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

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