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.
