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.

Abstract. We relate the spectral flow to the index for paths of selfadjoint Breuer-Fredholm operators affiliated to a semifinite von Neumann algebra, generalizing results of Robbin-Salamon and Pushnitski. Then we prove the vanishing of the von Neumann spectral flow for the tangential signature operator of a foliated manifold when the metric is varied. We conclude that the tangential signature of a foliated manifold with boundary does not depend on the metric. In the Appendix we reconsider integral formulas for the spectral flow of paths of bounded operators.

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.

We revisit traces of holomorphic families of pseudodifferential operators on a closed manifold in view of geometric applications. We then transpose the corresponding analytic constructions to two different geometric frameworks; the noncommutative torus and Hilbert modules. These traces are meromorphic functions whose residues at the poles as well as the constant term of the Laurent expansion at zero (the latter when the family at zero is a differential operator) can be expressed in terms of Wodzicki residues and extended Wodzicki residues involving logarithmic operators. They are therefore local and contain geometric information. For holomorphic families leading to zeta regularised traces, they relate to the heat-kernel asymptotic coefficients via an inverse Mellin mapping theorem. We revisit Atiyah's L 2index theorem by means of the (extended) Wodzicki residue and interpret the scalar curvature on the noncommutative two torus as an (extended) Wodzicki residue.

Let M be a closed manifold and α ∶ π1(M ) → Un a representation. We give a purely K-theoretic description of the associated element [α] in the K-theory of M with R Z-coefficients. To that end, it is convenient to describe the R Z-K-theory as a relative K-theory with respect to the inclusion of C in a finite von Neumann algebra B. We use the following fact: there is, associated with α, a finite von Neumann algebra B together with a flat bundle E → M with fibers B, such that Eα ⊗ E is canonically isomorphic with C n ⊗ E , where Eα denotes the flat bundle with fiber C n associated with α. We also discuss the spectral flow and rho type description of the pairing of the class [α] with the K-homology class of an elliptic selfadjoint (pseudo)-differential operator D of order 1.