  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 $\alpha$.
