• Title/Summary/Keyword: crossed products

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STABLE RANK OF TWISTED CROSSED PRODUCTS OF $C^{*}-ALGEBRAS$ BY ABELIAN GROUPS

  • Sudo, Takahiro
    • The Pure and Applied Mathematics
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    • v.10 no.2
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    • pp.103-118
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    • 2003
  • We estimate the stable rank of twisted crossed products of $C^{*}-algebras$ by topological Abelian groups. As an application we estimate the stable rank of twisted crossed products of $C^{*}-algebras$ by solvable Lie groups. In particular, we obtain the stable rank estimate of twisted group $C^{*}-algebras$ of solvable Lie groups by the (reduced) dimension and (generalized) rank of groups.

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K-THEORY OF CROSSED PRODUCTS OF C*-ALGEBRAS

  • SUDO TAKAHIRO
    • 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.

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ON GENERALIZED GRADED CROSSED PRODUCTS AND KUMMER SUBFIELDS OF SIMPLE ALGEBRAS

  • Bennis, Driss;Mounirh, Karim;Taraza, Fouad
    • Bulletin of the Korean Mathematical Society
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    • v.56 no.4
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    • pp.939-959
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    • 2019
  • Using generalized graded crossed products, we give necessary and sufficient conditions for a simple algebra over a Henselian valued field (under some hypotheses) to have Kummer subfields. This study generalizes some known works. We also study many properties of generalized graded crossed products and conditions for embedding a graded simple algebra into a matrix algebra of a graded division ring.

REDUCED CROSSED PRODUCTS BY SEMIGROUPS OF AUTOMORPHISMS

  • Jang, Sun-Young
    • 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.

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A characterization of crossed products without cohomology

  • Hong, Jeong-Hee
    • 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 $N_M (N)$.

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Topologically free actions and purely infinite $C^{*}$-crossed products

  • Jeong, Ja-A
    • 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.

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GROUP-FREENESS AND CERTAIN AMALGAMATED FREENESS

  • Cho, Il-Woo
    • 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 $n={\kappa}_1+{\kappa}_2\;for\;n,\;{\kappa}_1,\;{\kappa}_2{\in}\mathbb{N}$.