Cuntz Splice invariance for purely infinite graph algebras


  • Rasmus Bentmann



We show that the Cuntz Splice preserves the stable isomorphism class of a purely infinite graph $\mathrm{C}^*$-algebra with finitely many ideals.


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.

Bentmann, R., Kirchberg $X$-algebras with real rank zero and intermediate cancellation, J. Noncommut. Geom. 8 (2014), no. 4, 1061–1081.

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.

Crisp, T. and Gow, D., Contractible subgraphs and Morita equivalence of graph $C^*$-algebras, Proc. Amer. Math. Soc. 134 (2006), no. 7, 2003–2013.

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.

Cuntz, J. and Krieger, W., A class of $C^ast $-algebras and topological Markov chains, Invent. Math. 56 (1980), no. 3, 251–268.

Drinen, D. and Tomforde, M., The $C^*$-algebras of arbitrary graphs, Rocky Mountain J. Math. 35 (2005), no. 1, 105–135.

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.

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.

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.

Raeburn, I. and Szymański, W., Cuntz-Krieger algebras of infinite graphs and matrices, Trans. Amer. Math. Soc. 356 (2004), no. 1, 39–59.

Restorff, G., Classification of Cuntz-Krieger algebras up to stable isomorphism, J. Reine Angew. Math. 598 (2006), 185–210.

Rørdam, M., Classification of Cuntz-Krieger algebras, $K$-Theory 9 (1995), no. 1, 31–58.

Sørensen, A. P. W., Geometric classification of simple graph algebras, Ergodic Theory Dynam. Systems 33 (2013), no. 4, 1199–1220.




How to Cite

Bentmann, R. (2018). Cuntz Splice invariance for purely infinite graph algebras. MATHEMATICA SCANDINAVICA, 122(1), 91–106.