The Higson–Roe sequence for étale groupoids. I. Dual algebras and compatibility with the BC map
Moulay-Tahar Benameur
Université de Montpellier, FranceIndrava Roy
The Institute of Mathematical Sciences, Chennai, India
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Abstract
We introduce the dual Roe algebras for proper étale groupoid actions and deduce the expected Higson–Roe short exact sequence. When the action is co-compact, we show that the Roe -ideal of locally compact operators is Morita equivalent to the reduced -algebra of our groupoid, and we further identify the boundary map of the associated periodic six-term exact sequence with the Baum–Connes map, via a Paschke–Higson map for groupoids. For proper actions on continuous families of manifolds of bounded geometry, we associate with any -equivariant Dirac-type family, a coarse index class which generalizes the Paterson index class and also the Moore–Schochet Connes’ index class for laminations.
Cite this article
Moulay-Tahar Benameur, Indrava Roy, The Higson–Roe sequence for étale groupoids. I. Dual algebras and compatibility with the BC map. J. Noncommut. Geom. 14 (2020), no. 1, pp. 25–71
DOI 10.4171/JNCG/358