LEFT REGULAR REPRESENTATIONS OF GARSIDE CATEGORIES I. C*-ALGEBRAS AND GROUPOIDS

The paper’s Theorem 6.25 proves an inclusion-preserving bijection between closed I_C⋉Ω-invariant subspaces and admissible, C-invariant, max∞_S-closed pairs (T,D); it proceeds via (i) representing points of Ω by S-normal words with uniqueness, (ii) the maps X ↦ (T(X),D(X)) and (T,D) ↦ –(T,D), and (iii) a compactness/normal-form argument to show these maps are inverses. The candidate solution follows the same structure and uses the same key ingredients (normal forms, greedy head/normalization, compactness, and the max∞-closure criterion). Its only discrepancies are minor notational choices (writing S; for the left inverse hull and calling Ω “tight/cover-respecting” at once). Core logical steps and hypotheses align with the paper’s statements and lemmas, including Lemma 6.17 (normal-word representation and uniqueness), Lemma 6.22 (equivalence of admissibility/invariance/closure under the maps), Proposition 6.14 (greedy head), and Theorem 6.25 itself. Therefore, both are correct and substantially the same proof. See Theorem 6.25 for the classification, which is exactly as the model states ; uniqueness and representation by normal words ; the definition of (T(X),D(X)) and the characterization of admissibility/invariance/max∞-closedness via X ; and the greedy head/normalization facts used to propagate invariance .