Flat bundles, von Neumann algebras and K-theory with R/Z-coefficients

Let M be a closed manifold and α: π 1 (M) → Un a representation. We give a purely K-theoretic description of the associated element in the K-theory group of M with R/Z-coefficients ([α] ε K 1 (M; R/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 α, a finite von Neumann algebra B together with a flat bundle ε→ M with fibers B, such that Eα ε is canonically isomorphic with Cnε, 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.