• Journal of the Korean Mathematical Society
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• v.36 no.1
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• pp.97-107
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• 1999

Given a C-dynamical system (A, G, $\alpha$) with a locally compact group G, two kinds of C-algebras are made from it, called the full C-crossed product and the reduced C-crossed product. In this paper, we extend the theory of the classical C-crossed product to the C-dynamical system (A, G, $\alpha$) with a left-cancellative semigroup M with unit. We construct a new C-algebra A $\alpha$rM, the reduced crossed product of A by the semigroup M under the action $\alpha$ and investigate some properties of A $\alpha$rM.

• Bulletin of the Korean Mathematical Society
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• v.60 no.2
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• pp.529-539
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• 2023

In this paper, we study nuclearity of semigroup crossed products for quasi-lattice ordered groups. We show the relationships among nuclearity of the semigroup crossed product, amenability of the quasi-lattice ordered group and nuclearity of the underlying C^*-algebra.

• Bulletin of the Korean Mathematical Society
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• v.58 no.5
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• pp.1301-1314
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• 2021

In this paper, firstly, we work on the presentation of the crossed product of groups of general types. After that we find the generating pictures (Second Homotopy Group) of this product by looking the relations from a geometric viewpoint. Finally, we give some applications.

• Journal of the Korean Mathematical Society
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• v.45 no.3
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• pp.597-609
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• 2008

In this paper, we will consider certain amalgamated free product structure in crossed product algebras. Let M be a von Neumann algebra acting on a Hilbert space Hand G, a group and let ${\alpha}$ : G${\rightarrow}$ AutM be an action of G on M, where AutM is the group of all automorphisms on M. Then the crossed product $\mathbb{M}=M{\times}{\alpha}$ G of M and G with respect to ${\alpha}$ is a von Neumann algebra acting on $H{\bigotimes}{\iota}^2(G)$, generated by M and $(u_g)_g{\in}G$, where $u_g$ is the unitary representation of g on ${\iota}^2(G)$. We show that $M{\times}{\alpha}(G_1\;*\;G_2)=(M\;{\times}{\alpha}\;G_1)\;*_M\;(M\;{\times}{\alpha}\;G_2)$. We compute moments and cumulants of operators in $\mathbb{M}$. By doing that, we can verify that there is a close relation between Group Freeness and Amalgamated Freeness under the crossed product. As an application, we can show that if $F_N$ is the free group with N-generators, then the crossed product algebra $L_M(F_n){\equiv}M\;{\times}{\alpha}\;F_n$ satisfies that $$L_M(F_n)=L_M(F_{{\kappa}1})\;*_M\;L_M(F_{{\kappa}2})$$, whenerver $n={\kappa}_1+{\kappa}_2\;for\;n,\;{\kappa}_1,\;{\kappa}_2{\in}\mathbb{N}$.

• Journal of the Korean Mathematical Society
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• v.52 no.4
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• pp.869-889
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• 2015

Let ${\Gamma}^+$ be the positive cone in a totally ordered abelian group ${\Gamma}$, and ${\alpha}$ an action of ${\Gamma}^+$ by extendible endomorphisms of a $C^*$-algebra A. Suppose I is an extendible ${\alpha}$-invariant ideal of A. We prove that the partial-isometric crossed product $\mathcal{I}:=I{\times}^{piso}_{\alpha}{\Gamma}^+$ embeds naturally as an ideal of $A{\times}^{piso}_{\alpha}{\Gamma}^+$, such that the quotient is the partial-isometric crossed product of the quotient algebra. We claim that this ideal $\mathcal{I}$ together with the kernel of a natural homomorphism $\phi:A{\times}^{piso}_{\alpha}{\Gamma}^+{\rightarrow}A{\times}^{iso}_{\alpha}{\Gamma}^+$ gives a composition series of ideals of $A{\times}^{piso}_{\alpha}{\Gamma}^+$ studied by Lindiarni and Raeburn.

• Communications of the Korean Mathematical Society
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• v.17 no.2
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• pp.253-260
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• 2002

For an outer action $\alpha$ of a finite group G on a factor M, it was proved that H is a, normal subgroup of G if and only if there exists a finite group F and an outer action $\beta$ of F on the crossed product algebra M $\times$$_{\alpha}$ G = (M $\times$$_{\alpha}$ F. We generalize this to infinite group actions. For an outer action $\alpha$ of a discrete group, we obtain a Galois correspondence for crossed product algebras related to normal subgroups. When $\alpha$ satisfies a certain condition, we also obtain a Galois correspondence for fixed point algebras. Furthermore, for a minimal action $\alpha$ of a compact group G and a closed normal subgroup H, we prove $M^{G}$ = ( $M^{H}$)$^{{beta}(G/H)}$for a minimal action $\beta$ of G/H on $M^{H}$.f G/H on $M^{H}$.TEX> H/.

• Bulletin of the Korean Mathematical Society
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• v.31 no.2
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• pp.167-172
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• 1994

For a given $C^{*}$-dynamical system (A, G, .alpha.) with a G-simple $C^{*}$-algebra A (that is A has no proper .alpha.-invariant ideal) many authors have studied the simplicity of a $C^{*}$-crossed product A $x_{\alpha{r}}$ G. In [1] topological freeness of an action is shown to guarantee the simplicity of the reduced $C^{*}$-crossed product A $x_{\alpha{r}}$ G when A is G-simple. In this paper we investigate the pure infiniteness of a simple $C^{*}$-crossed product A $x_{\alpha}$ G of a purely infinite simple $C^{*}$-algebra A and a topologically free action .alpha. of a finite group G, and find a sufficient condition in terms of the action on the spectrum of the multiplier algebra M(A) of A. Showing this we also prove that some extension of a topologically free action is still topologically free.

• Journal of the Korean Mathematical Society
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• v.32 no.2
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• pp.183-193
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• 1995

Let N be a $II_1$ factor and G be a finite group acting outerly on N. Then the crossed product algebra $M = N \rtimes G$ is also a $II_1$ factor and $N' \cap M = CI$, i.e. N is irreducible in M. Moreover, N is regular in M, in other words, M is generated by the normalizer $N_M (N)$.

• The Pure and Applied Mathematics
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• v.14 no.4
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• pp.307-315
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• 2007

We study crossed products of the free group and semigroup $C^*-algebras$ by actions of ${\mathbb{R}$, i.e., flows, and estimate and compute their stable rank.

• The Pure and Applied Mathematics
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• v.12 no.1
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• pp.1-15
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• 2005

We study continuous fields and K-groups of crossed products of C＊-algebras. It is shown under a reasonable assumption that there exist continuous fields of C＊ -algebras between crossed products of C＊ -algebras by amenable locally compact groups and tensor products of C＊ -algebras with their group C＊ -algebras, and their K-groups are the same under the additional assumptions.