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

We construct equivariant KK  -theory with coefficients in RR and R/ZR/Z as suitable inductive limits over II1II1-factors. We show that the Kasparov product, together with its usual functorial properties, extends to KK-theory with real coefficients.Let Γ be a group. We define a Γ-algebra A to be K-theoretically free and proper (KFP) if the group trace tr of Γ acts as the unit element in KKRΓ(A,A). We show that free and proper Γ-algebras (in the sense of Kasparov) have the (KFP) property. Moreover, if Γ is torsion free and satisfies the KKΓKKΓ-form of the Baum–Connes conjecture, then every Γ-algebra satisfies (KFP).If α:Γ→Unα:Γ→Un is a unitary representation and A   satisfies property (KFP), we construct in a canonical way a rho class ραA∈KKR/Z1,Γ(A,A). This construction generalizes the Atiyah–Patodi–Singer K  -theory class with R/ZR/Z-coefficients associated to α.