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

Abstract

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'.