Question

Here I am referring to Kasparov's KK-theory, a bivariant functor on the category of separable C* algebras. It is well known that $KK(A, \mathbb{C})$ is K-homology and $KK(\mathbb{C}, B)$ is K-theory, both of which can be calculated for a huge collection of C* algebras (often by topological methods).

I am wondering if anybody knows how to actually calculate $KK(A,B)$ when it is not simply isomorphic to K-theory or K-homology (so I guess I also have to exclude $C_0(\mathbb{R})$. It seems that most of the time the interest in KK groups is not actually calculating them but in constructing specific elements (such as the Dirac / dual Dirac elements in proofs of the Baum-Connes conjecture).

It occurred to me that I have never actually seen anybody explicitly calculate $KK(A, B)$ in nontrivial cases. I am left wondering if this is because nobody knows how to do it or if it just isn't particularly useful to do so. Explanations or references are both appreciated.

Answer 1

There is Rosenberg-Schochet universal coefficient theorem, which says $KK(A,B)\simeq Ext(K_{\ast}(A),K_{\ast+1}(B))\oplus Hom(K_{\ast}(A),K_{\ast}(B))$ (not canonically) when $A$ is $KK$-equivalent to a commutative algebra. It was proved in The Künneth theorem and the universal coefficient theorem for Kasparov’s generalized K-functor, Duke Math. J., 55(1987), 431–474.

Answer 2

Hi Paul, I do not know if this is interesting (enough) for you, but in 'The operator K-functor and extensions of C*-algebras' Kasparov shows that for suitable A and B (I guess A,B seperable, A nuclear) there exists an isomorphism from $\mathrm{KK}(A,B)$ onto a group $\mathrm{Ext}(A,C_0(\mathbb{R})\otimes B)$ of 'stable equivalence classes of extensions' $$0\xrightarrow{}{} C_0(\mathbb{R})\otimes B\otimes\mathcal{K}\xrightarrow{}{} C \xrightarrow{}{} A \xrightarrow{}{} 0\text{ .}$$. Maybe the RHS is not concrete enough for this to qualify as an answer.