Exotic groupoid $C^*$-algebras associated to double groupoids
DOI:
https://doi.org/10.7146/math.scand.a-149438Abstract
We consider a class of partial action groupoids called double groupoids which are constructed from pairs of subgroups satisfying similar conditions to those of a matched pair of groups. If the double groupoid is étale, then we show that whenever the partially acting group admit exotic ideal completions in the sense of Brown and Guentner, the corresponding double groupoid also admit exotic $C^*$-completions.
References
Abadie, F., On partial actions and groupoids, Proc. Amer. Math. Soc. 132 (2004), no. 4, 1037–1047. https://doi.org/10.1090/S0002-9939-03-07300-3
Alekseev, V., and Finn-Sell, M., Non-amenable principal groupoids with weak containment, Int. Math. Res. Not. IMRN 8 (2018), no. 4, 2332–2340. https://doi.org/10.1093/imrn/rnw305
Anantharaman-Delaroche, C., and Renault, J., Amenable groupoids, Monographies de L'Enseignement Mathématique 36, L'Enseignement Mathématique, Geneva, 2000.
Baaj, S., Skandalis, G., and Vaes, S., Non-semi-regular quantum groups coming from number theory, Comm. Math. Phys. 235 (2003), no. 1, 139–167. https://doi.org/10.1007/s00220-002-0780-6
Bekka, M. B., Kaniuth, E., Lau, A. T., and Schlichting, G., On $C^*$-algebras associated with locally compact groups, Proc. Amer. Math. Soc. 124 (1996), no. 10, 3151–3158. https://doi.org/10.1090/S0002-9939-96-03382-5
Brannan, M., and Ruan, Z.-J., $L_p$-representations of discrete quantum groups, J. Reine Angew. Math. 732 (2017), 165–210. https://doi.org/10.1515/crelle-2014-0140
Brown, N. P., and Guentner, E. P., New $C^*$-completions of discrete groups and related spaces, Bull. Lond. Math. Soc. 45 (2013), no. 6, 1181-1193. https://doi.org/10.1112/blms/bdt044
Bruce, C., and Li, X., Algebraic actions I. C*-algebras and groupoids, J. Funct. Anal. 286 (2024), no. 4, Paper No. 110263, 57 pp. https://doi.org/10.1016/j.jfa.2023.110263
Buss, A., Echterhoff, S., and Willett, R., Exotic crossed products, Operator algebras and applications—the Abel Symposium 2015, 67–114, Abel Symp., 12, Springer, Cham, 2017.
Buss, A., Echterhoff, S., and Willett, R., Exotic crossed products and the Baum-Connes conjecture, J. Reine Angew. Math. 740 (2018), 111–159. https://doi.org/10.1515/crelle-2015-0061
Christensen, J., and Neshveyev, S., Isotropy fibers of ideals in groupoid $ C^*$-algebras, Adv. Math. 447 (2024), Paper No. 109696, 32 pp. https://doi.org/10.1016/j.aim.2024.109696
Christensen, J., and Neshveyev, S., (Non)exotic completions of the group algebras of isotropy groups, Int. Math. Res. Not. IMRN (2022), no. 19, 15155–15186. https://doi.org/10.1093/imrn/rnab127
Delvaux, L., and Van Daele, A., Algebraic quantum hypergroups, Adv. Math. 226 (2011), no. 2, 1134–1167. https://doi.org/10.1016/j.aim.2010.07.015
Delvaux, L., and Van Daele, A., Algebraic quantum hypergroups II. Constructions and examples, Internat. J. Math. 22 (2011), no. 3, 407–434. https://doi.org/10.1142/S0129167X11006830
Desmedt, P., Quaegebeur, J., and Vaes, S., Amenability and the bicrossed product construction, Illinois J. Math. 46 (2002), no. 4, 1259–1277. http://projecteuclid.org/euclid.ijm/1258138478
Exel, R., On Kumjian's $C^*$-diagonals and the opaque ideal, arXiv:2110.09445v2.
Exel, R., Partial dynamical systems, Fell bundles and applications, Mathematical Surveys and Monographs, 224, American Mathematical Society, Providence, RI, 2017. https://doi.org/10.1090/surv/224
Exel, R., and Pitts, D. R., Characterizing groupoid $C^*$-algebras of non-Hausdorff étale groupoids, Lecture Notes in Mathematics, 2306, Springer, Cham, 2022. https://doi.org/10.1007/978-3-031-05513-3
Exel, R., Pitts, D. R., and Zarikian, V., Exotic ideals in free transformation group $C^*$-algebras, Bull. Lond. Math. Soc. 55 (2024), no. 3, 1489–1510. https://doi.org/10.1112/blms.12805
Hahn, P., Haar measure for measure groupoids, Trans. Amer. Math. Soc. 242 (1978), 1–33. https://doi.org/10.2307/1997726
Heinig, D., de Laat, T., and Siebenand, T., Group $C^*$-algebras of locally compact groups acting on trees, Int. Math. Res. Not. IMRN (2024), no. 10, 8520–8539. https://doi.org/10.1093/imrn/rnad259
Kaliszewski, S., Landstad, M. B., and Quigg, J., Exotic group $C^*$-algebras in noncommutative duality, New York J. Math. 19 (2013), 689–711. http://nyjm.albany.edu:8000/j/2013/19_689.html
Kyed, D., and Sołtan, P. M., Property (T) and exotic quantum group norms, J. Noncommut. Geom. 6 (2012), no. 4, 773–800. https://doi.org/10.4171/JNCG/105
de Laat, T., and Siebenand, T., Exotic group $C^*$-algebras of simple Lie groups with real rank one, Ann. Inst. Fourier (Grenoble) 71 (2021), no. 5, 2117–2136. https://doi.org/10.5802/aif.3441
Landstad, M. B., and Van Daele, A., Finite quantum hypergroups, arXiv:2209.13282v1.
Landstad, M. B., and Van Daele, A., Polynomial functions for locally compact group actions, arXiv:2309.08319v1.
Landstad, M. B., and Van Daele, A., Topological quantum hypergroups arising from two closed sub-groups, in preparation.
Muhly, P. S., Coordinates in operator algebras, 1997, https://operatoralgebras.org/resources-resources/Groupoids-Book-Muhly.pdf
Okayasu, R., Free group $C^*$-algebras associated with $ell _p$, Internat. J. Math. 25 (2014), no. 7, 1450065, 12 pp. https://doi.org/10.1142/S0129167X14500657
Palmstrøm, M., Exotic $C^*$-completions of étale groupoids, arXiv:2311.12428v4.
Paterson, A., Groupoids, inverse semigroups, and their operator algebras, Progress in Mathematics, 170. Birkhäuser Boston, Inc., Boston, MA, 1999. https://doi.org/10.1007/978-1-4612-1774-9
Ramsay, A., Topologies on measured groupoids, J. Funct. Anal. 47 (1982), no. 3, 314-343. https://doi.org/10.1016/0022-1236(82)90110-0
Renault, J., A groupoid approach to $C^ast $-algebras, Lecture Notes in Mathematics, 793. Springer, Berlin, 1980.
Renault, J., and Williams, D. P., Amenability of groupoids arising from partial semigroup actions and topological higher rank graphs, Trans. Amer. Math. Soc. 369 (2017), no. 4, 2255–2283. https://doi.org/10.1090/tran/6736
Ruan, Z.-J., and Wiersma, M., On exotic group $C^*$-algebras, J. Funct. Anal. 271 (2016), no. 2, 437–453. https://doi.org/10.1016/j.jfa.2016.03.002
Samei, E., and Wiersma, M., Exotic $C^*$-algebras of geometric groups, J. Funct. Anal. 286 (2024), no. 2, Paper No. 110228, 32 pp. https://doi.org/10.1016/j.jfa.2023.110228
Sanov, I. N., A property of a representation of a free group, Doklady Akad. Nauk SSSR (N.S.) 57 (1947), 657–659.
Sims, A., Hausdorff étale groupoids and their $C^*$-algebras, In Operator algebras and dynamics: groupoids, crossed products, and Rokhlin dimension., Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser/Springer, Cham, 2020.
Vaes, S., and Vainerman, L., Extensions of locally compact quantum groups and the bicrossed product construction, Adv. Math. 175 (2003), no. 1, 1–101. https://doi.org/10.1016/S0001-8708(02)00040-3
Wiersma, M., Constructions of exotic group $C^*$-algebras, Illinois J. Math. 60 (2016), no. 3–4, 655–667. https://projecteuclid.org/euclid.ijm/1506067285
Willett, R., A non-amenable groupoid whose maximal and reduced $C^*$-algebras are the same, Münster J. Math. 8 (2015), no. 1, 241–252. https://doi.org/10.17879/65219671638
Williams, D. P., A tool kit for groupoid $C^*$-algebras, Mathematical Surveys and Monographs, 241, American Mathematical Society, Providence, RI, 2019. https://doi.org/10.1090/surv/241
Williams, D. P. Crossed products of $C^ast $-algebras, Mathematical Surveys and Monographs, 134, American Mathematical Society, Providence, RI, 2007. https://doi.org/10.1090/surv/134