Bivariant K-theory with R/Z-coefficients and rho classes of unitary representations

We construct equivariant $KK$-theory with coefficients in $\R$ and $\R/\Z$ as suitable inductive limits over ${\rm II}_1$-factors. We show that the Kasparov product, together with its usual functorial properties, extends to $KK$-theory with real coefficients.

Let $\Gamma$ be a group. We define a $\Gamma$-algebra $A$  to be \emph{$K$-theoretically free and proper} (KFP) if  the group trace ${\bf tr}$ of $\Gamma$ acts as the unit element in $KK^\Gamma_\R(A,A)$. We show that free and proper $\Gamma$-algebras (in the sense of Kasparov) have the (KFP) property. Moreover, if $\Gamma$ is torsion free and satisfies the $KK^\Gamma$-form of the Baum-Connes conjecture, then every $\Gamma$-algebra satisfies (KFP). 

If $\alpha:\Gamma\to U_n$ is a unitary representation and $A$ satisfies property (KFP), we construct in a canonical way a rho class $\rho_\alpha^A\in KK_{\R/\Z}^{1,\Gamma}(A,A)$. This construction generalizes the Atiyah-Patodi-Singer $K$-theory class with $\R/\Z$ coefficients associated to $\alpha$.