# Flat bundles, von Neumann algebras and $K$-theory with $R-$-coefficients

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1 IMJ - Institut de Mathématiques de Jussieu

Abstract : Let $M$ be a closed manifold and $\alpha : \pi 1M\to U n$ 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 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 $\alpha$, a finite von Neumann algebra $B$ 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 of the class $\alpha$ with the $K$-homology class of an elliptic selfadjoint pseudo-differential operator $D$ of order $1$.

keyword : R-Z K-theory von Neumann algebras flat bundles

Author: ** Paolo Antonini - Sara Azzali - Georges Skandalis - **

Source: https://hal.archives-ouvertes.fr/