AF 𝐶*-algebras from non-AF groupoids

Mitscher, Ian; Spielberg, Jack

doi:10.1090/tran/8723

AF $C^*$-algebras from non-AF groupoids
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by Ian Mitscher and Jack Spielberg PDF

Trans. Amer. Math. Soc. 375 (2022), 7323-7371 Request permission

Abstract:

We construct ample groupoids from certain categories of paths, and prove that their $C^*$-algebras coincide with the continued fraction approximately finite dimensional (AF) algebras of Effros and Shen. The proof relies on recent classification results for simple nuclear $C^*$-algebras. The groupoids are not principal. This provides examples of Cartan subalgebras in the continued fraction AF algebras that are isomorphic, but not conjugate, to the standard diagonal subalgebras.

References

Bruce Blackadar, $K$-theory for operator algebras, 2nd ed., Mathematical Sciences Research Institute Publications, vol. 5, Cambridge University Press, Cambridge, 1998. MR 1656031
Ola Bratteli and Akitaka Kishimoto, Noncommutative spheres. III. Irrational rotations, Comm. Math. Phys. 147 (1992), no. 3, 605–624. MR 1175495
Kenneth R. Davidson, $C^*$-algebras by example, Fields Institute Monographs, vol. 6, American Mathematical Society, Providence, RI, 1996. MR 1402012, DOI 10.1090/fim/006
D. Drinen, Viewing AF-algebras as graph algebras, Proc. Amer. Math. Soc. 128 (2000), no. 7, 1991–2000. MR 1657723, DOI 10.1090/S0002-9939-99-05286-7
Siegfried Echterhoff, Wolfgang Lück, N. Christopher Phillips, and Samuel Walters, The structure of crossed products of irrational rotation algebras by finite subgroups of $\textrm {SL}_2(\Bbb Z)$, J. Reine Angew. Math. 639 (2010), 173–221. MR 2608195, DOI 10.1515/CRELLE.2010.015
Edward G. Effros and Chao Liang Shen, Approximately finite $C^{\ast }$-algebras and continued fractions, Indiana Univ. Math. J. 29 (1980), no. 2, 191–204. MR 563206, DOI 10.1512/iumj.1980.29.29013
D. G. Evans, On higher rank graphs C*-algebras, Ph.D. Thesis, Cardiff University, 2002.
D. Gwion Evans, On the $K$-theory of higher rank graph $C^*$-algebras, New York J. Math. 14 (2008), 1–31. MR 2383584
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, DOI 10.1016/j.jfa.2012.03.024
G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 5th ed., The Clarendon Press, Oxford University Press, New York, 1979. MR 568909
A. Kumjian, An involutive automorphism of the Bunce-Deddens algebra, C. R. Math. Rep. Acad. Sci. Canada 10 (1988), no. 5, 217–218. MR 962104
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, 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
Marcelo Laca, Nadia S. Larsen, and Sergey Neshveyev, On Bost-Connes types systems for number fields, J. Number Theory 129 (2009), no. 2, 325–338. MR 2473881, DOI 10.1016/j.jnt.2008.09.008
Xin Li and Jean Renault, Cartan subalgebras in $\textrm {C}^*$-algebras. Existence and uniqueness, Trans. Amer. Math. Soc. 372 (2019), no. 3, 1985–2010. MR 3976582, DOI 10.1090/tran/7654
Ian Erroll Mitscher, Representing Certain Continued Fraction AF Algebras as C*-Algebras of Categories of Paths and Non-AF Groupoids, ProQuest LLC, Ann Arbor, MI, 2020. Thesis (Ph.D.)–Arizona State University. MR 4106401
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
Sergey Neshveyev, KMS states on the $C^\ast$-algebras of non-principal groupoids, J. Operator Theory 70 (2013), no. 2, 513–530. MR 3138368, DOI 10.7900/jot.2011sep20.1915
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, 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
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
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, The ideal structure of groupoid crossed product $C^\ast$-algebras, J. Operator Theory 25 (1991), no. 1, 3–36. With an appendix by Georges Skandalis. MR 1191252
Jean Renault, Cartan subalgebras in $C^*$-algebras, Irish Math. Soc. Bull. 61 (2008), 29–63. MR 2460017, DOI 10.33232/BIMS.0061.29.63
M. Rørdam, F. Larsen, and N. Laustsen, An introduction to $K$-theory for $C^*$-algebras, London Mathematical Society Student Texts, vol. 49, Cambridge University Press, Cambridge, 2000. MR 1783408
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
Jack Spielberg, Groupoids and $C^*$-algebras for left cancellative small categories, Indiana Univ. Math. J. 69 (2020), no. 5, 1579–1626. MR 4151331, DOI 10.1512/iumj.2020.69.7969
Şerban Strătilă and Dan Voiculescu, Representations of AF-algebras and of the group $U(\infty )$, Lecture Notes in Mathematics, Vol. 486, Springer-Verlag, Berlin-New York, 1975. MR 0458188, DOI 10.1007/BFb0082276
Aaron Tikuisis, Stuart White, and Wilhelm Winter, Quasidiagonality of nuclear $C^\ast$-algebras, Ann. of Math. (2) 185 (2017), no. 1, 229–284. MR 3583354, DOI 10.4007/annals.2017.185.1.4
Jean-Louis Tu, La conjecture de Baum-Connes pour les feuilletages moyennables, $K$-Theory 17 (1999), no. 3, 215–264 (French, with English and French summaries). MR 1703305, DOI 10.1023/A:1007744304422
Jason Tyler, Every AF-algebra is Morita equivalent to a graph algebra, Bull. Austral. Math. Soc. 69 (2004), no. 2, 237–240. MR 2051359, DOI 10.1017/S0004972700035978
H. S. Wall, Analytic Theory of Continued Fractions, D. Van Nostrand Co., Inc., New York, N. Y., 1948. MR 0025596
Dana P. Williams, A tool kit for groupoid $C^*$-algebras, Mathematical Surveys and Monographs, vol. 241, American Mathematical Society, Providence, RI, 2019. MR 3969970, DOI 10.1016/j.physletb.2019.06.021
Wilhelm Winter and Joachim Zacharias, The nuclear dimension of $C^\ast$-algebras, Adv. Math. 224 (2010), no. 2, 461–498. MR 2609012, DOI 10.1016/j.aim.2009.12.005