Karoubi-Villamayor K-theory, weakly stable C ∗ -categoroids, and KK-theory Georgian

Abstract. The aim of the paper is to give, according to Karoubi–Villamayor K-groups, an interpretation of Kasparov KK-groups. It continues the study ofKK-theory by the methods ofK-theory, focusing attention on the problem posed and discussed in the author’s papers published in 2000 and 2001. But the methods used in those papers are based on the excision property and Morita invariance of algebraic and topological K-theories on the category of C∗-algebras, which are not applicable to Karoubi–Villamayor K-theory, since excision holds only for some sub class of short exact sequences of C∗-algebras. In this paper we introduce and study a weak stability property of the C∗-category Rep(A,B), which is the key to our problem