On the descent homomorphsim of Kasparov equivariant KK theory

Question

Hello,

I have recently read about the construction of the descent map in Kasparov KK theory, which, for a group $G$ and two $G$-equivariant $C^*$ algebra $A$ and $B$ send $KK_i^G(A,B)$ to $KK_i(A \rtimes_{red} G, B \rtimes_{red} G)$.

I havn't checked all the detail, but it seems to me that the same construction should also give a map from $KK_i^G(A,B)$ to $KK_i(A \rtimes_{max} G, B \rtimes_{max} G)$ : One can also construct bimodule over $( . \rtimes_{max} G)$ from a $G$-equivariant bimodule. But as this one is never mentioned anywhere, I guess something goes wrong with it.

My question is "why this second map does not appears in literature ?"

Is the construction of the application impossible, and then what exactly goes wrong ? Or the construction is possible but this map doesn't bring more information than the classical descent homomorphism ? Or the map exists, is interesting but simply wasn't needed in the application as we have no idea of what is the $K$-theory of $A \rtimes_{max} G$ when it is different from $A \rtimes_{red} G$ ?

Note : I'm mostly interested in the case where $G$ is a discrete (countable) group.

Thank you !

Answer 1

This descent map sometimes appears in the literature, see e.g. my contribution to:

Mislin, Guido & Valette, Alain (2003), Proper Group Actions and the Baum-Connes Conjecture, Basel: Birkäuser, ISBN 0-8176-0408-1.

One reason to consider it, is that the full crossed product has better functoriality properties than the reduced crossed product, as can be seen already for group $C^*$-algebras (the full group $C^*$-algebra is functorial for arbitrary group homomorphisms).

There are some points in the construction which are easier for reduced crossed products, e.g. those dealing with positivity of scalar products in Hilbert $C^*$-modules; I make a remark on that on p.95 of the above-mentioned book.

Answer 2

The descent homomorphisms were both proved for the reduced crossed product as well as for the universal (maximal) crossed product in Kasparov's paper in Inventiones Math. from 1988.

As you have recognized, the proofs are indeed quite similar.