A new uniqueness theorem for the tight C*-algebra of an inverse semigroup

The paper’s Theorem 3.3 states the uniqueness result for tight C*-algebras of inverse semigroups and proves it by a slight generalization of the Brown–Nagy–Reznikoff–Sims–Williams uniqueness theorem (Theorem 2.1) applied to the open subgroupoid G_tight(S_Iso) inside Iso(G_tight(S))^∘, together with Proposition 3.2 establishing density of units with full isotropy and the needed inclusions. This yields that injectivity can be tested on C*_r(G_tight(S_Iso)), i.e., on the subalgebra generated by {T_s : s ∈ SIso} (and, under weak containment, the full tight algebra version) . By contrast, the model’s argument incorrectly identifies the subalgebra generated by {T_s : s ∈ SIso} with C*_r(Iso(G)^∘) and then appeals directly to the Brown et al theorem; this identification is not valid in general (the paper explicitly positions G_tight(S_Iso) as an open subgroupoid of Iso(G)^∘ and notes they need not be equal in general), so the model’s proof requires an extra assumption it did not state and thus fails as written .