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arXiv · 2609.37679

Roe algebras and coarse index maps for spaces with proper actions of {é}tale groupoids. I

Kai Mao
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Abstract

This is the first in a series of papers extending Roe algebras and coarse index theory to the groupoid-equivariant setting. We introduce some techniques to develop a framework of Roe algebras and their K-theory for spaces equipped with proper actions of {é}tale groupoids. For a fixed {é}tale groupoid G and G-C* -algebra A, we construct a functor KC(-; G, A) from the category of locally compact Hausdorff proper G-spaces and equivariant proper continuous maps to the category of graded abelian groups, which provides a natural receptacle for an equivariant coarse index map. We study the existence of universal modules, an analog of ample modules in non-equivariant setting, and develop a decomposition technique to establish their existence. As an application, we prove that groupoid simplicial complexes satisfying suitable hypotheses admit universal modules.

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Kai Mao. 2026-09-29. Roe algebras and coarse index maps for spaces with proper actions of {é}tale groupoids. I. https://arxiv.org/abs/2609.37679

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