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

TL;DR

This paper establishes a relative version of nuclearity equivalence in C*-algebras, utilizing weak containment of C*-correspondences, and demonstrates KK-equivalence between full and reduced amalgamated free products under a strengthened nuclearity condition.

Contribution

It introduces a relative analogue of nuclearity and proves KK-equivalence for amalgamated free products, advancing understanding of C*-algebraic structures.

Findings

Proved a relative analogue of nuclearity and CPAP equivalence.

Established KK-equivalence between full and reduced amalgamated free products.

Utilized weak containment for C*-correspondences in proofs.

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'.