Topological Property (T) for groupoids
Annales de l'Institut Fourier, Online first, 52 p.

We introduce a notion of topological Property (T) for étale groupoids. This simultaneously generalizes Kazhdan’s Property (T) for groups and geometric Property (T) for coarse spaces. One main goal is to use this Property (T) to prove the existence of so-called Kazhdan projections in both maximal and reduced groupoid C * -algebras, and explore applications of this to exactness, K-exactness, and the Baum–Connes conjecture. We also study various examples, and discuss the relationship with other notions of Property (T) for groupoids and with a-T-menability.

Nous définissons une notion de propriété (T) pour les groupoïdes étales. Elle généralise à la fois la propriété (T) de Kazhdan pour les groupes, et la propriété (T) géométrique pour les espaces grossiers. Notre but principal est l’application de cette propriété (T) à l’existence de projecteurs de type Kazhdan dans les C * -algèbres réduites et maximales des groupoïdes, dont nous explorons les conséquences sur l’exactitude, l’exactitude en K-théorie, et sur la validité de la conjecture de Baum–Connes. Nous étudions aussi divers exemples, et comparons cette notion à d’autres versions de la propriété (T) ainsi qu’à la a-T-moyennabilité.

Received:
Revised:
Accepted:
Online First:
DOI: 10.5802/aif.3513
Classification: 22A22,  46L85,  46L80,  51F99
Keywords: Property (T), Topological groupoids, Coarse geometry, Expander
Dell’Aiera, Clément 1; Willett, Rufus 2

1 ENS Lyon, UMPA Department of Mathematics 46 allée d’Italie 69342 Lyon Cedex 07 (France)
2 Department of Mathematics University of Hawai‘i at Mānoa 2565 McCarthy Mall Honolulu, HI 96822 (USA)
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Dell’Aiera, Clément; Willett, Rufus. Topological Property (T) for groupoids. Annales de l'Institut Fourier, Online first, 52 p.

[1] Akemann, Charles; Walter, Martin Unbounded neagtive definite functions, Can. J. Math., Volume 33 (1981) no. 4, pp. 862-871

[2] Anantharaman-Delaroche, Claire Cohomology of property T groupoids and applications, Ergodic Theory Dyn. Syst., Volume 25 (2005) no. 4, pp. 977-1013

[3] Bader, Uri; Furman, Alex; Gelander, Tsachik; Monod, Nicolas Property (T) and rigidity for actions on Banach spaces, Acta Math., Volume 198 (2007) no. 1, pp. 57-105

[4] Bekka, Bachir Property (T) for C * -algebras, Bull. Lond. Math. Soc., Volume 38 (2006) no. 5, pp. 857-867

[5] Bekka, Bachir; Harpe, Pierre de la; Valette, Alain Kazhdan’s Property (T), Cambridge University Press, 2008

[6] Brown, Nathanial; Ozawa, Narutaka C * -Algebras and Finite-Dimensional Approximations, Graduate Studies in Mathematics, 88, American Mathematical Society, 2008

[7] Dixmier, Jacques C * -Algebras, North-Holland, 1977

[8] Dong, Zhe; Ruan, Zhong-Jin A Hilbert module approach to the Haagerup property, Integral Equations Oper. Theory, Volume 73 (2012), pp. 431-454

[9] Drutu, Cornelia; Nowak, Piotr Kazhdan projections, random walks and ergodic theorems (2015) (to appear in J. Reine Angew. Math.) | arXiv

[10] Higson, Nigel The Baum-Connes conjecture, Proceedings of the International Congress of Mathematicians, Volume II (1998), pp. 637-646

[11] Higson, Nigel; Lafforgue, Vincent; Skandalis, Georges Counterexamples to the Baum-Connes conjecture, Geom. Funct. Anal., Volume 12 (2002), pp. 330-354

[12] Kazhdan, David Connection of the dual space of a group with the structure of its closed subgroups, Funct. Anal. Appl., Volume 1 (1967), pp. 63-65

[13] Lafforgue, Vincent Un renforcement de la propriété (T), Duke Math. J., Volume 143 (2008) no. 3, pp. 559-602

[14] Lubotzky, Alexander Discrete Groups, Expanding Graphs and Invariant Measures, Birkhäuser, 1994

[15] Lupini, Martino A von Neumann algebra characterization of property (T) for groupoids (2017) (to appear in J. Aust. Math. Soc.) | arXiv

[16] Renault, Jean A groupoid approach to C * -algebras, Lecture Notes in Mathematics, 793, Springer, 1980

[17] Renault, Jean C * -algebras and dynamical systems (27 Colóquio Brasilieiro de Mathemática), Publicações Mathemáticas do IMPA, Instituto Nacional de Matemática Pura e Aplicada, 2009 | Zbl

[18] Roe, John Lectures on Coarse Geometry, University Lecture Series, 31, American Mathematical Society, 2003

[19] Sims, Aidan Hausdorff étale groupoids and their C * -algebras (2017) | arXiv

[20] Skandalis, Georges; Tu, Jean-Louis; Yu, Guoliang The coarse Baum-Connes conjecture and groupoids, Topology, Volume 41 (2002), pp. 807-834

[21] Špakula, Ján; Willett, Rufus A metric approach to limit operators, Trans. Am. Math. Soc., Volume 369 (2017), pp. 263-308

[22] Tu, Jean-Louis La conjecture de Baum-Connes pour les feuilletages moyennables, K-Theory, Volume 17 (1999), pp. 215-264

[23] Valette, Alain Minimal projections, integrable representations and property (T), Arch. Math., Volume 43 (1984) no. 5, pp. 397-406

[24] Willett, Rufus A non-amenable groupoid whose maximal and reduced C * -algebras are the same, Münster J. Math., Volume 8 (2015), pp. 241-252

[25] Willett, Rufus; Yu, Guoliang Geometric Property (T), Chin. Ann. Math., Ser. B, Volume 35 (2014) no. 5, pp. 761-800

[26] Zimmer, Robert On the cohomology of ergodic actions of semisimple Lie groups and discrete subgroups, Am. J. Math., Volume 103 (1981) no. 5, pp. 937-951

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