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

Abstract

Key words C * -algebras, KK-theory, amalgamated free products, absorbing homomorphisms, countable direct sums of matrix algebras MSC (2000) 19K35 J. Cuntz has conjectured the existence of two cyclic six terms exact sequences relating the KK-groups of the amalgamated free product A1 * B A2 to the KK-groups of A1, A2 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 A1 and A2 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 UM(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 A1 * B A2 is split.

KK-equivalence for amalgamated free product C*-algebras

“…As we mentioned in the introduction, our KKequivalence and K-nuclearity results with Thomsen's result [17] imply the following: Note that the second exact sequence of KK-groups is new even in the full case. We also obtain the next corollary from Theorem A and [4]. Finally, we would like to point out that a similar result holds for HNN extensions.…”