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

We construct equivariant $KK$-theory with coefficients in $\mathbb{R}$ and $\mathbb{R}/\mathbb{Z}$ 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 $K$-theoretically free and proper (KFP) if the group trace ${\bf tr}$ of $\Gamma$ acts as the unit element in $KK^{\Gamma}_{\mathbb{R}}(A,A)$. 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 $\rho_\alpha^A\in KK_{\mathbb{R}/\mathbb{Z}}^{1,\Gamma}(A,A)$. This construction generalizes the Atiyah-Patodi-Singer $K$-theory class with $\mathbb{R}/\mathbb{Z}$ coefficients associated to $\alpha$.