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.

中文翻译：

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étalegroupoids的Higson-Roe序列。I.对偶代数和与BC映射的兼容性

我们引入双重Roe代数来实现适当的étalegroupoid行为，并推论出预期的Higson-Roe短精确序列。当动作是紧致的时，我们证明局部紧凑算子的Roe $ C ^ * $-理想化是Morita等于我们的类群的化简后的$ C ^ * $-代数，并且我们进一步确定了通过类群的Paschke-Higson映射，与Baum-Connes映射关联的周期性六项精确序列。为了对有界几何流形的连续族进行适当的处​​理，我们将与任何$ G $等价的Dirac型族相关联，这是一个粗略的索引类，可以将Paterson索引类以及Moore-Schochet Connes的索引类进行泛化。