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Let $M$ be a closed manifold and $\alpha : \pi_1(M)\to U_n$ a representation.We give a purely $K$-theoretic description of the associated element $[\alpha]$in the $K$-theory of $M$ with $\R/\Z$-coefficients. To that end, it isconvenient to describe the $\R/\Z$-$K$-theory as a relative $K$-theory withrespect to the inclusion of $\C$ in a finite von Neumann algebra $B$. We usethe following fact: there is, associated with $\alpha$, a finite von Neumannalgebra $B$ together with a flat bundle $\cE\to M$ with fibers $B$, such that$E_\a\otimes \cE$ is canonically isomorphic with $\C^n\otimes \cE$, where$E_\alpha$ denotes the flat bundle with fiber $\C^n$ associated with $\alpha$.We also discuss the spectral flow and rho type description of the pairing ofthe class $[\alpha]$ with the $K$-homology class of an elliptic selfadjoint(pseudo)-differential operator $D$ of order 1.

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