Kasparov’s KK-theory which is based on so called Kasparov-modules can be described much more algebraically using universal extensions and classifying maps for extensions. While Kasparov’s formalism is ideally suited for applications to problems such as the Baum-Connes conjecture, the algebraic approach also has its virtues.
In the C*-algebra framework it leads in many cases to simple arguments in computations, in proving key properties of KK-theory (e.g. Bott periodicity, long exact sequences, Thom isomorphism, Pimsner-Voiculescu sequence, K-theory for free group C*-algebras etc.) and notably in determining index maps. Moreover however, since the approach is essentially algebraic, it can be used to define and study versions of KK-theory for categories of algebras other than C*-algebras. These versions have nice properties and are computable just as Kasparov’s theory. Notably such a theory with good properties exists for locally convex algebras. This covers for instance algebras of smooth functions on manifolds or algebras of pseudodifferential operators – extending in particular topological K-theory for such algebras to the corresponding K-homology and bivariant theories. Particularly important versions of such a theory admit a natural bivariant Chern-Connes character into bivariant periodic cyclic homology (this character is in fact is a rational isomorphism for a large class of algebras). The bivariant K-groups are defined and can be computed also for purely algebraic algebras such as ‘Leavitt path algebras’ or even for more exotic algebras such as the Weyl algebra and algebras of differential operators.