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

✍ Scribed by Rune Eliasen

• Publisher: John Wiley and Sons
• Year: 2007
• Tongue: English
• Weight: 171 KB
• Volume: 280
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200410480

✦ Synopsis

Abstract

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)