• 대한수학회논문집
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• 제15권3호
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• pp.521-531
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• 2000

Graph C*-algebras C*(E) are the universal C*-algebras generated by partial isometries satisfying the Cuntz-Krieger relations determined by directed graphs E, and it is known that a simple graph C*-algebra is extremally rich in sense that it contains enough extreme consider a sufficient condition on a graph for which the associated graph algebra(possibly nonsimple) is extremally rich. We also present examples of nonextremally rich prime graph C*-algebras with finitely many ideals and with real rank zero.

• 대한수학회지
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• 제47권3호
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• pp.601-631
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• 2010

In [6] and [7], we introduced graph von Neumann algebras which are the (groupoid) crossed product algebras of von Neumann algebras and graph groupoids via groupoid actions. We showed that such crossed product algebras have the graph-depending amalgamated reduced free probabilistic properties. In this paper, we will consider a scalar-valued $W^*$-probability on a given graph von Neumann algebra. We show that a diagonal graph $W^*$-probability space (as a scalar-valued $W^*$-probability space) and a graph W¤-probability space (as an amalgamated $W^*$-probability space) are compatible. By this compatibility, we can find the relation between amalgamated free distributions and scalar-valued free distributions on a graph von Neumann algebra. Under this compatibility, we observe the scalar-valued freeness on a graph von Neumann algebra.

• 대한수학회보
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• 제44권4호
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• pp.651-661
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• 2007

Graph algebras establish a connection between directed graphs without multiple edges and special universal algebras of type (2,0). We say that a graph G satisfies an identity $s{\approx}t$ if the corresponding graph algebra $\underline{A(G)}$ satisfies $s{\approx}t$. A graph G=(V,E) is called an $(xy)x{\approx}x(yy)$ graph if the graph algebra $\underline{A(G)}$ satisfies the equation $(xy)x{\approx}x(yy)$. An identity $s{\approx}t$ of terms s and t of any type ${\tau}$ is called a hyperidentity of an algebra $\underline{A}$ if whenever the operation symbols occurring in s and t are replaced by any term operations of $\underline{A}$ of the appropriate arity, the resulting identities hold in $\underline{A}$. In this paper we characterize $(xy)x{\approx}x(yy)$ graph algebras, identities and hyperidentities in $(xy)x{\approx}x(yy)$ graph algebras.

• Kyungpook Mathematical Journal
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• 제45권4호
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• pp.561-570
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• 2005

Weak Crossed Product Algebras correspond to certain graphs called lower subtractive graphs. The properties of such algebras can be obtained by studying this kind of graphs ([4], [5]). In [1], the author showed that a weak crossed product is Frobenius and its restricted subalgebra is symmetric if and only if its associated graph has a unique maximal vertex. A special construction of these graphs came naturally and was known as standard lower subtractive graph. It was a deep question that when such a special graph possesses unique maximal vertex? This work is to answer the question partially and to give a particular characterization for such graphs at which the corresponding algebras are isomorphic. A graph that follows the mentioned characterization is called flexible. Flexibility is to some extend a generalization of the so-called Coxeter groups and its weak Bruhat ordering.

• 대한수학회보
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• 제59권3호
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• pp.567-608
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• 2022

We show that certain extensions of classifiable C^*-algebras are strongly classified by the associated six-term exact sequence in K-theory together with the positive cone of K₀-groups of the ideal and quotient. We use our results to completely classify all unital graph C^*-algebras with exactly one non-trivial ideal.

• 대한수학회보
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• 제52권4호
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• pp.1201-1211
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• 2015

s-sequences and d-sequences are fundamental sequences intensively studied in many fields of algebra. In this paper we are interested in dealing with monomial sequences associated to graphs in order to establish conditions for which they are s-sequences and/or d-sequences.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제19권2호
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• pp.199-209
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• 2012

For a locally compact higher rank graph ${\Lambda}$, we construct a two-sided path space ${\Lambda}^{\Delta}$ with shift homeomorphism ${\sigma}$ and its corresponding path groupoid ${\Gamma}$. Then we find equivalent conditions of aperiodicity, cofinality and irreducibility of ${\Lambda}$ in (${\Lambda}^{\Delta}$, ${\sigma}$), ${\Gamma}$, and the groupoid algebra $C^*({\Gamma})$.

• 대한수학회지
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• 제46권3호
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• pp.657-673
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• 2009

The fixed point algebra $C^*(E)^{\gamma}$ of a gauge action $\gamma$ on a graph $C^*$-algebra $C^*(E)$ and its AF subalgebras $C^*(E)^{\gamma}_{\upsilon}$ associated to each vertex v do play an important role for the study of dynamical properties of $C^*(E)$. In this paper, we consider the stability of $C^*(E)^{\gamma}$ (an AF algebra is either stable or equipped with a (nonzero bounded) trace). It is known that $C^*(E)^{\gamma}$ is stably isomorphic to a graph $C^*$-algebra $C^*(E_{\mathbb{Z}}\;{\times}\;E)$ which we observe being stable. We first give an explicit isomorphism from $C^*(E)^{\gamma}$ to a full hereditary $C^*$-subalgebra of $C^*(E_{\mathbb{N}}\;{\times}\;E)({\subset}\;C^*(E_{\mathbb{Z}}\;{\times}\;E))$ and then show that $C^*(E_{\mathbb{N}}\;{\times}\;E)$ is stable whenever $C^*(E)^{\gamma}$ is so. Thus $C^*(E)^{\gamma}$ cannot be stable if $C^*(E_{\mathbb{N}}\;{\times}\;E)$ admits a trace. It is shown that this is the case if the vertex matrix of E has an eigenvector with an eigenvalue $\lambda$ > 1. The AF algebras $C^*(E)^{\gamma}_{\upsilon}$ are shown to be nonstable whenever E is irreducible. Several examples are discussed.

• 대한수학회지
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• 제52권2호
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• pp.239-268
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• 2015

Let Q be a finite quiver and $G{\subseteq}Aut(\mathbb{k}Q)$ a finite abelian group. Assume that $\hat{Q}$ and ${\Gamma}$ are the generalized Mckay quiver and the valued graph corresponding to (Q, G) respectively. In this paper we discuss the relationship between indecomposable $\hat{Q}$-representations and the root system of Kac-Moody algebra $g({\Gamma})$. Moreover, we may lift G to $\bar{G}{\subseteq}Aut(g(\hat{Q}))$ such that $g({\Gamma})$ embeds into the fixed point algebra $g(\hat{Q})^{\bar{G}}$ and $g(\hat{Q})^{\bar{G}}$ as a $g({\Gamma})$-module is integrable.

• 한국콘텐츠학회논문지
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• 제11권9호
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• pp.63-75
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• 2011

분산 이동 실시간 시스템을 개발하기 위해 pi-Calculus, Mobile Ambients Calculus, Bigraph 등의 수많은 프로세스 대수가 존재한다. 하지만 시스템이 방대해지고 복잡해질수록 시스템을 구성하는 프로세스들의 통신과 이동 역시 방대해지고 복잡해지므로 프로세스 대수로 이를 이해하는데 어려움이 존재한다. 그러므로 방대하고 복잡한 시스템을 체계적으로 이해할 수 있는 방법이 필요하다. 본 논문에서는 방대하고 복잡한 시스템을 프로세스들의 통신과 이동의 순서화된 추상화 방법 즉, 계층적으로 구조화된 격자(Lattice)들의 형태인 프리즘(Prism)으로 다룬다. 이는 액티브 온톨로지(Active Ontology)에서 확장된 새로운 개념인 행위 온톨로지(Behavior Ontology)를 기반으로 한다. 프리즘은 시스템을 체계적으로 이해하기 위해 시스템을 계층적으로 구성된 행위적 속성을 지닌 격자들 관점에서 분석하는 것을 허용한다. 이러한 방법은 통신과 이동의 복잡함을 의미적이고 계층적으로 구성된 행위의 구조로 체계적으로 이해할 수 있게 한다.