Roe algebras and coarse index maps for spaces with proper actions of étale groupoids. I

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 étale groupoids. For a fixed étale 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.

Roe algebras and coarse index maps for spaces with proper actions of étale groupoids. II
Kai Mao
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BibTeX RIS

BibTeX

@misc{mao2026algebras,
  title = {{Roe algebras and coarse index maps for spaces with proper actions of étale groupoids. I}},
  author = {Kai Mao},
  howpublished = {HAL},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.37679},
  eprint = {2609.37679},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-05769372v1},
  note = {CC BY-NC-ND 4.0},
}

RIS

TY  - UNPB
TI  - Roe algebras and coarse index maps for spaces with proper actions of étale groupoids. I
AU  - Kai Mao
PY  - 2026
DA  - 2026/09/29
PB  - HAL
DO  - 10.48550/arxiv.2609.37679
UR  - https://hal.science/hal-05769372v1
AB  - 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 étale groupoids. For a fixed étale 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.
ER  -

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