Cuntz-Krieger algebras of infinite graphs and matrices

Raeburn, Iain; Szymański, Wojciech

doi:10.1090/S0002-9947-03-03341-5

Cuntz-Krieger algebras of infinite graphs and matrices
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by Iain Raeburn and Wojciech Szymański

Trans. Amer. Math. Soc. 356 (2004), 39-59

DOI: https://doi.org/10.1090/S0002-9947-03-03341-5

Published electronically: August 21, 2003

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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

Teresa Bates, David Pask, Iain Raeburn, and Wojciech Szymański, The $C^*$-algebras of row-finite graphs, New York J. Math. 6 (2000), 307–324. MR 1777234
Bruce Blackadar, Shape theory for $C^\ast$-algebras, Math. Scand. 56 (1985), no. 2, 249–275. MR 813640, DOI 10.7146/math.scand.a-12100
Bruce Blackadar, $K$-theory for operator algebras, 2nd ed., Mathematical Sciences Research Institute Publications, vol. 5, Cambridge University Press, Cambridge, 1998. MR 1656031
Man Duen Choi and Edward G. Effros, Separable nuclear $C^*$-algebras and injectivity, Duke Math. J. 43 (1976), no. 2, 309–322. MR 405117
A. Connes, Classification of injective factors. Cases $II_{1},$ $II_{\infty },$ $III_{\lambda },$ $\lambda \not =1$, Ann. of Math. (2) 104 (1976), no. 1, 73–115. MR 454659, DOI 10.2307/1971057
A. Connes, An analogue of the Thom isomorphism for crossed products of a $C^{\ast }$-algebra by an action of $\textbf {R}$, Adv. in Math. 39 (1981), no. 1, 31–55. MR 605351, DOI 10.1016/0001-8708(81)90056-6
Joachim Cuntz, Simple $C^*$-algebras generated by isometries, Comm. Math. Phys. 57 (1977), no. 2, 173–185. MR 467330, DOI 10.1007/BF01625776
Joachim Cuntz and Wolfgang Krieger, A class of $C^{\ast }$-algebras and topological Markov chains, Invent. Math. 56 (1980), no. 3, 251–268. MR 561974, DOI 10.1007/BF01390048
Joachim Cuntz and Wolfgang Krieger, A class of $C^{\ast }$-algebras and topological Markov chains, Invent. Math. 56 (1980), no. 3, 251–268. MR 561974, DOI 10.1007/BF01390048
Ruy Exel and Marcelo Laca, Cuntz-Krieger algebras for infinite matrices, J. Reine Angew. Math. 512 (1999), 119–172. MR 1703078, DOI 10.1515/crll.1999.051
Ruy Exel and Marcelo Laca, The $K$-theory of Cuntz-Krieger algebras for infinite matrices, $K$-Theory 19 (2000), no. 3, 251–268. MR 1756260, DOI 10.1023/A:1007815705271
Ruy Exel, Marcelo Laca, and John Quigg, Partial dynamical systems and $C^*$-algebras generated by partial isometries, J. Operator Theory 47 (2002), no. 1, 169–186. MR 1905819
Neal J. Fowler, Marcelo Laca, and Iain Raeburn, The $C^*$-algebras of infinite graphs, Proc. Amer. Math. Soc. 128 (2000), no. 8, 2319–2327. MR 1670363, DOI 10.1090/S0002-9939-99-05378-2
Jacob v. B. Hjelmborg, Purely infinite and stable $C^*$-algebras of graphs and dynamical systems, Ergodic Theory Dynam. Systems 21 (2001), no. 6, 1789–1808. MR 1869070, DOI 10.1017/S0143385701001857
Astrid an Huef and Iain Raeburn, The ideal structure of Cuntz-Krieger algebras, Ergodic Theory Dynam. Systems 17 (1997), no. 3, 611–624. MR 1452183, DOI 10.1017/S0143385797079200
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, 331–343. MR 1868695, DOI 10.2140/pjm.2001.200.331
Eberhard Kirchberg, Exact $\textrm {C}^*$-algebras, tensor products, and the classification of purely infinite algebras, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 943–954. MR 1403994
A. Kumjian, Notes on $C^\ast$-algebras of graphs, Operator algebras and operator theory (Shanghai, 1997) Contemp. Math., vol. 228, Amer. Math. Soc., Providence, RI, 1998, pp. 189–200. MR 1667662, DOI 10.1090/conm/228/03289
Alex Kumjian and David Pask, $C^*$-algebras of directed graphs and group actions, Ergodic Theory Dynam. Systems 19 (1999), no. 6, 1503–1519. MR 1738948, DOI 10.1017/S0143385799151940
Alex Kumjian, David Pask, and Iain Raeburn, Cuntz-Krieger algebras of directed graphs, Pacific J. Math. 184 (1998), no. 1, 161–174. MR 1626528, DOI 10.2140/pjm.1998.184.161
Alex Kumjian, David Pask, Iain Raeburn, and Jean Renault, Graphs, groupoids, and Cuntz-Krieger algebras, J. Funct. Anal. 144 (1997), no. 2, 505–541. MR 1432596, DOI 10.1006/jfan.1996.3001
M. H. Mann, Iain Raeburn, and C. E. Sutherland, Representations of finite groups and Cuntz-Krieger algebras, Bull. Austral. Math. Soc. 46 (1992), no. 2, 225–243. MR 1183780, DOI 10.1017/S0004972700011862
Kengo Matsumoto, $K$-theory for $C^*$-algebras associated with subshifts, Math. Scand. 82 (1998), no. 2, 237–255. MR 1646513, DOI 10.7146/math.scand.a-13835
David Pask, Cuntz-Krieger algebras associated to directed graphs, Operator algebras and quantum field theory (Rome, 1996) Int. Press, Cambridge, MA, 1997, pp. 85–92. MR 1491109, DOI 10.1006/jfan.1996.3001
David Pask and Iain Raeburn, On the $K$-theory of Cuntz-Krieger algebras, Publ. Res. Inst. Math. Sci. 32 (1996), no. 3, 415–443. MR 1409796, DOI 10.2977/prims/1195162850
Eberhard Kirchberg and N. Christopher Phillips, Embedding of exact $C^*$-algebras in the Cuntz algebra $\scr O_2$, J. Reine Angew. Math. 525 (2000), 17–53. MR 1780426, DOI 10.1515/crll.2000.065
Claudia Pinzari, The ideal structure of Cuntz-Krieger-Pimsner algebras and Cuntz-Krieger algebras over infinite matrices, Operator algebras and quantum field theory (Rome, 1996) Int. Press, Cambridge, MA, 1997, pp. 136–150. MR 1491114, DOI 10.1063/1.118673
Marc A. Rieffel, Dimension and stable rank in the $K$-theory of $C^{\ast }$-algebras, Proc. London Math. Soc. (3) 46 (1983), no. 2, 301–333. MR 693043, DOI 10.1112/plms/s3-46.2.301
Jonathan Rosenberg and Claude Schochet, The Künneth theorem and the universal coefficient theorem for Kasparov’s generalized $K$-functor, Duke Math. J. 55 (1987), no. 2, 431–474. MR 894590, DOI 10.1215/S0012-7094-87-05524-4
Wojciech Szymański, Bimodules for Cuntz-Krieger algebras of infinite matrices, Bull. Austral. Math. Soc. 62 (2000), no. 1, 87–94. MR 1775890, DOI 10.1017/S0004972700018505
Wojciech Szymański and Shuang Zhang, Infinite simple $C^*$-algebras and reduced cross products of abelian $C^*$-algebras and free groups, Manuscripta Math. 92 (1997), no. 4, 487–514. MR 1441490, DOI 10.1007/BF02678208
Wojciech Szymański and Shuang Zhang, $K$-theory of certain $C^*$-algebras associated with free products of cyclic groups, J. Operator Theory 45 (2001), no. 2, 251–264. MR 1841096
Yasuo Watatani, Graph theory for $C^{\ast }$-algebras, Operator algebras and applications, Part 1 (Kingston, Ont., 1980) Proc. Sympos. Pure Math., vol. 38, Amer. Math. Soc., Providence, R.I., 1982, pp. 195–197. MR 679705