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Homology and K-theory of dynamical systems I. Torsion-free ample groupoids

Part of: $K$-theory and operator algebras Dynamical systems with hyperbolic behavior Selfadjoint operator algebras

Published online by Cambridge University Press:  04 June 2021

VALERIO PROIETTI Open the ORCID record for VALERIO PROIETTI [Opens in a new window]  and
MAKOTO YAMASHITA
VALERIO PROIETTI*
Affiliation:
Research Center for Operator Algebras, Department of Mathematics, and Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, East China Normal University, Shanghai200241, China
MAKOTO YAMASHITA
Affiliation:
Department of Mathematics, University of Oslo, P.O. box 1053, Blindern, 0316Oslo, Norway (e-mail: makotoy@math.uio.no)
*
e-mail: proiettivalerio@math.ecnu.edu.cn
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Abstract

Given an ample groupoid, we construct a spectral sequence with groupoid homology with integer coefficients on the second sheet, converging to the K-groups of the (reduced) groupoid C $^*$ -algebra, provided the groupoid has torsion-free stabilizers and satisfies a strong form of the Baum–Connes conjecture. The construction is based on the triangulated category approach to the Baum–Connes conjecture developed by Meyer and Nest. We also present a few applications to topological dynamics and discuss the HK conjecture of Matui.

Keywords

Baum–Connes conjecturegroupoid homologyK-theoryspectral sequence

MSC classification

Primary: 46L85: Noncommutative topology
Secondary: 19K35: Kasparov theory ($KK$-theory) 37D99: None of the above, but in this section
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 8 , August 2022 , pp. 2630 - 2660
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

Anantharaman-Delaroche, C. and Renault, J.. Amenable Groupoids (Monographies de L’Enseignement Mathématique, 36). L’Enseignement Mathématique, Geneva, 2000. With a foreword by Georges Skandalis and Appendix B by E. Germain.Google Scholar
Anderson, J. E. and Putnam, I. F.. Topological invariants for substitution tilings and their associated ${C}^{\ast }$ -algebras. Ergod. Th. & Dynam. Sys. 18(3) (1998), 509–537.CrossRefGoogle Scholar
Bauval, A.. $\mathrm{RKK}(\mathrm{X})$ -nucléarité (d’après G. Skandalis). $K$ -Theory 13(1) (1998), 23–40.Google Scholar
Bekka, M. E. B., Cherix, P.-A. and Valette, A.. Proper affine isometric actions of amenable groups. Novikov Conjectures, Index Theorems and Rigidity. Vol. 2. Cambridge University Press, Cambridge, 1995, pp. 1–4 (in English). Based on a conference of the Mathematisches Forschungsinstitut Oberwolfach in September 1993.Google Scholar
Blackadar, B.. ${K}$ -Theory for Operator Algebras (Mathematical Sciences Research Institute Publications, 5). Springer-Verlag, New York, NY, 1986.CrossRefGoogle Scholar
Blanchard, É.. Déformations de ${\mathrm{C}}^{\ast }$ -algèbres de Hopf. Bull. Soc. Math. France 124(1) (1996), 141–215.CrossRefGoogle Scholar
Blanchard, E. and Kirchberg, E.. Global Glimm halving for ${\mathrm{C}}^{\ast }$ -bundles. J. Operator Theory 52(2) (2004), 385–420.Google Scholar
Bönicke, C.. A Going-Down principle for ample groupoids and the Baum–Connes conjecture. Adv. Math. 372 (2020), 107314.CrossRefGoogle Scholar
Bratteli, O., Evans, D. E. and Kishimoto, A.. Crossed products of totally disconnected spaces by ${\mathrm{Z}}_2\ast {\mathrm{Z}}_2$ . Ergod. Th. & Dynam. Sys. 13(3) (1993), 445–484.CrossRefGoogle Scholar
Brown, K. S.. Cohomology of Groups (Graduate Texts in Mathematics, 87). Springer-Verlag, New York, NY, 1994. Corrected reprint of the 1982 original.Google Scholar
Christensen, J. D.. Ideals in triangulated categories: phantoms, ghosts and skeletal. Adv. Math. 136(2) (1998), 284–339.CrossRefGoogle Scholar
Crainic, M. and Moerdijk, I.. A homology theory for étale groupoids. J. Reine Angew. Math. 521 (2000), 25–46.Google Scholar
Deaconu, V.. Groupoids associated with endomorphisms. Trans. Amer. Math. Soc. 347(5) (1995), 1779–1786.CrossRefGoogle Scholar
Emerson, H. and Meyer, R.. Dualities in equivariant Kasparov theory. New York J. Math. 16 (2010), 245–313.Google Scholar
Exel, R. and Pardo, E.. Self-similar graphs, a unified treatment of Katsura and Nekrashevych ${C}^{\ast }$ -algebras. Adv. Math. 306 (2017), 1046–1129.CrossRefGoogle Scholar
Exel, R. and Renault, J.. Semigroups of local homeomorphisms and interaction groups. Ergod. Th. & Dynam. Sys. 27(6) (2007), 1737–1771.CrossRefGoogle Scholar
Farsi, C., Kumjian, A., Pask, D. and Sims, A.. Ample groupoids: equivalence, homology, and Matui’s HK conjecture. Münster J. Math. 12(2) (2019), 411–451.Google Scholar
Giordano, T., Putnam, I. F. and Skau, C. F.. Topological orbit equivalence and ${C}^{\ast }$ -crossed products. J. Reine Angew. Math. 469 (1995), 51–111.Google Scholar
Godement, R.. Topologie algébrique et théorie des faisceaux. Troisième édition revue et corrigée. Publications de l’Institut de Mathématique de l’Université de Strasbourg, XIII (Actualités Scientifiques et Industrielles, 1252). Hermann, Paris, 1973.Google Scholar
Juschenko, K. and Monod, N.. Cantor systems, piecewise translations and simple amenable groups. Ann. of Math. (2) 178(2) (2013), 775–787.CrossRefGoogle Scholar
Kaminkerad, J. and Putnam, I.. A proof of the gap labeling conjecture. Michigan Math. J. 51(3) (2003), 537–546.Google Scholar
Kasparov, G. G.. Equivariant $KK$ -theory and the Novikov conjecture. Invent. Math. 91(1) (1988), 147–201.CrossRefGoogle Scholar
Kasparov, G.. Elliptic and transversally elliptic index theory from the viewpoint of $KK$ -theory. J. Noncommut. Geom. 10(4) (2016), 1303–1378.CrossRefGoogle Scholar
Katsura, T.. A construction of actions on Kirchberg algebras which induce given actions on their $K$ -groups. J. Reine Angew. Math. 617 (2008), 27–65.Google Scholar
Kellendonk, J. and Putnam, I. F.. Tilings, ${C}^{\ast }$ -algebras, and $K$ -theory. Directions in Mathematical Quasicrystals. American Mathematical Society, Providence, RI, 2000, pp. 177–206.Google Scholar
Khoshkam, M. and Skandalis, G.. Crossed products of ${C}^{\ast }$ -algebras by groupoids and inverse semigroups. J. Operator Theory 51(2) (2004), 255–279.Google Scholar
Lafforgue, V.. $K$ -théorie bivariante pour les algèbres de Banach, groupoïdes et conjecture de Baum-Connes. Avec un appendice d’Hervé Oyono-Oyono. J. Inst. Math. Jussieu 6(3) (2007), 415–451.CrossRefGoogle Scholar
Lance, E. C.. Hilbert ${C}^{\ast }$ -Modules: A Toolkit for Operator Algebraists (London Mathematical Society Lecture Note Series, 210). Cambridge University Press, Cambridge, 1995.Google Scholar
Le Gall, P.-Y.. Théorie de Kasparov équivariante et groupoïdes. I. $K$ -Theory 16(4) (1999), 361–390.Google Scholar
Matsumoto, K.. Topological conjugacy of topological Markov shifts and Ruelle algebras. J. Operator Theory 82(2) (2019), 253–284.Google Scholar
Matui, H.. Homology and topological full groups of étale groupoids on totally disconnected spaces. Proc. Lond. Math. Soc. (3) 104(1) (2012), 27–56.CrossRefGoogle Scholar
Matui, H.. Some remarks on topological full groups of Cantor minimal systems II. Ergod. Th. & Dynam. Sys. 33(5) (2013), 1542–1549.CrossRefGoogle Scholar
Meyer, R.. Homological algebra in bivariant $K$ -theory and other triangulated categories. II. Tbilisi Math. J. 1 (2008), 165–210.CrossRefGoogle Scholar
Meyer, R. and Nest, R.. The Baum–Connes conjecture via localisation of categories. Topology 45(2) (2006), 209–259.CrossRefGoogle Scholar
Meyer, R. and Nest, R.. Homological algebra in bivariant $K$ -theory and other triangulated categories. I. Triangulated Categories. Cambridge University Press, Cambridge, 2010, pp. 236–289.CrossRefGoogle Scholar
Moerdijk, I.. Proof of a conjecture of A. Haefliger. Topology 37(4) (1998), 735–741.CrossRefGoogle Scholar
Muhly, P. S. and Williams, D. P.. Renault’s Equivalence Theorem for Groupoid Crossed Products (NYJM Monographs, 3). State University of New York, University at Albany, Albany, NY, 2008.Google Scholar
Ortega, E.. Homology of the Katsura–Exel–Pardo groupoid. J. Noncommut. Geom. 14(3) (2020), 913–935.CrossRefGoogle Scholar
Oyono-Oyono, H.. Baum–Connes conjecture and extensions. J. Reine Angew. Math. 532 (2001), 133–149.Google Scholar
Park, E. and Trout, J.. Representable $E$ -theory for ${C}_0(X)$ -algebras. J. Funct. Anal. 177(1) (2000), 178–202.CrossRefGoogle Scholar
Phillips, N. C.. Crossed products of the Cantor set by free minimal actions of ${\mathbb{Z}}^d$ . Comm. Math. Phys. 256(1) (2005), 1–42.CrossRefGoogle Scholar
Popescu, R.. Equivariant $E$ -theory for groupoids acting on ${C}^{\ast }$ -algebras. J. Funct. Anal. 209(2) (2004), 247–292. CrossRefGoogle Scholar
Proietti, V.. On $K$ -theory, groups, and topological dynamics. PhD Thesis, University of Copenhagen, 2018.Google Scholar
Putnam, I. F.. ${C}^{\ast }$ -algebras from Smale spaces. Canad. J. Math. 48(1) (1996), 175–195.CrossRefGoogle Scholar
Renault, J.. A Groupoid Approach to C*-Algebras (Lecture Notes in Mathematics, 793). Springer, Berlin, 1980.Google Scholar
Sadun, L. and Williams, R. F.. Tiling spaces are Cantor set fiber bundles. Ergod. Th. & Dynam. Sys. 23(1) (2003), 307–316.CrossRefGoogle Scholar
Savinien, J. and Bellissard, J.. A spectral sequence for the $K$ -theory of tiling spaces. Ergod. Th. & Dynam. Sys. 29(3) (2009), 997–1031.CrossRefGoogle Scholar
Scarparo, E.. Homology of odometers. Ergod. Th. & Dynam. Sys. 40(9) (2020), 2541–2551.CrossRefGoogle Scholar
Sims, A. and Williams, D. P.. Renault’s equivalence theorem for reduced groupoid ${C}^{\ast }$ -algebras. J. Operator Theory 68(1) (2012), 223–239.Google Scholar
Skandalis, G.. Une notion de nucléarité en $\mathrm{K}$ -théorie (d’après J. Cuntz). $K$ -Theory 1(6) (1988), 549–573.Google Scholar
Steinberg, B.. Modules over étale groupoid algebras as sheaves. J. Aust. Math. Soc. 97(3) (2014), 418–429.CrossRefGoogle Scholar
Thomsen, K.. ${C}^{\ast }$ -algebras of homoclinic and heteroclinic structure in expansive dynamics. Mem. Amer. Math. Soc. 206(970) (2010), x+122.Google Scholar
Tu, J.-L.. La conjecture de Baum–Connes pour les feuilletages moyennables. $K$ -Theory 17(3) (1999), 215–264.Google Scholar
Tu, J.-L.. La conjecture de Novikov pour les feuilletages hyperboliques. $K$ -Theory 16(2) (1999), 129–184.Google Scholar
Tu, J.-L.. Non-Hausdorff groupoids, proper actions and $\textrm{K}$ -theory. Doc. Math. 9 (2004), 565–597, Extended version available at the author’s website.Google Scholar
Tu, J.-L.. Groupoid cohomology and extensions. Trans. Amer. Math. Soc. 358(11) (2006), 4721–4747.CrossRefGoogle Scholar
Weibel, C. A.. An Introduction to Homological Algebra (Cambridge Studies in Advanced Mathematics, 38). Cambridge University Press, Cambridge, 1994.CrossRefGoogle Scholar
Yi, I.. Homology and Matui’s HK conjecture for groupoids on one-dimensional solenoids. Bull. Aust. Math. Soc. 101(1) (2020), 105–117.CrossRefGoogle Scholar