2404.15139
Ideals of étale groupoid algebras with coefficients in a sheaf with applications to topological dynamics
Gilles G. de Castro, Daniel Gonçalves, Benjamin Steinberg
correcthigh confidence
- Category
- Not specified
- Journal tier
- Specialist/Solid
- Processed
- Sep 28, 2025, 12:56 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper proves (i) every primitive ideal of Γc(G,O) is Ann(Indx(Mx)) for some stalk Mx (Theorem 4.1) and (ii) if each isotropy skew group ring Bx is left max, then primitive ideals are exactly Ann(Indx(M)) for simple Bx-modules M (Theorem 4.3); it also shows induction preserves simplicity (Theorem 4.2) and uses Theorem 3.5 (ideals are intersections of induced annihilators) together with the Disintegration Theorem 2.6 to pass between modules and sheaves . The candidate solution reaches the same classification, but via an adjunction-based route (Frobenius reciprocity and a counit map) that is not explicitly developed in the paper; it also states an equivalent annihilator criterion in bimodule form (matching Lemma 3.4). Thus, the results agree, while the proof strategies differ.
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions \textbf{Journal Tier:} specialist/solid \textbf{Justification:} The work extends Effros–Hahn-type classifications to groupoid convolution algebras with sheaf coefficients, providing a coherent ideal-theoretic picture with applications. Proofs rely on established machinery and are carefully executed. Minor expository enhancements would make the logical flow between induction, disintegration, and the primitive ideal results even clearer.