The KK-theory of amalgamated free products

In the presence of conditional expectations, we prove a long exact sequence in KK-theory for both the maximal and the vertex reduced amalgamated free product of unital C*-algebras that is valid even for non GNS-faithful conditional expectations. However, in the degenerated case, one has to introduce a new reduced amalgamated free product, that we call vertex-reduced. In the course of the proof we established the KK-equivalence between the full amalgamated free product and the vertex-reduced amalgamated free product. These results generalize and simplify the results obtained before by Germain and Thomsen. When the conditional expectations are extremely degenerated, i.e. when they are *-homomorphisms, our vertex-reduced amalgamated free product is isomorphic to the fiber direct sum. Hence our results also generalize a result of Cuntz. (C) 2020 Elsevier Inc. All rights reserved.