In mathematics, KK-theory is a common generalization both of K-homology and K-theory as an additive bivariant functor on separable C*-algebras.
It was influenced by Atiyah's concept of Fredholm modules for the Atiyah–Singer index theorem, and the classification of extensions of C*-algebras by Lawrence G. Brown, Ronald G. Douglas, and Peter Arthur Fillmore in 1977.
KK-theory was followed by a series of similar bifunctor constructions such as the E-theory and the bivariant periodic cyclic theory, most of them having more category-theoretic flavors, or concerning another class of algebras rather than that of the separable C*-algebras, or incorporating group actions.
It becomes an abelian group under the direct sum operation of bimodules as the addition, and the class of the degenerate modules as its neutral element.
There are various, but equivalent definitions of the KK-theory, notably the one due to Joachim Cuntz[3] that eliminates bimodule and 'Fredholm' operator F from the picture and puts the accent entirely on the homomorphism ρ.
More precisely it can be defined as the set of homotopy classes of *-homomorphisms from the classifying algebra qA of quasi-homomorphisms to the C*-algebra of compact operators of an infinite dimensional separable Hilbert space tensored with 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).
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 that induce KK-equivalences.