Semiprojectivity for certain purely infinite $C^*$-algebras
HTML articles powered by AMS MathViewer
- by Jack Spielberg PDF
- Trans. Amer. Math. Soc. 361 (2009), 2805-2830 Request permission
Abstract:
It is proved that classifiable simple separable nuclear purely infinite $C^*$-algebras having finitely generated $K$-theory and torsion-free $K_{1}$ are semiprojective. This is accomplished by exhibiting these algebras as $C^*$-algebras of infinite directed graphs.References
- 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, Semiprojectivity in simple $C^*$-algebras, Operator algebras and applications, Adv. Stud. Pure Math., vol. 38, Math. Soc. Japan, Tokyo, 2004, pp. 1–17. MR 2059799, DOI 10.2969/aspm/03810001
- Lawrence G. Brown, Philip Green, and Marc A. Rieffel, Stable isomorphism and strong Morita equivalence of $C^*$-algebras, Pacific J. Math. 71 (1977), no. 2, 349–363. MR 463928, DOI 10.2140/pjm.1977.71.349
- 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, $K$-theory for certain $C^{\ast }$-algebras, Ann. of Math. (2) 113 (1981), no. 1, 181–197. MR 604046, DOI 10.2307/1971137
- 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
- E. G. Effros and J. Kaminker, Homotopy continuity and shape theory for $C^\ast$-algebras, Geometric methods in operator algebras (Kyoto, 1983) Pitman Res. Notes Math. Ser., vol. 123, Longman Sci. Tech., Harlow, 1986, pp. 152–180. MR 866493
- 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
- James G. Glimm, On a certain class of operator algebras, Trans. Amer. Math. Soc. 95 (1960), 318–340. MR 112057, DOI 10.1090/S0002-9947-1960-0112057-5
- E. Kirchberg, The classification of purely infinite $C^*$-algebras using Kasparov’s theory, (preprint).
- B. Neubüser, Semiprojektivität und realisierunen von rein unendlichen $C^*$-algebren, preprint, Münster, 2000.
- 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
- Iain Raeburn, Graph algebras, CBMS Regional Conference Series in Mathematics, vol. 103, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2005. MR 2135030, DOI 10.1090/cbms/103
- Iain Raeburn and Wojciech Szymański, Cuntz-Krieger algebras of infinite graphs and matrices, Trans. Amer. Math. Soc. 356 (2004), no. 1, 39–59. MR 2020023, DOI 10.1090/S0002-9947-03-03341-5
- Mikael Rørdam, Classification of Cuntz-Krieger algebras, $K$-Theory 9 (1995), no. 1, 31–58. MR 1340839, DOI 10.1007/BF00965458
- Jack Spielberg, A functorial approach to the $C^*$-algebras of a graph, Internat. J. Math. 13 (2002), no. 3, 245–277. MR 1911104, DOI 10.1142/S0129167X02001319
- Jack Spielberg, Non-cyclotomic presentations of modules and prime-order automorphisms of Kirchberg algebras, J. Reine Angew. Math. 613 (2007), 211–230. MR 2377136, DOI 10.1515/CRELLE.2007.098
- Wojciech Szymański, On semiprojectivity of $C^*$-algebras of directed graphs, Proc. Amer. Math. Soc. 130 (2002), no. 5, 1391–1399. MR 1879962, DOI 10.1090/S0002-9939-01-06282-7
- Wojciech Szymański, The range of $K$-invariants for $C^*$-algebras of infinite graphs, Indiana Univ. Math. J. 51 (2002), no. 1, 239–249. MR 1896162, DOI 10.1512/iumj.2002.51.1920
- Shuang Zhang, Certain $C^\ast$-algebras with real rank zero and their corona and multiplier algebras. I, Pacific J. Math. 155 (1992), no. 1, 169–197. MR 1174483, DOI 10.2140/pjm.1992.155.169
Additional Information
- Jack Spielberg
- Affiliation: Department of Mathematics and Statistics, Arizona State University, Tempe, Arizona 85287-1804
- Email: jack.spielberg@asu.edu
- Received by editor(s): February 19, 2001
- Received by editor(s) in revised form: August 26, 2005
- Published electronically: January 26, 2009
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 361 (2009), 2805-2830
- MSC (2000): Primary 46L80; Secondary 46L85, 22A22
- DOI: https://doi.org/10.1090/S0002-9947-09-04928-9
- MathSciNet review: 2485409