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 C^* -ideal of locally compact operators is Morita equivalent to the reduced C^* -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 G -equivariant Dirac-type family, a coarse index class which generalizes the Paterson index class and also the Moore–Schochet Connes’ index class for laminations.
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