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Let M be a closed manifold and˛W 1 .M / ! U n a representation. We give a purely K-theoretic description of the associated element in the K-theory group of M with R=Z-coefficients (OE˛ 2 K 1 .M IR=Z/). To that end, it is convenient to describe the R=Z-K-theory as a relative K-theory of the unital inclusion of C into a finite von Neumann algebra B. We use the following fact: there is, associated with˛, a finite von Neumann algebra B together with a flat bundle E ! M with fibers B, such that E˛˝E is canonically isomorphic with C n˝E , where E˛denotes the flat bundle with fiber C n associated with . We also discuss the spectral flow and rho type description of the pairing of the class OE˛ with the K-homology class of an elliptic selfadjoint (pseudo)-differential operator D of order 1.

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We construct equivariant KK-theory with coefficients in R and R Z as suitable inductive limits over II1-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 KK Γ R (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 KK Γ -form of the Baum-Connes conjecture, then every Γ-algebra satisfies (KFP).If α ∶ Γ → Un is a unitary representation and A satisfies property (KFP), we construct in a canonical way a rho class ρ A α ∈ KK 1,Γ R Z (A, A). This construction generalizes the Atiyah-Patodi-Singer K-theory class with R Z coefficients associated to α.