Strong classification of purely infinite Cuntz-Krieger algebras

Carlsen, Toke; Restorff, Gunnar; Ruiz, Efren

doi:10.1090/btran/14

Strong classification of purely infinite Cuntz-Krieger algebras
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by Toke Meier Carlsen, Gunnar Restorff and Efren Ruiz HTML | PDF

Trans. Amer. Math. Soc. Ser. B 4 (2017), 1-30

Abstract:

In 2006, Restorff completed the classification of all Cuntz-Krieger algebras with finitely many ideals (i.e., those that are purely infinite) up to stable isomorphism. He left open the questions concerning strong classification up to stable isomorphism and unital classification. In this paper, we address both questions. We show that any isomorphism between the reduced filtered $K$-theory of two Cuntz-Krieger algebras with finitely many ideals lifts to a $^*$-isomorphism between the stabilized Cuntz-Krieger algebras. As a result, we also obtain strong unital classification.

References

Sara E. Arklint, Rasmus Bentmann, and Takeshi Katsura, The K-theoretical range of Cuntz-Krieger algebras, J. Funct. Anal. 266 (2014), no. 8, 5448–5466. MR 3177344, DOI 10.1016/j.jfa.2014.01.020
Sara E. Arklint, Rasmus Bentmann, and Takeshi Katsura, Reduction of filtered K-theory and a characterization of Cuntz-Krieger algebras, J. K-Theory 14 (2014), no. 3, 570–613. MR 3349327, DOI 10.1017/is014009013jkt281
P. Ara, M. A. Moreno, and E. Pardo, Nonstable $K$-theory for graph algebras, Algebr. Represent. Theory 10 (2007), no. 2, 157–178. MR 2310414, DOI 10.1007/s10468-006-9044-z
Sara Arklint, Gunnar Restorff, and Efren Ruiz, Filtrated K-theory for real rank zero $C^*$-algebras, Internat. J. Math. 23 (2012), no. 8, 1250078, 19. MR 2949216, DOI 10.1142/S0129167X12500784
Sara E. Arklint, Gunnar Restorff, and Efren Ruiz, Classification of real rank zero, purely infinite $C^*$-algebras with at most four primitive ideals, J. Funct. Anal. 271 (2016), no. 7, 1921–1947. MR 3535322, DOI 10.1016/j.jfa.2016.05.014
Mike Boyle and Danrun Huang, Poset block equivalence of integral matrices, Trans. Amer. Math. Soc. 355 (2003), no. 10, 3861–3886. MR 1990568, DOI 10.1090/S0002-9947-03-02947-7
Teresa Bates, Jeong Hee Hong, Iain Raeburn, and Wojciech Szymański, The ideal structure of the $C^*$-algebras of infinite graphs, Illinois J. Math. 46 (2002), no. 4, 1159–1176. MR 1988256
Rasmus Bentmann and Manuel Köhler, Universal coefficient theorems for $C^*$-algebras over finite topological spaces, arXiv e-prints (2011), arXiv:1101.5702v3.
Alexander Bonkat, Bivariante $K$-Theorie für Kategorien projektiver Systeme von $C^*$-Algebren, Ph.D. thesis, Westfälischen Wilhelms-Universität Münster, Jan 2004, Published in Preprintreihe SFB 478 Geometrische Strukturen in der Mathematik, Heft 319.
Mike Boyle, Flow equivalence of shifts of finite type via positive factorizations, Pacific J. Math. 204 (2002), no. 2, 273–317. MR 1907894, DOI 10.2140/pjm.2002.204.273
Lawrence G. Brown, Stable isomorphism of hereditary subalgebras of $C^*$-algebras, Pacific J. Math. 71 (1977), no. 2, 335–348. MR 454645, DOI 10.2140/pjm.1977.71.335
Toke Meier Carlsen, Søren Eilers, and Mark Tomforde, Index maps in the $K$-theory of graph algebras, J. K-Theory 9 (2012), no. 2, 385–406. MR 2922394, DOI 10.1017/is011004017jkt156
J. Cuntz, A class of $C^{\ast }$-algebras and topological Markov chains. II. Reducible chains and the Ext-functor for $C^{\ast }$-algebras, Invent. Math. 63 (1981), no. 1, 25–40. MR 608527, DOI 10.1007/BF01389192
Søren Eilers and Gunnar Restorff, On Rørdam’s classification of certain $C^*$-algebras with one non-trivial ideal, Operator Algebras: The Abel Symposium 2004, Abel Symp., vol. 1, Springer, Berlin, 2006, pp. 87–96. MR 2265044, DOI 10.1007/978-3-540-34197-0_{4}
Søren Eilers, Gunnar Restorff, and Efren Ruiz, Classification of extensions of classifiable $C^\ast$-algebras, Adv. Math. 222 (2009), no. 6, 2153–2172. MR 2562779, DOI 10.1016/j.aim.2009.07.014
Søren Eilers, Gunnar Restorff, and Efren Ruiz, Classification of graph $C^*$-algebras with no more than four primitive ideals, Operator algebra and dynamics, Springer Proc. Math. Stat., vol. 58, Springer, Heidelberg, 2013, pp. 89–129. MR 3142033, DOI 10.1007/978-3-642-39459-1_{5}
Søren Eilers, Gunnar Restorff, and Efren Ruiz, Classifying $C^*$-algebras with both finite and infinite subquotients, J. Funct. Anal. 265 (2013), no. 3, 449–468. MR 3056712, DOI 10.1016/j.jfa.2013.05.006
Søren Eilers, Gunnar Restorff, and Efren Ruiz, Strong classification of extensions of classifiable $C^*$-algebras, arXiv e-prints (2013), arXiv:1301.7695v1.
Søren Eilers, Gunnar Restorff, Efren Ruiz, and Adam P. W. Sørensen, The complete classification of unital graph $C^*$-algebras: Geometric and strong, arXiv e-prints (2016), arXiv:1611.07120v1.
Søren Eilers, Gunnar Restorff, Efren Ruiz, and Adam P. W. Sørensen, Geometric classification of graph $C^*$-algebras over finite graphs, arXiv e-prints (2016), arXiv:1604.05439v1.
Søren Eilers, Gunnar Restorff, Efren Ruiz, and Adam P. W. Sørensen, Invariance of the Cuntz splice, arXiv e-prints (2016), arXiv:1602.03709v2.
Søren Eilers and Mark Tomforde, On the classification of nonsimple graph $C^*$-algebras, Math. Ann. 346 (2010), no. 2, 393–418. MR 2563693, DOI 10.1007/s00208-009-0403-z
John Franks, Flow equivalence of subshifts of finite type, Ergodic Theory Dynam. Systems 4 (1984), no. 1, 53–66. MR 758893, DOI 10.1017/S0143385700002261
Danrung Huang, Flow equivalence of reducible shifts of finite type, Ergodic Theory Dynam. Systems 14 (1994), no. 4, 695–720. MR 1304139, DOI 10.1017/S0143385700008129
Danrun Huang, Flow equivalence of reducible shifts of finite type and Cuntz-Krieger algebras, J. Reine Angew. Math. 462 (1995), 185–217. MR 1329907, DOI 10.1515/crll.1995.462.185
Danrun Huang, The classification of two-component Cuntz-Krieger algebras, Proc. Amer. Math. Soc. 124 (1996), no. 2, 505–512. MR 1301504, DOI 10.1090/S0002-9939-96-03079-1
Eberhard Kirchberg, Das nicht-kommutative Michael-Auswahlprinzip und die Klassifikation nicht-einfacher Algebren, $C^*$-algebras (Münster, 1999) Springer, Berlin, 2000, pp. 92–141 (German, with English summary). MR 1796912
Ralf Meyer and Ryszard Nest, $\textrm {C}^*$-algebras over topological spaces: filtrated K-theory, Canad. J. Math. 64 (2012), no. 2, 368–408. MR 2953205, DOI 10.4153/CJM-2011-061-x
N. Christopher Phillips, A classification theorem for nuclear purely infinite simple $C^*$-algebras, Doc. Math. 5 (2000), 49–114. MR 1745197
Gunnar Restorff, Classification of Cuntz-Krieger algebras up to stable isomorphism, J. Reine Angew. Math. 598 (2006), 185–210. MR 2270572, DOI 10.1515/CRELLE.2006.074
Gunnar Restorff, Classification of non-simple $C^*$-algebras, Ph.D. thesis, University of Copenhagen, 2008, URL: http://www.math.ku.dk/noter/filer/phd08gr.pdf.
Mikael Rørdam, Classification of Cuntz-Krieger algebras, $K$-Theory 9 (1995), no. 1, 31–58. MR 1340839, DOI 10.1007/BF00965458
Mikael Rørdam, Classification of extensions of certain $C^*$-algebras by their six term exact sequences in $K$-theory, Math. Ann. 308 (1997), no. 1, 93–117. MR 1446202, DOI 10.1007/s002080050067
Gunnar Restorff and Efren Ruiz, On Rørdam’s classification of certain $C^*$-algebras with one non-trivial ideal. II, Math. Scand. 101 (2007), no. 2, 280–292. MR 2379290, DOI 10.7146/math.scand.a-15045
Wojciech Szymański, General Cuntz-Krieger uniqueness theorem, Internat. J. Math. 13 (2002), no. 5, 549–555. MR 1914564, DOI 10.1142/S0129167X0200137X