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Classification of extensions of classifiable $$C^{*}$$-algebras. (English) Zbl 1207.46055
Authors’ abstract: For a certain class of extensions $${\mathfrak e}: 0\rightarrow B \rightarrow E \rightarrow A \rightarrow 0$$ of $$C^*$$-algebras in which $$B$$ and $$A$$ belong to classifiable classes of $$C^*$$-algebras, we show that the functor which sends $${\mathfrak e}$$ to its associated six term exact sequence in $$K$$-theory and the positive cones of $$K_0 (B)$$ and $$K_0 (A)$$ is a classification functor. We give two independent applications addressing the classification of a class of $$C^*$$-algebras arising from substitutional shift spaces on one hand and of graph algebras on the other. The former application leads to the answer of a question of Carlsen and the first author concerning the completeness of stabilized Matsumoto algebras as an invariant of flow equivalence. The latter leads to the first classification result for nonsimple graph $$C^*$$-algebras.

##### MSC:
 46L35 Classifications of $$C^*$$-algebras 19K14 $$K_0$$ as an ordered group, traces 19K33 Ext and $$K$$-homology 19K35 Kasparov theory ($$KK$$-theory) 37B10 Symbolic dynamics 46M18 Homological methods in functional analysis (exact sequences, right inverses, lifting, etc.)
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##### References:
 [1] Blackadar, B., K-theory for operator algebras, Math. sci. res. inst. publ., vol. 5, (1998), Cambridge University Press Cambridge · Zbl 0913.46054 [2] Brown, L.G., Stable isomorphism of hereditary subalgebras of $$C^\ast$$-algebras, Pacific J. math., 71, 335-348, (1977) · Zbl 0362.46042 [3] Brown, L.G., Semicontinuity and multipliers of $$C^\ast$$-algebras, Canad. J. math., 40, 865-988, (1988) · Zbl 0647.46044 [4] T.M. Carlsen, Symbolic dynamics, partial dynamical systems, Boolean algebras and $$C^*$$-algebras generated by partial isometries, Preprintreihe Sonderforshungsbereich 478, Westfälische Wilhelms-Universität Münster, Heft 438 [5] Carlsen, T.M.; Eilers, S., Java applet, (2002) [6] Carlsen, T.M.; Eilers, S., Augmenting dimension group invariants for substitution dynamics, Ergodic theory dynam. systems, 24, 1015-1039, (2004) · Zbl 1060.37014 [7] Carlsen, T.M.; Eilers, S., Matsumoto K-groups associated to certain shift spaces, Doc. math., 9, 639-671, (2004) · Zbl 1062.37004 [8] Carlsen, T.M.; Eilers, S., Ordered K-groups associated to substitutional dynamics, J. funct. anal., 38, 99-117, (2006) · Zbl 1105.46046 [9] Carlsen, T.M.; Eilers, S., A graph approach to computing nondeterminacy in substitutional dynamical systems, RAIRO - theor. inform. appl., 41, 285-306, (2007) · Zbl 1165.37006 [10] Cuntz, J., K-theory for certain C*-algebras, Ann. of math. (2), 113, 181-197, (1981) · Zbl 0437.46060 [11] Dadarlat, M., Morphisms of simple tracially AF algebras, Internat. J. math., 15, 919-957, (2004) · Zbl 1071.46036 [12] Dadarlat, M.; Eilers, S., The bockstein map is necessary, Canad. math. bull., 42, 274-284, (1999) · Zbl 0946.46047 [13] Eilers, S.; Restorff, G., On Rørdam’s classification of certain $$C^\ast$$-algebras with one nontrivial ideal, (), 87-96 · Zbl 1118.46053 [14] S. Eilers, M. Tomforde, On the classification of nonsimple graph $$C^\ast$$-algebras, Math. Ann. (2009), doi:10.1007/s00208-009-0403-z, in press · Zbl 1209.46035 [15] Eilers, S.; Loring, T.A.; Pedersen, G.K., Morphisms of extensions of $$C^\ast$$-algebras: pushing forward the busby invariant, Adv. math., 147, 74-109, (1999) · Zbl 1014.46030 [16] Elliott, G.A., On the classification of inductive limits of sequences of semisimple finite-dimensional algebras, J. algebra, 38, 29-44, (1976) · Zbl 0323.46063 [17] G.A. Elliott, Towards a theory of classification, Adv. Math. (2009), doi:10.1016/j.aim.2009.07.018, in press [18] Elliott, G.A.; Kucerovsky, D., An abstract voiculescu-Brown-Douglas-fillmore absorption theorem, Pacific J. math., 198, 385-409, (2001) · Zbl 1058.46041 [19] Elliott, G.A.; Rørdam, M., Classification of certain infinite simple $$C^\ast$$-algebras. II, Comment. math. helv., 70, 615-638, (1995) · Zbl 0864.46038 [20] E. Kirchberg, The classification of purely infinite simple $$C^*$$-algebras using Kasparov’s theory, 1994, 3rd draft [21] Kirchberg, E.; Phillips, N.C., Embedding of exact $$C^\ast$$-algebras in the Cuntz algebra $$\mathcal{O}_2$$, J. reine angew. math., 525, 17-53, (2000) · Zbl 0973.46048 [22] Kishimoto, A.; Kumjian, A., The ext class of an approximately inner automorphism, Trans. amer. math. soc., 350, 4127-4148, (1998) · Zbl 0902.46047 [23] Kucerovsky, D.; Ng, P.W., The corona factorization property and approximate unitary equivalence, Houston J. math., 32, 531-550, (2006) · Zbl 1111.46050 [24] Lin, H., On the classification of $$C^\ast$$-algebras of real rank zero with zero $$K_1$$, J. operator theory, 35, 147-178, (1996) · Zbl 0849.46043 [25] Lin, H., A classification theorem for infinite Toeplitz algebras, (), 219-275 · Zbl 0946.46048 [26] Lin, H., A separable Brown-Douglas-fillmore theorem and weak stability, Trans. amer. math. soc., 356, 2889-2925, (2004) · Zbl 1053.46032 [27] Lin, H., Classification of simple $$C^*$$-algebras of tracial topological rank zero, Duke math. J., 125, 91-119, (2004) · Zbl 1068.46032 [28] Lin, H., Simple corona $$C^\ast$$-algebras, Proc. amer. math. soc., 132, 3215-3224, (2004) · Zbl 1049.46040 [29] Lin, H., Full extensions and approximate unitary equivalence, Pacific J. math., 229, 389-428, (2007) · Zbl 1152.46049 [30] Lin, H.; Nui, Z., Lifting KK-elements, asymptotical unitary equivalence and classification of simple C∗-algebras, Adv. math., 219, 1729-1769, (2008) · Zbl 1162.46033 [31] Lin, H.; Su, H., Classification of direct limits of generalized Toeplitz algebras, Pacific J. math., 181, 89-140, (1997) · Zbl 0905.46043 [32] Ng, P.W., The corona factorization property, (), 97-110 · Zbl 1110.46039 [33] Parry, B.; Sullivan, D., A topological invariant of flows on 1-dimensional spaces, Topology, 14, 297-299, (1975) · Zbl 0314.54045 [34] Pedersen, G.K., Pullback and pushout constructions in $$C^\ast$$-algebra theory, J. funct. anal., 167, 243-344, (1999) · Zbl 0944.46063 [35] Phillips, N.C., A classification theorem for nuclear purely infinite simple $$C^\ast$$-algebras, Doc. math., 5, 49-114, (2000) · Zbl 0943.46037 [36] Putnam, I.F., On the topological stable rank of certain transformation group $$C^\ast$$-algebras, Ergodic theory dynam. systems, 10, 197-207, (1990) · Zbl 0667.46045 [37] Raeburn, I., Graph algebras, CBMS reg. conf. ser. math., vol. 103, (2005), Conference Board of the Mathematical Sciences Washington, DC · Zbl 1079.46002 [38] Restorff, G.; Ruiz, E., On Rørdam’s classification of certain $$C^\ast$$-algebras with one nontrivial ideal II, Math. scand., 101, 280-292, (2007) · Zbl 1161.46036 [39] Rørdam, M., Classification of certain infinite simple $$C^\ast$$-algebras, J. funct. anal., 131, 415-458, (1995) · Zbl 0831.46063 [40] Rørdam, M., Classification of extensions of certain $$C^\ast$$-algebras by their six term exact sequences in K-theory, Math. ann., 308, 93-117, (1997) · Zbl 0874.46039 [41] Rørdam, M.; Larsen, F.; Laustsen, N., An introduction to K-theory for C∗-algebras, London math. soc. stud. texts, vol. 49, (2000), Cambridge University Press Cambridge · Zbl 0967.19001 [42] Rosenberg, J.; Schochet, C., The Künneth theorem and the universal coefficient theorem for Kasparov’s generalized K-functor, Duke math. J., 55, 431-474, (1987) · Zbl 0644.46051 [43] Ruiz, E., A classification theorem for direct limits of extensions of circle algebras by purely infinite $$C^*$$-algebras, J. operator theory, 58, 311-349, (2007) · Zbl 1150.46025
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