KK-theory - Properties

Properties

When one takes the C*-algebra C of the complex numbers as the first argument of KK as in KK(C, B) this additive group is naturally isomorphic to the K₀-group K₀(B) of the second argument B. In the Cuntz point of view, a K₀-class of B is nothing but a homotopy class of *-homomorphisms from the complex numbers to the stabilization of B. Similarly when one takes the algebra C₀(R) of the continuous functions on the real line decaying at infinity as the first argument, the obtained group KK(C₀(R), B) is naturally isomorphic to K₁(B).

An important property of KK-theory is the so-called Kasparov product, or the composition product,

,

which is bilinear with respect to the additive group structures. In particular each element of KK(A, B) gives a homomorphism of K_*(A) → K_*(B) and another homomorphism K*(B) → K*(A).

The product can be defined much more easily in the Cuntz picture given that there are natural maps from QA to A, and from B to K(H) ⊗ B which induce KK-equivalences.

The composition product gives a new category, whose objects are given by the separable C*-algebras while the morphisms between them are given by KK-groups. Moreover, any *-homomorphism of A into B induces an element of KK(A, B) and this correspondence gives a functor from the original category of the separable C*-algebras into . The approximately inner automorphisms of the algebras become identity morphisms in .

This functor is universal among the split-exact, homotopy invariant and stable additive functors on the category of the separable C*-algebras. Any such theory satisfies Bott periodicity in the appropriate sense since does.

The Kasparov product can be further generalized to the following form:

It contains as special cases not only the K-theoretic cup product, but also the K-theoretic cap, cross, and slant products and the product of extensions.