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

TL;DR

The paper provides a $K$-theoretic framework to describe elements associated with group representations in the $K$-theory of manifolds with $ /z$-coefficients, using von Neumann algebras and flat bundles.

Contribution

It introduces a purely $K$-theoretic description of representation classes in $K$-theory with $ /z$-coefficients via von Neumann algebra techniques.

Findings

Describes the $K$-theoretic element associated with a representation using von Neumann algebras.

Connects the $K$-theory classes with spectral flow and rho invariants.

Provides a new perspective on pairing $K$-theory classes with elliptic operators.

Abstract

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.