Classification of real rank zero, purely infinite C^⁎-algebras with at most four primitive ideals

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• 作者：Sara E. Arklint^a ; ^{arklint@math.ku.dk" class="auth_mail" title="E-mail the corresponding author} ; Gunnar Restorff^b ; ^{gunnarr@setur.fo" class="auth_mail" title="E-mail the corresponding author} ; Efren Ruiz^c ; ^{ruize@hawaii.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：46L35 ; 46L80
• 刊名：Journal of Functional Analysis
• 出版年：2016
• 出版时间：1 October 2016
• 年：2016
• 卷：271
• 期：7
• 页码：1921-1947
• 全文大小：529 K

文摘

Counterexamples to classification of purely infinite, nuclear, separable C^⁎$C^{⁎}$-algebras (in the ideal-related bootstrap class) and with primitive ideal space X using ideal-related K-theory occur for infinitely many finite primitive ideal spaces X  , the smallest of which being six spaces with four points. All constructed counterexamples for spaces with at least five points are based on the counterexamples constructed for these six four-point spaces. With real rank zero added to the assumptions imposed on the C^⁎$C^{⁎}$-algebras, ideal-related K-theory is known to be strongly complete for four of these six spaces. In this article, we close the two remaining cases: the pseudo-circle and the diamond space. We show that ideal-related K-theory is strongly complete for real rank zero, purely infinite, nuclear, separable C^⁎$C^{⁎}$-algebras that have the pseudo-circle as primitive ideal space. In the opposite direction, we construct a Cuntz–Krieger algebra with the diamond space as its primitive ideal space for which an automorphism on ideal-related K-theory does not lift.