Séminaire Physique mathématique ICJ

The class of a flat vector bundle and a noncommutative generalisation

par Sara Azzali (IFM, Potsdam)

Europe/Paris
Fokko du Cloux (Institut Camille Jordan)

Fokko du Cloux

Institut Camille Jordan

Université Lyon 1, Bât. Braconnier, 21 av. Claude Bernard, 69100 Villeurbanne
Description
Let G be the fundamental group of a closed manifold X and  \alpha: G —> U(n) a finite dimensional unitary representation, i.e. a flat unitary vector bundle over X.  To these data, Atiyah, Patodi and Singer associated a class [\alpha] in the K-theory group with R/Z-coefficients and investigated its relation to spectral rho invariants.  In this talk, we take an operator algebraic point of view on the class [\alpha] and generalise it to the noncommutative setting of a discrete group G suitably acting on a C^*-algebra A. The condition on the action is encoded by KK-theory with real coefficients, which will be introduced, and can be called K-theoretical free and properness.  We exhibit natural classes of G-algebras satisfying this property. Based on joint work with Paolo Antonini and Georges Skandalis.