The ideal intersection property for essential groupoid C∗-algebras

The paper’s main theorem explicitly states that, for étale groupoids with locally compact Hausdorff unit space, C*_ess(G) is simple if and only if G is minimal and has no essentially confined amenable sections of isotropy, under the mild hypotheses that G is Hausdorff, or σ-compact, or has compact unit space (Theorem A; proved via Theorems 7.2 and 7.10) . The candidate solution reproduces this equivalence by (i) reducing simplicity to minimality plus the ideal–intersection property (IIP), and (ii) using the paper’s characterisation of IIP in terms of the absence of essentially confined amenable isotropy sections (Theorem 6.1 and Theorems 7.2/7.10) . Minor differences include an unnecessary invocation of topological transitivity and some nonstandard notation around the local conditional expectation, but these do not affect correctness. Net: both the paper and model give the same result by essentially the same route.