We prove a long exact sequence in KK-theory for both full and reduced amalgamated free products in the presence of conditional expectations. In the course of the proof, we established the KK-equivalence between the full amalgamated free product of two unital C*-algebras and a newly defined reduced amalgamated free product that is valid even for non GNS-faithful conditional expectations. Our results unify, simplify and generalize all the previous results obtained before by Cuntz, Germain and Thomsen.

中文翻译：

---

合并自由产品的 KK 理论

我们在 KK 理论中证明了存在条件期望的完全和减少的合并自由产品的长精确序列。在证明过程中，我们建立了两个单位 C*-代数的完全合并自由乘积与新定义的减少合并自由乘积之间的 KK 等价，该乘积即使对于非 GNS 忠实条件期望也有效。我们的结果统一、简化和概括了 Cuntz、Germain 和 Thomsen 之前获得的所有结果。