The ideal property, the projection property, continuous fields and crossed products

Let A be the C∗-algebra associated to an arbitrary continuous field of C∗-algebras. We give a necessary and sufficient condition for A to have the ideal property and, if moreover A is separable, we give a necessary and sufficient condition for A to have the projection property. Some applications of these results are given. We also prove that “many” crossed products of commutative C∗-algebras by discrete, amenable groups have the projection property, generalizing some of our previous results.