Relative nuclearity for ${\mathrm{C}}^{\ast }$-algebras and $KK$-equivalences of amalgamated free products

We prove a relative analogue of equivalence between nuclearity and CPAP. In its proof, the notion of weak containment for C${}^{\ast }$-correspondences plays an important role. As an application we prove $KK$-equivalence between full and reduced amalgamated free products of C${}^{\ast }$-algebras under a strengthened variant of `relative nuclearity’.