Cuntz Splice invariance for purely infinite graph algebras
DOI:
https://doi.org/10.7146/math.scand.a-96633Abstract
We show that the Cuntz Splice preserves the stable isomorphism class of a purely infinite graph $\mathrm{C}^*$-algebra with finitely many ideals.
References
Bates, T., Hong, J. H., Raeburn, I., and Szymański, W., The ideal structure of the $C^*$-algebras of infinite graphs, Illinois J. Math. 46 (2002), no. 4, 1159–1176.
Bates, T. and Pask, D., Flow equivalence of graph algebras, Ergodic Theory Dynam. Systems 24 (2004), no. 2, 367–382. https://doi.org/10.1017/S0143385703000348
Bentmann, R., Kirchberg $X$-algebras with real rank zero and intermediate cancellation, J. Noncommut. Geom. 8 (2014), no. 4, 1061–1081. https://doi.org/10.4171/JNCG/178
Bentmann, R. and Meyer, R., A more general method to classify up to equivariant KK-equivalence, preprint arXiv:1405.6512 [math.OA], 2014.
Bowen, R. and Franks, J., Homology for zero-dimensional nonwandering sets, Ann. of Math. (2) 106 (1977), no. 1, 73–92. https://doi.org/10.2307/1971159
Crisp, T. and Gow, D., Contractible subgraphs and Morita equivalence of graph $C^*$-algebras, Proc. Amer. Math. Soc. 134 (2006), no. 7, 2003–2013. https://doi.org/10.1090/S0002-9939-06-08216-5
Cuntz, J., 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. https://doi.org/10.1007/BF01389192
Cuntz, J. and Krieger, W., A class of $C^ast $-algebras and topological Markov chains, Invent. Math. 56 (1980), no. 3, 251–268. https://doi.org/10.1007/BF01390048
Drinen, D. and Tomforde, M., The $C^*$-algebras of arbitrary graphs, Rocky Mountain J. Math. 35 (2005), no. 1, 105–135. https://doi.org/10.1216/rmjm/1181069770
Eilers, S., Restorff, G., and Ruiz, E., Classification of graph $C^*$-algebras with no more than four primitive ideals, in “Operator algebra and dynamics”, Springer Proc. Math. Stat., vol. 58, Springer, Heidelberg, 2013, pp. 89--129. https://doi.org/10.1007/978-3-642-39459-1_5
Eilers, S., Restorff, G., Ruiz, E., and Sørensen, A. P. W., Geometric classification of unital graph $C^*$-algebras of real rank zero, preprint arXiv:1505.0677 [math.OA], 2015.
Hong, J. H. and Szymański, W., Purely infinite Cuntz-Krieger algebras of directed graphs, Bull. London Math. Soc. 35 (2003), no. 5, 689–696. https://doi.org/10.1112/S0024609303002364
Kirchberg, E., Das nicht-kommutative Michael-Auswahlprinzip und die Klassifikation nicht-einfacher Algebren, in “$C^*$-algebras (Münster, 1999)'', Springer, Berlin, 2000, pp. 92--141.
Mac Lane, S., Homology, Classics in Mathematics, Springer-Verlag, Berlin, 1995.
Meyer, R. and Nest, R., $C^*$-algebras over topological spaces: the bootstrap class, Münster J. Math. 2 (2009), 215–252.
Raeburn, I., Graph algebras, CBMS Regional Conference Series in Mathematics, vol. 103, American Mathematical Society, Providence, RI, 2005. https://doi.org/10.1090/cbms/103
Raeburn, I. and Szymański, W., Cuntz-Krieger algebras of infinite graphs and matrices, Trans. Amer. Math. Soc. 356 (2004), no. 1, 39–59. https://doi.org/10.1090/S0002-9947-03-03341-5
Restorff, G., Classification of Cuntz-Krieger algebras up to stable isomorphism, J. Reine Angew. Math. 598 (2006), 185–210. https://doi.org/10.1515/CRELLE.2006.074
Rørdam, M., Classification of Cuntz-Krieger algebras, $K$-Theory 9 (1995), no. 1, 31–58. https://doi.org/10.1007/BF00965458
Sørensen, A. P. W., Geometric classification of simple graph algebras, Ergodic Theory Dynam. Systems 33 (2013), no. 4, 1199–1220. https://doi.org/10.1017/S0143385712000260
