Uniqueness theorems for topological higher-rank graph 𝐶*-algebras

Renault, Jean; Sims, Aidan; Williams, Dana; Yeend, Trent

doi:10.1090/proc/13745

Uniqueness theorems for topological higher-rank graph $C^*$-algebras
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by Jean Renault, Aidan Sims, Dana P. Williams and Trent Yeend PDF

Proc. Amer. Math. Soc. 146 (2018), 669-684 Request permission

Abstract:

We consider the boundary-path groupoids of topological higher-rank graphs. We show that all such groupoids are topologically amenable. We deduce that the $C^*$-algebras of topological higher-rank graphs are nuclear and prove versions of the gauge-invariant uniqueness theorem and the Cuntz–Krieger uniqueness theorem. We then provide a necessary and sufficient condition for simplicity of a topological higher-rank graph $C^*$-algebra, and a condition under which it is also purely infinite.

References

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, DOI 10.24033/bsmf.2304
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
Jonathan Brown, Lisa Orloff Clark, Cynthia Farthing, and Aidan Sims, Simplicity of algebras associated to étale groupoids, Semigroup Forum 88 (2014), no. 2, 433–452. MR 3189105, DOI 10.1007/s00233-013-9546-z
Toke M. Carlsen, Nadia S. Larsen, Aidan Sims, and Sean T. Vittadello, Co-universal algebras associated to product systems, and gauge-invariant uniqueness theorems, Proc. Lond. Math. Soc. (3) 103 (2011), no. 4, 563–600. MR 2837016, DOI 10.1112/plms/pdq028
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
Masatoshi Enomoto and Yasuo Watatani, A graph theory for $C^{\ast }$-algebras, Math. Japon. 25 (1980), no. 4, 435–442. MR 594544
Ruy Exel, Inverse semigroups and combinatorial $C^\ast$-algebras, Bull. Braz. Math. Soc. (N.S.) 39 (2008), no. 2, 191–313. MR 2419901, DOI 10.1007/s00574-008-0080-7
Cynthia Farthing, Paul S. Muhly, and Trent Yeend, Higher-rank graph $C^*$-algebras: an inverse semigroup and groupoid approach, Semigroup Forum 71 (2005), no. 2, 159–187. MR 2184052, DOI 10.1007/s00233-005-0512-2
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
Takeshi Katsura, A class of $C^\ast$-algebras generalizing both graph algebras and homeomorphism $C^\ast$-algebras. I. Fundamental results, Trans. Amer. Math. Soc. 356 (2004), no. 11, 4287–4322. MR 2067120, DOI 10.1090/S0002-9947-04-03636-0
Alex Kumjian and David Pask, Higher rank graph $C^\ast$-algebras, New York J. Math. 6 (2000), 1–20. MR 1745529
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
Alberto E. Marrero and Paul S. Muhly, Groupoid and inverse semigroup presentations of ultragraph $C^*$-algebras, Semigroup Forum 77 (2008), no. 3, 399–422. MR 2457327, DOI 10.1007/s00233-008-9046-8
Alan L. T. Paterson, Groupoids, inverse semigroups, and their operator algebras, Progress in Mathematics, vol. 170, Birkhäuser Boston, Inc., Boston, MA, 1999. MR 1724106, DOI 10.1007/978-1-4612-1774-9
Alan L. T. Paterson, Graph inverse semigroups, groupoids and their $C^\ast$-algebras, J. Operator Theory 48 (2002), no. 3, suppl., 645–662. MR 1962477
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, DOI 10.1016/j.jfa.2003.10.014
Arlan Ramsay, The Mackey-Glimm dichotomy for foliations and other Polish groupoids, J. Funct. Anal. 94 (1990), no. 2, 358–374. MR 1081649, DOI 10.1016/0022-1236(90)90018-G
Jean Renault, A groupoid approach to $C^{\ast }$-algebras, Lecture Notes in Mathematics, vol. 793, Springer, Berlin, 1980. MR 584266, DOI 10.1007/BFb0091072
Jean Renault, Représentation des produits croisés d’algèbres de groupoïdes, J. Operator Theory 18 (1987), no. 1, 67–97 (French). MR 912813
Jean N. Renault and Dana P. Williams, Amenability of groupoids arising from partial semigroup actions and topological higher rank graphs, Trans. Amer. Math. Soc. 369 (2017), no. 4, 2255–2283. MR 3592511, DOI 10.1090/tran/6736
David I. Robertson and Aidan Sims, Simplicity of $C^\ast$-algebras associated to higher-rank graphs, Bull. Lond. Math. Soc. 39 (2007), no. 2, 337–344. MR 2323468, DOI 10.1112/blms/bdm006
Aidan Sims, Benjamin Whitehead, and Michael F. Whittaker, Twisted $C^*$-algebras associated to finitely aligned higher-rank graphs, Doc. Math. 19 (2014), 831–866. MR 3262073
Jack Spielberg, Groupoids and $C^*$-algebras for categories of paths, Trans. Amer. Math. Soc. 366 (2014), no. 11, 5771–5819. MR 3256184, DOI 10.1090/S0002-9947-2014-06008-X
Mark Tomforde, A unified approach to Exel-Laca algebras and $C^\ast$-algebras associated to graphs, J. Operator Theory 50 (2003), no. 2, 345–368. MR 2050134
Sarah E. Wright, Aperiodicity conditions in topological $k$-graphs, J. Operator Theory 72 (2014), no. 1, 3–14. MR 3246978, DOI 10.7900/jot.2012aug20.196
S. Yamashita, Cuntz–Krieger type uniqueness theorem for topological higher-rank graph $C^*$-algebras, preprint 2009 (arXiv:0911.2978v1 [math.OA]).
T. Yeend, Topological higher-rank graphs, their groupoids, and operator algebras, Ph.D. thesis, University of Newcastle, 2005.
Trent Yeend, Topological higher-rank graphs and the $C^*$-algebras of topological 1-graphs, Operator theory, operator algebras, and applications, Contemp. Math., vol. 414, Amer. Math. Soc., Providence, RI, 2006, pp. 231–244. MR 2277214, DOI 10.1090/conm/414/07812
Trent Yeend, Groupoid models for the $C^*$-algebras of topological higher-rank graphs, J. Operator Theory 57 (2007), no. 1, 95–120. MR 2301938