Orbit-breaking in Deaconu-Renault groupoids and models for UCT Kirchberg algebras
Abstract
We generalize the concept of orbit-breaking from the case of a homeomorphism to the case of a surjective local homeomorphism. This method creates open subgroupoids of the Deaconu--Renualt groupoid associated to the surjective local homeomorphism. When the map satisfies natural assumptions (including being locally expanding) the resulting groupoid C*-algebras are unital UCT Kirchberg algebras. A specific class of Deaconu--Renault systems are constructed, each modelling the Cuntz algebra 𝒪2. The C*-algebras obtained from orbit-breaking for such systems come with explicit embeddings into 𝒪2. We also introduce a method for orbit-breaking after removing a fixed point. Using this method, we can realize all stable UCT Kirchberg algebras with each coming with an embeddings into 𝒪2⊗𝒦. Moreover, we show that, given a homomorphism between countable graded abelian groups, there is a *-homomorphism between orbit-breaking algebras obtained from a groupoid inclusion. In particular, we obtain explicit *-homomorphisms between stable UCT Kirchberg algebras from any group homomorphism between their K-groups.
Related
- Roe algebras and coarse index maps for spaces with proper actions of étale groupoids. II
- Discretisation in algebraic K-theory of independent ample groupoids and their Steinberg algebras
- Roe algebras and coarse index maps for spaces with proper actions of étale groupoids. I
- Zhang twists for semigroups and groupoids, with applications to Steinberg algebras
- Morita equivalence for Lp-operator algebras associated with étale groupoids
BibTeX
@misc{deeley2026orbitbreaking,
title = {{Orbit-breaking in Deaconu-Renault groupoids and models for UCT Kirchberg algebras}},
author = {Robin J. Deeley and Ian F. Putnam and Karen R. Strung},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31883},
eprint = {2609.31883},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31883},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Orbit-breaking in Deaconu-Renault groupoids and models for UCT Kirchberg algebras
AU - Robin J. Deeley
AU - Ian F. Putnam
AU - Karen R. Strung
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.31883
UR - https://arxiv.org/abs/2609.31883
AB - We generalize the concept of orbit-breaking from the case of a homeomorphism to the case of a surjective local homeomorphism. This method creates open subgroupoids of the Deaconu--Renualt groupoid associated to the surjective local homeomorphism. When the map satisfies natural assumptions (including being locally expanding) the resulting groupoid C^*-algebras are unital UCT Kirchberg algebras. A specific class of Deaconu--Renault systems are constructed, each modelling the Cuntz algebra 𝒪_2. The C^*-algebras obtained from orbit-breaking for such systems come with explicit embeddings into 𝒪_2. We also introduce a method for orbit-breaking after removing a fixed point. Using this method, we can realize all stable UCT Kirchberg algebras with each coming with an embeddings into 𝒪_2⊗𝒦. Moreover, we show that, given a homomorphism between countable graded abelian groups, there is a ^*-homomorphism between orbit-breaking algebras obtained from a groupoid inclusion. In particular, we obtain explicit ^*-homomorphisms between stable UCT Kirchberg algebras from any group homomorphism between their K-groups.
ER -