam-Rassias stability of the Jensen's functional equations in a Hilbert module over a unital $C^{*}$-algebra.

• Korean Journal of Mathematics
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• v.27 no.2
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• pp.445-463
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• 2019

Let C[0, T] denote the space of continuous real-valued functions on [0, T]. On the space C[0, T], we introduce a Banach algebra of series of functions which are generalized Fourier-Stieltjes transforms of measures of finite variation on the product of simplex and Euclidean space. We evaluate analytic Feynman integrals of the functions in the Banach algebra which play significant roles in the Feynman integration theory and quantum mechanics.

• 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 $\overline{D}$.