KK-theory of A-valued semi-circular systems

Equivariant KK-theory and noncommutative index theory

The Universal Property of Equivariant Kk-theory

Let G be a locally compact, σ-compact group. We prove that the equivariant KK-theory, KK, is the universal category for functors from G-algebras to abelian groups which are stable, homotopy invariant and split-exact. This is a generalization of Higsons characterisation of (non-equivariant) KK-theory.

Algebraic K-theory of Fredholm Modules and Kk-theory

Kasparov KK-groups KK(A,B) are represented as homotopy groups of the Pedersen-Weibel nonconnective algebraic K-theory spectrum of the additive category of Fredholm (A,B)-bimodules for A and B, respectively, a separable and σ-unital trivially graded real or complex C∗-algebra acted upon by a fixed compact metrizable group.

Axiomatic Kk-theory for Real C*-algebras

We establish axiomatic characterizations of K-theory and KK-theory for real C*-algebras. In particular, let F be an abelian group-valued functor on separable real C*-algebras. We prove that if F is homotopy invariant, stable, and split exact, then F factors through the category KK. Also, if F is homotopy invariant, stable, half exact, continuous, and satisfies an appropriate dimension axiom, th...

L-index theorems, KK-theory, and connections

Let M be a compact manifold. and D a Dirac type differential operator on M . Let A be a C ∗-algebra. Given a bundle W of A-modules over M (with connection), the operator D can be twisted with this bundle. One can then use a trace on A to define numerical indices of this twisted operator. We prove an explicit formula for this index. Our result does complement the Mishchenko-Fomenko index theorem...