A characterization of KK-theory

Abstract

We characterize the KK-groups of G. G. Kasparov, along with the Kasparov product KK(A, B) × KK(B, C) → KK(A, C), from the point of view of category theory (in a very elementary sense): the product is regarded as a law of composition in a category and we show that this category is the universal one with "homotopy invariance", "stability" and "split exactness". The third property is a weakened type of half-exactness: it amounts to the fact that the KK-groups transform split exact sequences of C*-algebras to split exact sequences of abelian groups. The method is borrowed from Joachim Cuntz’s approach to KK-theory, in which cycles for KK(A, B) are regarded as generalized homomorphisms from A to B: the results follow from an analysis of the Kasparov product in this light. © 1987 by Pacific Journal of Mathematics.