A new approach to bivariant K-theory

by 1968- Dumitras?cu, Constantin Dorin

We construct a new bivariant theory, that we call KE-theory, which is intermediate between the KK-theory of Gennadi Kasparov, and the E-theory of Alain Connes and Nigel Higson. It has an associative product, and there are natural transformations KKG ? KEG and KEG ? EG which preserve the product structures of the three theories. We obtain in this way an explicit description of the map between KK-theory and E-theory, abstractly known to exist due to their characterization using category theory. In an effort to further elucidate the relationship with the other two bivariant theories, we study some of the functoriality properties of the KE-theory groups and of the product. The thesis concludes with an example: we show that the new theory recovers ordinary K-theory. All the C?- algebras that we consider are separable, graded, and admit an action of a locally compact ?-compact Hausdorff group. The thesis adviser was Prof. Nigel Higson.