Pullbacks of Groupoid C^*-Algebras over Disjoint Invariant Sets

Abstract

We establish a pullback theorem for C^*-algebras of locally compact Hausdorff étale groupoids. The theorem shows that, when the unit space is covered by two closed invariant subsets whose complements are disjoint open invariant subsets, the corresponding groupoid C^*-algebras form a pullback diagram in the category of 𝕋-C^*-algebras and 𝕋-equivariant *-homomorphisms, for the gauge actions induced by a ℤ-valued cocycle and its restrictions. We then prove a collection of boundary-path decomposition theorems for graphs, relative graphs, and topological graphs. We show that admissible decompositions of graphs, together with their analogues in the relative and topological settings, induce corresponding decompositions of boundary path spaces. Combining these decomposition theorems with the groupoid pullback theorem, we recover previously known pullback theorems for graph C^*-algebras, relative graph C^*-algebras, and topological graph C^*-algebras. Thus these pullback phenomena are explained by a single groupoid-theoretic mechanism.

Symmetry and Invariant Theory of Quadratic Dynamics over Eight-Dimensional Composition Algebras
Daniele Corradetti
arXiv7 Oct 2026PreprintCC BY 4.0Full textAlgebra & number theory Dynamical systemsarXiv
A characterization of simplicity of reduced groupoid C*-algebras
Paolo Boldrini
arXiv18 Sep 2026PreprintarXiv licenceFull textAnalysis & PDEs Dynamical systemsarXiv
Operator systems in étale groupoid C*-algebras
Georgios Baziotis, Douglas Farenick, Evgenios T. A. Kakariadis +3
arXiv25 Oct 2026PreprintCC BY 4.0Full textAnalysis & PDEsarXiv
Von Neumann algebras as reduced twisted groupoid C^*-algebras
Alcides Buss, Luiz Felipe Garcia, Tomás Pacheco
arXiv9 Sep 2026PreprintarXiv licenceFull textAnalysis & PDEsarXiv
Quartic polynomial differential systems with three disjoint invariant circles
Chen Cao, Jaume Llibre
arXiv3 Nov 2026PreprintarXiv licenceFull textDynamical systemsarXiv

Cite

BibTeX RIS

BibTeX

@misc{castro2026pullbacks,
  title = {{Pullbacks of Groupoid $C^*$-Algebras over Disjoint Invariant Sets}},
  author = {Gilles G. de Castro and Eun Ji Kang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.09616},
  eprint = {2609.09616},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.09616},
}

RIS

TY  - UNPB
TI  - Pullbacks of Groupoid C^*-Algebras over Disjoint Invariant Sets
AU  - Gilles G. de Castro
AU  - Eun Ji Kang
PY  - 2026
DA  - 2026/09/09
PB  - arXiv
DO  - 10.48550/arxiv.2609.09616
UR  - https://arxiv.org/abs/2609.09616
AB  - We establish a pullback theorem for C^*-algebras of locally compact Hausdorff étale groupoids. The theorem shows that, when the unit space is covered by two closed invariant subsets whose complements are disjoint open invariant subsets, the corresponding groupoid C^*-algebras form a pullback diagram in the category of 𝕋-C^*-algebras and 𝕋-equivariant *-homomorphisms, for the gauge actions induced by a ℤ-valued cocycle and its restrictions. We then prove a collection of boundary-path decomposition theorems for graphs, relative graphs, and topological graphs. We show that admissible decompositions of graphs, together with their analogues in the relative and topological settings, induce corresponding decompositions of boundary path spaces. Combining these decomposition theorems with the groupoid pullback theorem, we recover previously known pullback theorems for graph C^*-algebras, relative graph C^*-algebras, and topological graph C^*-algebras. Thus these pullback phenomena are explained by a single groupoid-theoretic mechanism.
ER  -

Abstract

Related

Cite