Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Cuntz-Krieger algebras of infinite graphs and matrices


Authors: Iain Raeburn and Wojciech Szymanski
Journal: Trans. Amer. Math. Soc. 356 (2004), 39-59
MSC (2000): Primary 46L05
DOI: https://doi.org/10.1090/S0002-9947-03-03341-5
Published electronically: August 21, 2003
MathSciNet review: 2020023
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that the Cuntz-Krieger algebras of infinite graphs and infinite $\{0,1\}$-matrices can be approximated by those of finite graphs. We then use these approximations to deduce the main uniqueness theorems for Cuntz-Krieger algebras and to compute their $K$-theory. Since the finite approximating graphs have sinks, we have to calculate the $K$-theory of Cuntz-Krieger algebras of graphs with sinks, and the direct methods we use to do this should be of independent interest.


References [Enhancements On Off] (What's this?)

  • 1. T. Bates, D. Pask, I. Raeburn and W. Szymanski, The $C^*$-algebras of row-finite graphs, New York J. Math. 6 (2000), 307-324. MR 2001k:46084
  • 2. B. Blackadar, Shape theory for $C^*$-algebras, Math. Scand. 56 (1985), 249-275. MR 87b:46074
  • 3. B. Blackadar, $K$-Theory for Operator Algebras, second edition, MSRI Publ., vol. 5, Cambridge Univ. Press, 1998. MR 99g:46104
  • 4. M.-D. Choi and E. G. Effros, Separable nuclear $C^*$-algebras and injectivity, Duke Math. J. 43 (1976), 309-32. MR 53:8912
  • 5. A. Connes, Classification of injective factors, Ann. of Math. 104 (1976), 73-115. MR 56:12908
  • 6. A. Connes, An analogue of the Thom isomorphism for crossed products of a $C^*$-algebra by an action of $\mathbb{R} $, Adv. Math. 39 (1981), 31-55. MR 82j:46084
  • 7. J. Cuntz, Simple $C^*$-algebras generated by isometries, Comm. Math. Phys. 57 (1977), 173-185. MR 57:7189
  • 8. J. Cuntz, A class of $C^*$-algebras and topological Markov chains II: Reducible chains and the Ext-functor for $C^*$-algebras, Invent. Math. 63 (1981), 25-40. MR 82f:46073b
  • 9. J. Cuntz and W. Krieger, A class of $C^*$-algebras and topological Markov chains, Invent. Math. 56 (1980), 251-268. MR 82f:46073a
  • 10. R. Exel and M. Laca, Cuntz-Krieger algebras for infinite matrices, J. Reine Angew. Math. 512 (1999), 119-172. MR 2000i:46064
  • 11. R. Exel and M. Laca, The $K$-theory of Cuntz-Krieger algebras for infinite matrices, $K$-Theory 19 (2000), 251-268. MR 2001c:46123
  • 12. R. Exel, M. Laca and J. Quigg, Partial dynamical systems and $C^*$-algebras generated by partial isometries, J. Operator Theory 47 (2002), no. 1, 169-186. MR 2003f:46108
  • 13. N. J. Fowler, M. Laca and I. Raeburn, The $C^*$-algebras of infinite graphs, Proc. Amer. Math. Soc. 128 (2000), 2319-2327. MR 2000k:46079
  • 14. J. v. B. Hjelmborg, Purely infinite and stable $C^*$-algebras of graphs and dynamical systems, Ergod. Th. & Dynam. Sys. 21 (2001), 1789-1808. MR 2002h:46112
  • 15. A. an Huef and I. Raeburn, The ideal structure of Cuntz-Krieger algebras, Ergod. Th. & Dynam. Sys. 17 (1997), 611-624. MR 98k:46098
  • 16. J. A. Jeong, G. H. Park and D. Y. Shin, Stable rank and real rank of graph $C^*$-algebras, Pacific J. Math. 200 (2001), no. 2, 231-343. MR 2002j:46064
  • 17. E. Kirchberg, Exact $C^*$-algebras, tensor products, and the classification of purely infinite algebras, Proc. Internat. Congress of Math. (Zürich, 1994), vol. 2, Birkhäuser, Basel, 1995, pages 943-954. MR 97g:46074
  • 18. A. Kumjian, Notes on $C^*$-algebras of graphs, in Contemp. Math., vol. 228, Amer. Math. Soc., Providence, 1998, pages 189-200. MR 99m:46137
  • 19. A. Kumjian and D. Pask, $C^*$-algebras of directed graphs and group actions, Ergod. Th. & Dynam. Sys. 19 (1999), no. 6, 1503-1519. MR 2000m:46125
  • 20. A. Kumjian, D. Pask and I. Raeburn, Cuntz-Krieger algebras of directed graphs, Pacific J. Math. 184 (1998), 161-174. MR 99i:46049
  • 21. A. Kumjian, D. Pask, I. Raeburn and J. Renault, Graphs, groupoids, and Cuntz-Krieger algebras, J. Funct. Anal. 144 (1997), 505-541. MR 98g:46083
  • 22. M. H. Mann, I. Raeburn and C. E. Sutherland, Representations of finite groups and Cuntz-Krieger algebras, Bull. Austral. Math. Soc. 46 (1992), 225-243. MR 93k:46046
  • 23. K. Matsumoto, $K$-theory for $C^*$-algebras associated with subshifts, Math. Scand. 82 (1998), 237-255. MR 2000e:46087
  • 24. D. Pask, Cuntz-Krieger algebras associated to directed graphs, in Operator Algebras and Quantum Field Theory (S. Doplicher, R. Longo, J. E. Roberts and L. Zsido, eds.), International Press, 1997, pages 85-92. MR 98m:46072
  • 25. D. Pask and I. Raeburn, On the $K$-theory of Cuntz-Krieger algebras, Publ. RIMS, Kyoto Univ. 32 (1996), 415-443. MR 97m:46111
  • 26. N. C. Phillips, A classification theorem for nuclear purely infinite simple $C^*$-algebras, Doc. Math. 5 (2000), 49-114. MR 2001d:46086b
  • 27. C. Pinzari, The ideal structure of Cuntz-Krieger algebras and Cuntz-Krieger algebras over infinite matrices, in Operator Algebras and Quantum Field Theory (S. Doplicher, R. Longo, J. E. Roberts and L. Zsido, eds.), International Press, 1997, pages 136-150. MR 98m:46074
  • 28. M. A. Rieffel, Dimension and stable rank in the $K$-theory of $C^*$-algberas, Proc. London Math. Soc. 46 (1983), 301-333. MR 84g:46085
  • 29. J. Rosenberg and C. Schochet, The Künneth theorem and the universal coefficient theorem for Kasparov's generalized $K$-functor, Duke Math. J. 55 (1987), 431-474. MR 88i:46091
  • 30. W. Szymanski, Bimodules for Cuntz-Krieger algebras of infinite matrices, Bull. Austral. Math. Soc. 62 (2000), no. 1, 87-94. MR 2001g:46151
  • 31. W. Szymanski and S. Zhang, Infinite simple $C^*$-algebras and reduced cross products of abelian $C^*$-algebras and free groups, Manuscripta Math. 92 (1997), 487-514. MR 98a:46073
  • 32. W. Szymanski and S. Zhang, $K$-theory of certain $C^*$-algebras associated with free products of cyclic groups, J. Operator Theory 45 (2001), no. 2, 251-264. MR 2002e:46088
  • 33. Y. Watatani, Graph theory for $C^*$-algebras, in Operator algebras and their applications (R. V. Kadison, ed.), Proc. Sympos. Pure Math., vol. 38 Part I, Amer. Math. Soc., Providence, 1982, pages 195-197. MR 84a:46124

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 46L05

Retrieve articles in all journals with MSC (2000): 46L05


Additional Information

Iain Raeburn
Affiliation: Department of Mathematics, University of Newcastle, New South Wales 2308, Australia
Email: iain@frey.newcastle.edu.au

Wojciech Szymanski
Affiliation: Department of Mathematics, University of Newcastle, New South Wales 2308, Australia
Email: wojciech@frey.newcastle.edu.au

DOI: https://doi.org/10.1090/S0002-9947-03-03341-5
Received by editor(s): December 15, 1999
Published electronically: August 21, 2003
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society