Equivariant Kasparov theory and generalized homomorphisms

Let G be a locally compact group. We describe elements of KK^G (A,B) by equivariant homomorphisms, following Cuntz's treatment in the non-equivariant case. This yields another proof for the universal property of KK^G: It is the universal split exact stable homotopy functor. To describe a Kasparov triple (E, phi, F) by an equivariant homomorphism, we have to arrange for the Fredholm operator F to be equivariant. This can be done if A is of the form K(L^2G) otimes A' and more generally if the group action on A is proper in the sense of Rieffel and Exel.Comment: 22 pages, final version, will appear in K-Theory added references and a few additional explanations to the tex

Generalized fixed point algebras and square-integrable group actions

We analzye Rieffel's construction of generalized fixed point algebras in the setting of group actions on Hilbert modules. Let G be a locally compact group acting on a C*-algebra B. We construct a Hilbert module F over the reduced crossed product of G and B, using a pair (E, R), where E is an equivariant Hilbert module over B and R is a dense subspace of E with certain properties. The generalized fixed point algebra is the C*-algebra of compact operators on F. Any Hilbert module over the reduced crossed product arises by this construction for a pair (E, R) that is unique up to isomorphism. A necessary condition for the existence of R is that E be square-integrable. The consideration of square-integrable representations of Abelian groups on Hilbert space shows that this condition is not sufficient and that different choices for R may yield different generalized fixed point algebras. If B is proper in Kasparov's sense, there is a unique R with the required properties. Thus the generalized fixed point algebra only depends on E.Comment: 19 page

Categorical aspects of bivariant K-theory

This survey article on bivariant Kasparov theory and E-theory is mainly intended for readers with a background in homotopical algebra and category theory. We approach both bivariant K-theories via their universal properties and equip them with extra structure such as a tensor product and a triangulated category structure. We discuss the construction of the Baum-Connes assembly map via localisation of categories and explain how this is related to the purely topological construction by Davis and Lueck