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 !