KK ‐theory of amalgamated free products of C *‐algebras

J. Cuntz has conjectured the existence of two cyclic six terms exact sequences relating the KK ‐groups of the amalgamated free product A _{1 ?? B} A ₂ to the KK ‐groups of A ₁, A ₂ and B. First we establish automatic existence of strict and absorbing homomorphisms. Then we use this result to verify the conjecture when B is a countable direct sum of matrix algebras and the embeddings of B into A ₁ and A ₂ are quasiunital. Inspired by the proof we achieve the following nice classification result: A separable C *‐algebra B is a countable direct sum of matrix algebras if and only if the unitary group of the multiplier algebra U M (B) is compact in the strict topology. Finally we prove the conjecture when the amalgamated free product has the property that any asymptotically split extension of A _{1 ?? B} A ₂ is split. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)