Isotropy fibers of ideals in groupoid \(\mathrm{C}^\ast\)-algebras.(English)Zbl 1551.46044
Summary: Given a locally compact étale groupoid and an ideal \(I\) in its groupoid \(\mathrm{C}^\ast\)-algebra, we show that \(I\) defines a family of ideals in group \(\mathrm{C}^\ast\)-algebras of the isotropy groups and then study to which extent \(I\) is determined by this family. As an application we obtain the following results: (a) prove that every proper ideal is contained in an induced primitive ideal; (b) describe the maximal ideals; (c) classify the primitive ideals for a class of graded groupoids with essentially central isotropy.
MSC:
| 46L05 | General theory of \(C^*\)-algebras |
| 22D25 | \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations |
| 22A22 | Topological groupoids (including differentiable and Lie groupoids) |
Cite
References:
| [1] | Alekseev, V.; Finn-Sell, M., Non-amenable principal groupoids with weak containment, Int. Math. Res. Not., 8, 2332-2340, 2018, MR 3801485 ·Zbl 1406.22016 |
| [2] | Archbold, R. J.; Spielberg, J. S., Topologically free actions and ideals in discrete \(C^\ast \)-dynamical systems, Proc. Edinb. Math. Soc. (2), 37, 1, 119-124, 1994, MR 1258035 ·Zbl 0799.46076 |
| [3] | Batty, C. J.K., Simplexes of extensions of states of \(C^\ast \)-algebras, Trans. Am. Math. Soc., 272, 1, 237-246, 1982, MR 0656488 ·Zbl 0494.46057 |
| [4] | Bekka, B.; de la Harpe, P.; Valette, A., Kazhdan’s Property (T), New Mathematical Monographs, vol. 11, 2008, Cambridge University Press: Cambridge University Press Cambridge, MR 2415834 ·Zbl 1146.22009 |
| [5] | Breuillard, E.; Kalantar, M.; Kennedy, M.; Ozawa, N., \( C^\ast \)-simplicity and the unique trace property for discrete groups, Publ. Math. Inst. Hautes Études Sci., 126, 35-71, 2017, MR 3735864 ·Zbl 1391.46071 |
| [6] | Brown, J. H.; Nagy, G.; Reznikoff, S.; Sims, A.; Williams, D., Cartan subalgebras in C^⁎-algebras of Hausdorff étale groupoids, Integral Equ. Oper. Theory, 85, 1, 109-126, 2016 ·Zbl 1360.46046 |
| [7] | Christensen, J., The structure of KMS weights on étale groupoid \(C^\ast \)-algebras, J. Noncommut. Geom., 17, 2, 663-691, 2023, MR 4592883 ·Zbl 1526.46037 |
| [8] | Christensen, J.; Neshveyev, S., (Non)exotic completions of the group algebras of isotropy groups, Int. Math. Res. Not., 19, 15155-15186, 2022, MR 4490951 ·Zbl 1506.46053 |
| [9] | Clark, L. O.; Exel, R.; Pardo, E.; Sims, A.; Starling, C., Simplicity of algebras associated to non-Hausdorff groupoids, Trans. Am. Math. Soc., 372, 5, 3669-3712, 2019, MR 3988622 ·Zbl 1491.16032 |
| [10] | Crytser, D.; Nagy, G., Simplicity criteria for étale groupoid \(C^\ast \)-algebras, J. Oper. Theory, 83, 1, 95-138, 2020, MR 4043708 ·Zbl 1463.53088 |
| [11] | Effros, E. G.; Hahn, F., Locally Compact Transformation Groups and \(C^\ast \)-Algebras, vol. 75, 1967, American Mathematical Society: American Mathematical Society Providence, R.I., MR 0227310 ·Zbl 0166.11802 |
| [12] | Exel, R., Inverse semigroups and combinatorial \(C^\ast \)-algebras, Bull. Braz. Math. Soc. (N. S.), 39, 2, 191-313, 2008, MR 2419901 ·Zbl 1173.46035 |
| [13] | Gootman, E. C.; Rosenberg, J., The structure of crossed product \(C^\ast \)-algebras: a proof of the generalized Effros-Hahn conjecture, Invent. Math., 52, 3, 283-298, 1979, MR 0537063 ·Zbl 0396.22009 |
| [14] | Higson, N.; Lafforgue, V.; Skandalis, G., Counterexamples to the Baum-Connes conjecture, Geom. Funct. Anal., 12, 2, 330-354, 2002, MR 1911663 ·Zbl 1014.46043 |
| [15] | Ionescu, M.; Williams, D. P., Irreducible representations of groupoid \(C^\ast \)-algebras, Proc. Am. Math. Soc., 137, 4, 1323-1332, 2009, MR 2465655 ·Zbl 1170.46060 |
| [16] | Ionescu, M.; Williams, D. P., The generalized Effros-Hahn conjecture for groupoids, Indiana Univ. Math. J., 58, 6, 2489-2508, 2009, MR 2603756 ·Zbl 1213.46065 |
| [17] | Kalantar, M.; Scarparo, E., Boundary maps and covariant representations, Bull. Lond. Math. Soc., 54, 5, 1944-1961, 2022, MR 4523573 ·Zbl 1534.46047 |
| [18] | Kawabe, T., Uniformly recurrent subgroups and the ideal structure of reduced crossed products, 2017, preprint, available at |
| [19] | Kennedy, M.; Kim, S.-J.; Li, X.; Raum, S.; Ursu, D., The ideal intersection property for essential groupoid C^⁎-algebras, 2021, preprint, available at |
| [20] | Kirchberg, E.; Wassermann, S., Exact groups and continuous bundles of \(C^\ast \)-algebras, Math. Ann., 315, 2, 169-203, 1999, MR 1721796 ·Zbl 0946.46054 |
| [21] | Kwaśniewski, B. K.; Meyer, R., Essential crossed products for inverse semigroup actions: simplicity and pure infiniteness, Doc. Math., 26, 271-335, 2021, MR 4246403 ·Zbl 1472.46071 |
| [22] | Neshveyev, S., KMS states on the \(C^\ast \)-algebras of non-principal groupoids, J. Oper. Theory, 70, 2, 513-530, 2013, MR 3138368 ·Zbl 1299.46067 |
| [23] | Neshveyev, S.; Schwartz, G., Non-Hausdorff étale groupoids and \(C^\ast \)-algebras of left cancellative monoids, Münster J. Math., 16, 1, 147-175, 2023, MR 4563262 ·Zbl 1518.46037 |
| [24] | Ozawa, N., Lecture on the Furstenberg boundary and C^⁎-simplicity, 2014, available at |
| [25] | Phillips, N. C., Crossed products of the Cantor set by free minimal actions of \(\mathbb{Z}^d\), Commun. Math. Phys., 256, 1, 1-42, 2005, MR 2134336 ·Zbl 1084.46056 |
| [26] | Renault, J., A Groupoid Approach to \(C^\ast \)-Algebras, Lecture Notes in Mathematics, vol. 793, 1980, Springer: Springer Berlin, MR 584266 ·Zbl 0433.46049 |
| [27] | Renault, J., The ideal structure of groupoid crossed product \(C^\ast \)-algebras, J. Oper. Theory, 25, 1, 3-36, 1991, With an appendix by Georges Skandalis. MR 1191252 ·Zbl 0786.46050 |
| [28] | Sauvageot, J.-L., Idéaux primitifs induits dans les produits croisés, J. Funct. Anal., 32, 3, 381-392, 1979, MR 0538862 ·Zbl 0434.46036 |
| [29] | Sims, A.; Williams, D. P., The primitive ideals of some étale groupoid C^⁎-algebras, Algebr. Represent. Theory, 19, 2, 255-276, 2016 ·Zbl 1357.46051 |
| [30] | Starling, C., A new uniqueness theorem for the tight C^⁎-algebra of an inverse semigroup, C. R. Math. Rep. Acad. Sci. Canada, 44, 4, 88-112, 2022 ·Zbl 07908847 |
| [31] | Willett, R., A non-amenable groupoid whose maximal and reduced \(C^\ast \)-algebras are the same, Münster J. Math., 8, 1, 241-252, 2015, MR 3549528 ·Zbl 1369.46064 |
| [32] | Williams, D. P., The topology on the primitive ideal space of transformation group \(C^\ast \)-algebras and C.C.R. transformation group \(C^\ast \)-algebras, Trans. Am. Math. Soc., 266, 2, 335-359, 1981, MR 0617538 ·Zbl 0474.46057 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
