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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304930&_mathId=si1.gif&_user=111111111&_pii=S0022247X16304930&_rdoc=1&_issn=0022247X&md5=a247e0a9980bc1682b15ce3bc33933e4" title="Click to view the MathML source">C^⁎class="mathContainer hidden">class="mathCode">$C^{⁎}$-algebra is either AF or purely infinite. In this paper, we address the question of whether this is the case for labeled graph class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304930&_mathId=si1.gif&_user=111111111&_pii=S0022247X16304930&_rdoc=1&_issn=0022247X&md5=a247e0a9980bc1682b15ce3bc33933e4" title="Click to view the MathML source">C^⁎class="mathContainer hidden">class="mathCode">$C^{⁎}$-algebras which include all graph class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304930&_mathId=si1.gif&_user=111111111&_pii=S0022247X16304930&_rdoc=1&_issn=0022247X&md5=a247e0a9980bc1682b15ce3bc33933e4" title="Click to view the MathML source">C^⁎class="mathContainer hidden">class="mathCode">$C^{⁎}$-algebras and Matsumoto algebras of subshifts. There have been various class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304930&_mathId=si1.gif&_user=111111111&_pii=S0022247X16304930&_rdoc=1&_issn=0022247X&md5=a247e0a9980bc1682b15ce3bc33933e4" title="Click to view the MathML source">C^⁎class="mathContainer hidden">class="mathCode">$C^{⁎}$-algebra constructions associated with subshifts and some of them are known to have the crossed products class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304930&_mathId=si2.gif&_user=111111111&_pii=S0022247X16304930&_rdoc=1&_issn=0022247X&md5=c97f10e645e60962858537ba602a985f" title="Click to view the MathML source">C(X)×_TZclass="mathContainer hidden">class="mathCode">$C(X){\times }_T\mathbb{Z}$ of Cantor minimal subshifts class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304930&_mathId=si3.gif&_user=111111111&_pii=S0022247X16304930&_rdoc=1&_issn=0022247X&md5=032d0d34d53a676c98b6287192182372" title="Click to view the MathML source">(X,T)class="mathContainer hidden">class="mathCode">$(X,T)$ as their quotient algebras.

We show that such a simple crossed product class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304930&_mathId=si2.gif&_user=111111111&_pii=S0022247X16304930&_rdoc=1&_issn=0022247X&md5=c97f10e645e60962858537ba602a985f" title="Click to view the MathML source">C(X)×_TZclass="mathContainer hidden">class="mathCode">$C(X){\times }_T\mathbb{Z}$ can be realized as a labeled graph class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304930&_mathId=si1.gif&_user=111111111&_pii=S0022247X16304930&_rdoc=1&_issn=0022247X&md5=a247e0a9980bc1682b15ce3bc33933e4" title="Click to view the MathML source">C^⁎class="mathContainer hidden">class="mathCode">$C^{⁎}$-algebra. Since this class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304930&_mathId=si1.gif&_user=111111111&_pii=S0022247X16304930&_rdoc=1&_issn=0022247X&md5=a247e0a9980bc1682b15ce3bc33933e4" title="Click to view the MathML source">C^⁎class="mathContainer hidden">class="mathCode">$C^{⁎}$-algebra is known to be an class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304930&_mathId=si5.gif&_user=111111111&_pii=S0022247X16304930&_rdoc=1&_issn=0022247X&md5=53e66d0fd1722c9c0afe99c63d4aabf1" title="Click to view the MathML source">ATclass="mathContainer hidden">class="mathCode">$A\mathbb{T}$ algebra and has class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304930&_mathId=si6.gif&_user=111111111&_pii=S0022247X16304930&_rdoc=1&_issn=0022247X&md5=09168ef6d11c910ecdd7674464319ac4" title="Click to view the MathML source">Zclass="mathContainer hidden">class="mathCode">$\mathbb{Z}$ as its class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304930&_mathId=si7.gif&_user=111111111&_pii=S0022247X16304930&_rdoc=1&_issn=0022247X&md5=a0c9639afd62171e240f27d2c7af500c" title="Click to view the MathML source">K₁class="mathContainer hidden">class="mathCode">$K_1$-group, our result provides a family of simple finite non-AF unital labeled graph class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304930&_mathId=si1.gif&_user=111111111&_pii=S0022247X16304930&_rdoc=1&_issn=0022247X&md5=a247e0a9980bc1682b15ce3bc33933e4" title="Click to view the MathML source">C^⁎class="mathContainer hidden">class="mathCode">$C^{⁎}$-algebras.