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Groupoids and $ C^*$-algebras for categories of paths


Author: Jack Spielberg
Journal: Trans. Amer. Math. Soc. 366 (2014), 5771-5819
MSC (2010): Primary 46L05; Secondary 20L05
DOI: https://doi.org/10.1090/S0002-9947-2014-06008-X
Published electronically: June 3, 2014
MathSciNet review: 3256184
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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we describe a new method of defining $ C^*$-algebras from oriented combinatorial data, thereby generalizing the construction of algebras from directed graphs, higher-rank graphs, and ordered groups. We show that only the most elementary notions of concatenation and cancellation of paths are required to define versions of Cuntz-Krieger and Toeplitz-Cuntz-Krieger algebras, and the presentation by generators and relations follows naturally. We give sufficient conditions for the existence of an AF core, hence of the nuclearity of the $ C^*$-algebras, and for aperiodicity, which is used to prove the standard uniqueness theorems.


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  • [1] Claire Anantharaman-Delaroche, Purely infinite $ C^*$-algebras arising from dynamical systems, Bull. Soc. Math. France 125 (1997), no. 2, 199-225 (English, with English and French summaries). MR 1478030 (99i:46051)
  • [2] C. Anantharaman-Delaroche and J. Renault, Amenable groupoids, Monographies de L'Enseignement Mathématique [Monographs of L'Enseignement Mathématique], vol. 36, L'Enseignement Mathématique, Geneva, 2000. With a foreword by Georges Skandalis and Appendix B by E. Germain. MR 1799683 (2001m:22005)
  • [3] R. J. Archbold and J. S. Spielberg, Topologically free actions and ideals in discrete $ C^*$-dynamical systems, Proc. Edinburgh Math. Soc. (2) 37 (1994), no. 1, 119-124. MR 1258035 (94m:46101), https://doi.org/10.1017/S0013091500018733
  • [4] Victor Arzumanian and Jean Renault, Examples of pseudogroups and their $ C^*$-algebras, Operator algebras and quantum field theory (Rome, 1996) Int. Press, Cambridge, MA, 1997, pp. 93-104. MR 1491110 (99a:46101)
  • [5] Ola Bratteli, Inductive limits of finite dimensional $ C^{\ast } $-algebras, Trans. Amer. Math. Soc. 171 (1972), 195-234. MR 0312282 (47 #844)
  • [6] N. Brownlowe, A. Sims and S. Vittadello, Co-universal $ C^*$-algebras associated to generalised graphs, Israel J. Math. 193 (2013), no. 1, 399-440. MR 3038557
  • [7] Ian Chiswell, Introduction to $ \Lambda $-trees, World Scientific Publishing Co. Inc., River Edge, NJ, 2001. MR 1851337 (2003e:20029)
  • [8] John Crisp and Marcelo Laca, Boundary quotients and ideals of Toeplitz $ C^*$-algebras of Artin groups, J. Funct. Anal. 242 (2007), no. 1, 127-156. MR 2274018 (2007k:46117), https://doi.org/10.1016/j.jfa.2006.08.001
  • [9] Joachim Cuntz, Simple $ C^*$-algebras generated by isometries, Comm. Math. Phys. 57 (1977), no. 2, 173-185. MR 0467330 (57 #7189)
  • [10] 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 (82f:46073a), https://doi.org/10.1007/BF01390048
  • [11] Menassie Ephrem, $ C^*$-algebra of the $ \mathbb{Z}^n$-tree, New York J. Math. 17 (2011), 1-20. MR 2781905 (2012h:46087)
  • [12] D. Gwion Evans and Aidan Sims, When is the Cuntz-Krieger algebra of a higher-rank graph approximately finite-dimensional?, J. Funct. Anal. 263 (2012), no. 1, 183-215. MR 2920846, https://doi.org/10.1016/j.jfa.2012.03.024
  • [13] R. Exel, Tight representations of semilattices and inverse semigroups, Semigroup Forum 79 (2009), no. 1, 159-182. MR 2534230 (2011b:20163), https://doi.org/10.1007/s00233-009-9165-x
  • [14] R. Exel, Semigroupoid $ C^\ast $-algebras, J. Math. Anal. Appl. 377 (2011), no. 1, 303-318. MR 2754831 (2012a:46094), https://doi.org/10.1016/j.jmaa.2010.10.061
  • [15] R. Exel, Non-Hausdorff étale groupoids, Proc. Amer. Math. Soc. 139 (2011), no. 3, 897-907. MR 2745642 (2012b:46148), https://doi.org/10.1090/S0002-9939-2010-10477-X
  • [16] Takeshi Katsura, A class of $ C^*$-algebras generalizing both graph algebras and homeomorphism $ C^*$-algebras. IV. Pure infiniteness, J. Funct. Anal. 254 (2008), no. 5, 1161-1187. MR 2386934 (2008m:46143), https://doi.org/10.1016/j.jfa.2007.11.014
  • [17] Alex Kumjian and David Pask, Higher rank graph $ C^\ast $-algebras, New York J. Math. 6 (2000), 1-20. MR 1745529 (2001b:46102)
  • [18] Alex Kumjian, David Pask, and Aidan Sims, Generalised morphisms of $ k$-graphs: $ k$-morphs, Trans. Amer. Math. Soc. 363 (2011), no. 5, 2599-2626. MR 2763728 (2012a:46099), https://doi.org/10.1090/S0002-9947-2010-05152-9
  • [19] Marcelo Laca and Iain Raeburn, Semigroup crossed products and the Toeplitz algebras of nonabelian groups, J. Funct. Anal. 139 (1996), no. 2, 415-440. MR 1402771 (97h:46109), https://doi.org/10.1006/jfan.1996.0091
  • [20] Marcelo Laca and Jack Spielberg, Purely infinite $ C^*$-algebras from boundary actions of discrete groups, J. Reine Angew. Math. 480 (1996), 125-139. MR 1420560 (98a:46085)
  • [21] Peter Lewin and Aidan Sims, Aperiodicity and cofinality for finitely aligned higher-rank graphs, Math. Proc. Cambridge Philos. Soc. 149 (2010), no. 2, 333-350. MR 2670219 (2012b:46106), https://doi.org/10.1017/S0305004110000034
  • [22] Paul S. Muhly, Jean N. Renault, and Dana P. Williams, Equivalence and isomorphism for groupoid $ C^\ast $-algebras, J. Operator Theory 17 (1987), no. 1, 3-22. MR 873460 (88h:46123)
  • [23] A. Nica, $ C^*$-algebras generated by isometries and Wiener-Hopf operators, J. Operator Theory 27 (1992), no. 1, 17-52. MR 1241114 (94m:46094)
  • [24] David Pask, Iain Raeburn, Mikael Rørdam, and Aidan Sims, Rank-two graphs whose $ C^*$-algebras are direct limits of circle algebras, J. Funct. Anal. 239 (2006), no. 1, 137-178. MR 2258220 (2007e:46053), https://doi.org/10.1016/j.jfa.2006.04.003
  • [25] Alan L. T. Paterson, Groupoids, inverse semigroups, and their operator algebras, Progress in Mathematics, vol. 170, Birkhäuser Boston Inc., Boston, MA, 1999. MR 1724106 (2001a:22003)
  • [26] John C. Quigg, Discrete $ C^*$-coactions and $ C^*$-algebraic bundles, J. Austral. Math. Soc. Ser. A 60 (1996), no. 2, 204-221. MR 1375586 (97c:46086)
  • [27] Iain Raeburn, Graph algebras, CBMS Regional Conference Series in Mathematics, vol. 103, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 2005. MR 2135030 (2005k:46141)
  • [28] Iain Raeburn, Aidan Sims, and Trent Yeend, The $ C^*$-algebras of finitely aligned higher-rank graphs, J. Funct. Anal. 213 (2004), no. 1, 206-240. MR 2069786 (2005e:46103), https://doi.org/10.1016/j.jfa.2003.10.014
  • [29] Jean Renault, A groupoid approach to $ C^{\ast } $-algebras, Lecture Notes in Mathematics, vol. 793, Springer, Berlin, 1980. MR 584266 (82h:46075)
  • [30] Aidan Sims, Gauge-invariant ideals in the $ C^*$-algebras of finitely aligned higher-rank graphs, Canad. J. Math. 58 (2006), no. 6, 1268-1290. MR 2270926 (2007j:46095), https://doi.org/10.4153/CJM-2006-045-2
  • [31] Jack Spielberg, A functorial approach to the $ C^*$-algebras of a graph, Internat. J. Math. 13 (2002), no. 3, 245-277. MR 1911104 (2004e:46084), https://doi.org/10.1142/S0129167X02001319
  • [32] Jack Spielberg, Graph-based models for Kirchberg algebras, J. Operator Theory 57 (2007), no. 2, 347-374. MR 2329002 (2008f:46073)
  • [33] J. Spielberg, $ C^*$-algebras for categories of paths associated to the Baumslag-Solitar groups, J. London Math. Soc. (2) 86 (2012), no. 3, 728-754, doi: 10.1112/jlms/jds025. MR 3000828
  • [34] Stephen Willard, General topology, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1970. MR 0264581 (41 #9173)

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Additional Information

Jack Spielberg
Affiliation: School of Mathematical and Statistical Sciences, Arizona State University, P.O. Box 871804, Tempe, Arizona 85287-1804
Email: jack.spielberg@asu.edu

DOI: https://doi.org/10.1090/S0002-9947-2014-06008-X
Keywords: Cuntz-Krieger algebra, Toeplitz Cuntz-Krieger algebra, groupoid, aperiodicity
Received by editor(s): January 12, 2012
Received by editor(s) in revised form: February 23, 2012, and October 23, 2012
Published electronically: June 3, 2014
Dedicated: Dedicated to the memory of Bill Arveson
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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