Relative nuclearity for C⁎C⁎-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⁎-correspondences plays an important role. As an application we prove KK-equivalence between full and reduced amalgamated free products of C⁎-algebras under a strengthened variant of ‘relative nuclearity’.