Editing Noncommutative topology (section)

== K-theory ==
One of the major examples of this idea is the generalization of [[topological K-theory]] to noncommutative C*-algebras in the form of [[operator K-theory]].

A further development in this is a [[functor#Bifunctors|bivariant]] version of K-theory called [[KK-theory]], which has a composition product 

<math>KK(A,B)\times KK(B,C)\rightarrow KK(A,C)</math>

of which the [[ring (mathematics)|ring]] structure in ordinary K-theory is a special case. The product gives the structure of a [[Category (topology)|category]] to KK. It has been related to [[Correspondence (algebraic geometry)|correspondences]] of [[algebraic varieties]].<ref>{{citation
 | last1 = Connes | first1 = Alain | author1-link = Alain Connes
 | last2 = Consani | first2 = Caterina | author2-link = Caterina Consani
 | last3 = Marcolli | first3 = Matilde | author3-link = Matilde Marcolli
 | arxiv = math.QA/0512138
 | doi = 10.1016/j.aim.2007.03.006 | doi-access=free
 | issue = 2
 | journal = [[Advances in Mathematics]]
 | mr = 2349719
 | pages = 761–831
 | title = Noncommutative geometry and motives: the thermodynamics of endomotives
 | volume = 214
 | year = 2007}}</ref>