CONJUGACY OF LOCAL HOMEOMORPHISMS VIA GROUPOIDS AND C*-ALGEBRAS

The candidate reproduces the equivalence in Theorem 3.1 for second-countable Deaconu–Renault systems using the same three pillars as the paper: (i) a C0-level characterisation (Proposition 3.4), (ii) a groupoid/cocycle characterisation (Proposition 3.8), and (iii) a C*-algebraic characterisation via weighted actions plus groupoid reconstruction (Lemma 3.10 and Proposition 3.12). The paper’s statements and proofs explicitly appear in 3.4, 3.8, 3.10, 3.12 and the summary Theorem 3.1, and the candidate’s route mirrors these steps with only minor stylistic differences (e.g., an explicit “edge test” to pin down ψ on Z(X,1,0,σX(X))). One small nuance is that, where the paper requires a “separating” group in some implications, the candidate instantiates Γ = ℝ and argues the diagonal as the joint fixed-point algebra directly; this is consistent with the theorem as stated over ℝ. Overall, the arguments agree line-by-line with the paper’s logic and dependencies.