Nuclearity for partial crossed products by exact discrete groups

We show that the partial crossed product of a commutative C∗-algebra by an exact discrete group is nuclear if the full and reduced partial crossed products coincide. This generalises a result by Matsumura for global actions. In general, we prove that a partial action of an exact discrete group on a C∗-algebra A has Exel's approximation property if and only if the full and reduced crossed products by the diagonal partial action on A⊗maxAop coincide. We apply our results to show that the weak containment property implies nuclearity in the case of semigroup C∗-algebras and C∗-algebras of separated graphs.