With a flat unitary vector bundle E_a over a closed manifold M one can associate a class a in the K-theory of M with R/Z-coefficients. This class encodes the fact that a flat bundle admits a multiple kE_a which is trivial, and measures the difference between the flat and the trivial connections on kE_a.
The construction of the class a is due to by Atiyah, Patodi and Singer, who investigated it in connection with spectral invariants of Dirac operators: they proved that the pairing of a with the K-homology class represented by a Dirac operator gives the value (modulo Z) of the relative eta invariant of D.
In this talk, we start by the description of the K-theory groups with real and with R/Z-coefficients, which can be done by means of von Neumann algebras. In fact, the R/Z-K-theory is a relative construction, with respect to the inclusion of the complex numbers into (any) II_1-factor. In this model, we give a canonical construction of the Atiyah-Patodi-Singer class a associated with a flat bundle.
Inbetween the ingredients of the construction, Atiyahs L2 index theorem for coverings plays an interesting role, and suggests a generalization of the class a for certain actions of a discrete group on a noncommutative C*-algebra.
This is joint work with Paolo Antonini and Georges Skandalis.