Geodesic
Non-splitting in Kirchberg's Ideal-related KK-Theory
Canadian mathematical bulletin, Tome 54 (2011) no. 1, pp. 68-81

Voir la notice de l'article provenant de la source Cambridge University Press

A. Bonkat obtained a universal coefficient theorem in the setting of Kirchberg's ideal-related $KK$ -theory in the fundamental case of a ${{C}^{*}}$ -algebra with one specified ideal. The universal coefficient sequence was shown to split, unnaturally, under certain conditions. Employing certain $K$ -theoretical information derivable from the given operator algebras using a method introduced here, we shall demonstrate that Bonkat's $\text{UCT}$ does not split in general. Related methods lead to information on the complexity of the $K$ -theory which must be used to classify $*$ -isomorphisms for purely infinite ${{C}^{*}}$ -algebras with one non-trivial ideal.
DOI : 10.4153/CMB-2010-083-7
Mots-clés : 46L35, KK-theory, UCT 
Eilers, Søren; Restorff, Gunnar; Ruiz, Efren. Non-splitting in Kirchberg's Ideal-related KK-Theory. Canadian mathematical bulletin, Tome 54 (2011) no. 1, pp. 68-81. doi: 10.4153/CMB-2010-083-7
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[1] [1] Bonkat, A., Bivariante K-Theorie für Kategorien projektiver Systeme von C*-Algebren. Ph.D. thesis, Westfälische Wilhelms-Universität, 2002. http://wwwmath1.uni-muenster.de/sfb/about/publ/heft319.ps. Google Scholar

[2] [2] Cuntz, J., A class of C*-algebras and topological Markov chains. II. Reducible chains and the Ext-functor for C*-algebras. Invent. Math. 63(1981), no. 1, 25–40. doi=10.1007/BF01389192 Google Scholar

[3] [3] Dădărlat, M. and Eilers, S., Compressing coefficients while preserving ideals in the K-theory for C*-algebras. K-Theory 14(1998), no. 3, 281–304. doi=10.1023/A:1007744626135 Google Scholar

[4] [4] Dădărlat, M. and Loring, T., A universal multicoefficient theorem for the Kasparov groups. Duke Math. J. 84(1996), no. 2, 355–377. doi=10.1215/S0012-7094-96-08412-4 Google Scholar

[5] [5] Eilers, S., Loring, T., and Pedersen, G., Stability of anticommutation relations: an application of noncommutative CW complexes. J. Reine Angew. Math. 499(1998), 101–143. Google Scholar

[6] [6] Eilers, S. and Restorff, G., On Rørdam's classification of certain C*-algebras with one nontrivial ideal. In: Operator algebras: The Abel symposium 2004, Abel Symp., 1, Springer, Berlin, 2006, pp. 87–96. Google Scholar

[7] [7] Higson, N., A characterization of KK-theory. Pacific J. Math. 126(1987), 253–276. Google Scholar

[8] [8] Kasparov, G. G., The operator K-functor and extensions of C*-algebras. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 44(1980), no. 3, 571–636, 719. Google Scholar

[9] [9] Kirchberg, E., Das nicht-kommutative Michael-Auswahlprinzip und die Klassifikation nicht-einfacher Algebren. In: C*-algebras (Münster, 1999), Springer, Berlin, 2000, 92–141. Google Scholar

[10] [10] Kirchberg, E. and Phillips, N. C., Embedding of exact C*-algebras in the Cuntz algebra . J. Reine Angew. Math. 525(2000), 17–53. doi=10.1515/crll.2000.065 Google Scholar

[11] [11] Meyer, R. and Nest, R., C*-Algebras over topological spaces: Filtrated K-theory. arXiv:0810.0096v2[math.OA] Google Scholar

[12] [12] Restorff, G., Classification of Cuntz-Krieger algebras up to stable isomorphism. J. Reine Angew. Math. 598(2006), 185–210. doi=10.1515/CRELLE.2006.074 Google Scholar

[13] [13] Restorff, G., Classification of non-simple C*-algebras. Ph. D. thesis, Department of Mathematical Sciences, University of Copenhagen, 2008. http://www.math.ku.dk/»restorff/papers/thesis.pdf Google Scholar

[14] [14] Restorff, G. and Ruiz, E., On Rørdam's classification of certain C*-algebras with one nontrivial ideal, II. Math. Scand. 101(2007), 280–292. Google Scholar

[15] [15] Rørdam, M., Classification of extensions of certain C*-algebras by their six term exact sequences in K-theory. Math. Ann. 308(1997), no. 1, 93–117. doi=10.1007/s002080050067 Google Scholar

[16] [16] Rosenberg, J. and Schochet, C., The Künneth theorem and the universal coefficient theorem for Kasparov's generalized K-functor. Duke Math. J. 55(1987), no. 2, 431–474. doi=10.1215/S0012-7094-87-05524-4 Google Scholar

[17] [17] Schochet, C., The UCT, the Milnor sequence, and a canonical decomposition of the Kasparov groups. K-Theory 10(1996), no. 1, 49–72. doi=10.1007/BF00534888 Google Scholar

[18] [18] Schochet, C., Correction to: “The UCT, the Milnor sequence, and a canonical decomposition of the Kasparov groups”. K-Theory 14(1998), no. 2, 197–199. doi=10.1023/A:1007736102864 Google Scholar

[19] [19] Schochet, C., The fine structure of the Kasparov groups. II. Topologizing the UCT. J. Funct. Anal. 194(2002), no. 2, 263–287. doi=10.1006/jfan.2002.3949 Google Scholar

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