Finite simple labeled graph C^⁎-algebras of Cantor minimal subshifts

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• 作者：Ja A Jeong^a ; ¹ ; ^{jajeong@snu.ac.kr" class="auth_mail" title="E-mail the corresponding author} ; Eun Ji Kang^a ; ¹ ; ^{kkang33@snu.ac.kr" class="auth_mail" title="E-mail the corresponding author} ; Sun Ho Kim^b ; ² ; ^{sunho.kim.math@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Gi Hyun Park^c ; ³ ; ^{ghpark@hs.ac.kr" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Labeled graph C⁎C⁎-algebra ; Finite C⁎C⁎-algebra ; Cantor minimal system
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 February 2017
• 年：2017
• 卷：446
• 期：1
• 页码：395-410
• 全文大小：452 K

文摘

It is well known that a simple graph C^⁎$C^{⁎}$-algebra is either AF or purely infinite. In this paper, we address the question of whether this is the case for labeled graph C^⁎$C^{⁎}$-algebras which include all graph C^⁎$C^{⁎}$-algebras and Matsumoto algebras of subshifts. There have been various C^⁎$C^{⁎}$-algebra constructions associated with subshifts and some of them are known to have the crossed products C(X)×_TZ$C(X){\times }_T\mathbb{Z}$ of Cantor minimal subshifts (X,T)$(X,T)$ as their quotient algebras.

We show that such a simple crossed product C(X)×_TZ$C(X){\times }_T\mathbb{Z}$ can be realized as a labeled graph C^⁎$C^{⁎}$-algebra. Since this C^⁎$C^{⁎}$-algebra is known to be an AT$A\mathbb{T}$ algebra and has ac4" title="Click to view the MathML source">Z$\mathbb{Z}$ as its K₁$K_1$-group, our result provides a family of simple finite non-AF unital labeled graph C^⁎$C^{⁎}$-algebras.