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Pré-Publication, Document De Travail Année : 2013

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

Paolo Antonini
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  • PersonId : 944167
Sara Azzali
  • Fonction : Auteur
  • PersonId : 944168
Georges Skandalis
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  • PersonId : 943906

Résumé

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 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 $1$.
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Dates et versions

hal-00849933 , version 1 (01-08-2013)

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Paolo Antonini, Sara Azzali, Georges Skandalis. Flat bundles, von Neumann algebras and $K$-theory with $\R/\Z$-coefficients. 2013. ⟨hal-00849933⟩
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