Nuclearity for partial crossed products by exact discrete groups

The paper’s main theorem states: for a partial action α of an exact discrete group G on a commutative C*-algebra A, if the full and reduced partial crossed products coincide (Aoα G = Aoα,r G), then Aoα G is nuclear, or equivalently α has Exel’s approximation property (AP) . The paper defines AP for Fell bundles and notes that nuclearity of the full cross-sectional algebra implies AP . It also provides a general diagonal characterisation of AP for exact groups (Theorem 4.13) and proves the key cp-map/expectation step (Corollary 4.9) used to deduce nuclearity under the weak containment hypothesis . The candidate solution leverages precisely these ingredients in a different order: it cites Buss–Ferraro–Sehnem to get AP from Aoα G = Aoα,r G in the commutative, exact case, and then invokes the classical Exel–Ng result that AP plus nuclear unit fibre (here A is commutative, hence nuclear) implies nuclearity of the full cross-sectional C*-algebra. This is consistent with the paper’s equivalence nuclearity ⇔ AP in this setting. The only minor issue is a side remark suggesting that diagonal full=reduced follows directly from full=reduced for A using exactness; the paper does not assert that implication and instead proves a distinct diagonal characterisation of AP (Theorem 4.13) . Overall, the paper’s argument is correct, and the model’s solution is also correct, following a standard AP→nuclearity route.