WEYL GROUPS OF GROUPOID C*-ALGEBRAS

The paper proves that for an effective, expansive, ample étale groupoid G, the Weyl group WG is a countable discrete group (Theorem 2.1.14) by: (i) identifying WG with Aut(G) using the Aut(G)⋉Z(G) description (Theorem 1.5.7 and its corollaries), (ii) embedding Aut(G) into Aut(Bisc(G)) and showing this latter group is countable via the images of a finite generating set of compact open bisections afforded by expansiveness, and (iii) showing discreteness by isolating points with the norm inequality ∥·∥r ≥ ∥·∥∞ for characteristic functions of bisections and a 1/2–gap argument. These steps appear verbatim in the paper’s statements and proof snippets (Definition 2.1.10; Proposition 2.1.13; Theorem 2.1.14; and the norm step 1/2 > ∥φΦ(χU) − φΨ(χU)∥r ≥ ∥·∥∞) . The identification WG ≅ Aut(G) under effectiveness is established via the groupoid reconstruction isomorphism Aut(G)⋉Z(G) ≃ Aut(C*_r(G); C0(G(0))) with Aut_{C0(G(0))}(C*_r(G)) ≃ Z(G) (so the quotient is Aut(G)) and summarized in the paper’s overview . The candidate’s solution follows the same blueprint (expansiveness ⇒ finite F whose inverse-semigroup closure is a countable open basis; WG ≅ Aut(G); inject Aut(G) into automorphisms of the countable, finitely generated inverse semigroup; and the same 1/2–gap neighborhood argument). The only substantive flaw is a sign error: the candidate wrote ∥·∥r ≤ ∥·∥∞ when the needed (and used in the paper) inequality is ∥·∥r ≥ ∥·∥∞ for these indicator functions; fixing this restores the discreteness argument exactly as in the paper . With that minor correction, the model’s proof is essentially the same as the paper’s and correct.