Kasparov product

Summary

The Kasparov product is a bilinear, associative pairing $KK(A, D) \times KK(D, B) \to KK(A, B)$ in KK-theory. It composes generalized morphisms, so that separable C*-algebras with the groups $KK(A, B)$ as morphisms form an additive category. It contains as special cases the pairing of K-theory with K-homology, the index of elliptic operators twisted by bundles, and the composition of $*$-homomorphisms. Its existence is the main technical result of Kasparov's theory.