A twist over a minimal étale groupoid that is topologically nontrivial over the interior of the isotropy
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- by Becky Armstrong, Abraham C. S. Ng, Aidan Sims and Yumiao Zhou;
- Proc. Amer. Math. Soc. 153 (2025), 1849-1866
- DOI: https://doi.org/10.1090/proc/17159
- Published electronically: February 27, 2025
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Abstract:
We present an example of a twist over a minimal Hausdorff étale groupoid such that the restriction of the twist to the interior of the isotropy is not topologically trivial; that is, the restricted twist is not induced by a continuous $2$-cocycle.References
- Becky Armstrong, A uniqueness theorem for twisted groupoid C*-algebras, J. Funct. Anal. 283 (2022), no. 6, Paper No. 109551, 33. MR 4433049, DOI 10.1016/j.jfa.2022.109551
- Becky Armstrong, Jonathan H. Brown, Lisa Orloff Clark, Kristin Courtney, Ying-Fen Lin, Kathryn McCormick, and Jacqui Ramagge, The local bisection hypothesis for twisted groupoid $\rm C^*$-algebras, Semigroup Forum 107 (2023), no. 3, 609–623. MR 4683164, DOI 10.1007/s00233-023-10392-9
- Becky Armstrong, Nathan Brownlowe, and Aidan Sims, Simplicity of twisted $C^*$-algebras of Deaconu-Renault groupoids, J. Noncommut. Geom. 18 (2024), no. 1, 265–312. MR 4705670, DOI 10.4171/jncg/527
- Are Austad and Eduard Ortega, $C^*$-uniqueness results for groupoids, Int. Math. Res. Not. IMRN 4 (2022), 3057–3073. MR 4381939, DOI 10.1093/imrn/rnaa225
- Selçuk Barlak and Sven Raum, Cartan subalgebras in dimension drop algebras, J. Inst. Math. Jussieu 20 (2021), no. 3, 725–755. MR 4260640, DOI 10.1017/S147474801900032X
- T. Bice, L.O. Clark, Y.-F. Lin, and K. McCormick, Cartan semigroups and twisted groupoid C*-algebras, Preprint, arXiv:2407.05024, 2024.
- Christian Bönicke, K-theory and homotopies of twists on ample groupoids, J. Noncommut. Geom. 15 (2021), no. 1, 195–222. MR 4248211, DOI 10.4171/jncg/399
- Nicolas Bourbaki, General topology. Chapters 1–4, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1989. Translated from the French; Reprint of the 1966 edition. MR 979294, DOI 10.1007/978-3-642-61703-4
- Jonathan H. Brown, Ruy Exel, Adam H. Fuller, David R. Pitts, and Sarah A. Reznikoff, Intermediate $C^*$-algebras of Cartan embeddings, Proc. Amer. Math. Soc. Ser. B 8 (2021), 27–41. MR 4199728, DOI 10.1090/bproc/66
- Jonathan H. Brown, Adam H. Fuller, David R. Pitts, and Sarah A. Reznikof, Graded $C^\ast$-algebras and twisted groupoid $C^\ast$-algebras, New York J. Math. 27 (2021), 205–252. MR 4209533
- Anna Duwenig, Elizabeth Gillaspy, and Rachael Norton, Analyzing the Weyl construction for dynamical Cartan subalgebras, Int. Math. Res. Not. IMRN 20 (2022), 15721–15755. MR 4498163, DOI 10.1093/imrn/rnab114
- A. Duwenig, E. Gillaspy, R. Norton, S. Reznikoff, and S. Wright, Cartan subalgebras for non-principal twisted groupoid $C^*$-algebras, J. Funct. Anal. 279 (2020), no. 6, 108611, 40. MR 4096726, DOI 10.1016/j.jfa.2020.108611
- Jacob Feldman and Calvin C. Moore, Ergodic equivalence relations, cohomology, and von Neumann algebras, Bull. Amer. Math. Soc. 81 (1975), no. 5, 921–924. MR 425075, DOI 10.1090/S0002-9904-1975-13888-2
- Jacob Feldman and Calvin C. Moore, Ergodic equivalence relations, cohomology, and von Neumann algebras. I, Trans. Amer. Math. Soc. 234 (1977), no. 2, 289–324. MR 578656, DOI 10.1090/S0002-9947-1977-0578656-4
- Jacob Feldman and Calvin C. Moore, Ergodic equivalence relations, cohomology, and von Neumann algebras. II, Trans. Amer. Math. Soc. 234 (1977), no. 2, 325–359. MR 578730, DOI 10.1090/S0002-9947-1977-0578730-2
- Alexander Kumjian, On $C^\ast$-diagonals, Canad. J. Math. 38 (1986), no. 4, 969–1008. MR 854149, DOI 10.4153/CJM-1986-048-0
- Xin Li, Every classifiable simple $\rm C^*$-algebra has a Cartan subalgebra, Invent. Math. 219 (2020), no. 2, 653–699. MR 4054809, DOI 10.1007/s00222-019-00914-0
- 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
- Shigeyuki Morita, Geometry of differential forms, Translations of Mathematical Monographs, vol. 201, American Mathematical Society, Providence, RI, 2001. Translated from the two-volume Japanese original (1997, 1998) by Teruko Nagase and Katsumi Nomizu; Iwanami Series in Modern Mathematics. MR 1851352, DOI 10.1090/mmono/201
- Paul S. Muhly and Dana P. Williams, Continuous trace groupoid $C^*$-algebras. II, Math. Scand. 70 (1992), no. 1, 127–145. MR 1174207, DOI 10.7146/math.scand.a-12390
- Ali I. Raad, A generalization of Renault’s theorem for Cartan subalgebras, Proc. Amer. Math. Soc. 150 (2022), no. 11, 4801–4809. MR 4489313, DOI 10.1090/proc/16003
- 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
- N. Seaton, Equivariant-sheaf cohomology and the generalised-twist group of a locally compact, Hausdorff groupoid, In preparation.
- N. Seaton, The Brauer group for Fell algebras, In preparation.
- Aidan Sims, Gábor Szabó, and Dana Williams, Operator algebras and dynamics: groupoids, crossed products, and Rokhlin dimension, Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser/Springer, Cham, [2020] ©2020. Lecture notes from the Advanced Course held at Centre de Recerca Matemàtica (CRM) Barcelona, March 13–17, 2017; Edited by Francesc Perera. MR 4321941, DOI 10.1007/978-3-030-39713-5
Bibliographic Information
- Becky Armstrong
- Affiliation: School of Mathematics and Statistics, Victoria University of Wellington, Wellington 6012, New Zealand
- MR Author ID: 1278315
- ORCID: 0000-0002-2432-7003
- Email: becky.armstrong@vuw.ac.nz
- Abraham C. S. Ng
- Affiliation: School of Mathematics and Statistics, The University of Sydney, New South Wales 2006, Australia
- MR Author ID: 1341810
- ORCID: 0000-0002-1701-7904
- Email: abraham.ng@sydney.edu.au
- Aidan Sims
- Affiliation: School of Mathematics and Applied Statistics, University of Wollongong, New South Wales 2522, Australia
- MR Author ID: 671497
- ORCID: 0000-0002-1965-6451
- Email: asims@uow.edu.au
- Yumiao Zhou
- Affiliation: School of Mathematics and Applied Statistics, University of Wollongong, New South Wales 2522, Australia
- ORCID: 0009-0008-7652-5328
- Email: ymchou1989@outlook.com
- Received by editor(s): May 19, 2024
- Received by editor(s) in revised form: September 5, 2024
- Published electronically: February 27, 2025
- Additional Notes: This research was funded by a University of Wollongong AEGiS Connect Grant; the Australian Research Council grants DP180100595 and DP200100155; the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC 2044 – 390685587, Mathematics Münster – Dynamics – Geometry – Structure; the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 427320536 – SFB 1442; the ERC Advanced Grant 834267 – AMAREC; and the Marsden Fund of the Royal Society of New Zealand (grant number 21-VUW-156)
- Communicated by: Matthew Kennedy
- © Copyright 2025 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 153 (2025), 1849-1866
- MSC (2020): Primary 18B40; Secondary 22A22
- DOI: https://doi.org/10.1090/proc/17159
- MathSciNet review: 4881379