# Kasparov theory with real coefficients and secondary invariants

## Meeting Details

Abstract: The $KK$-theory groups with real coefficients can be useful tools to encode certain properties of discrete group actions on $C^*$-algebras. The commutative instance concerns the study of elliptic operators on Galois coverings of manifolds. Atiyah's $L^2$-index theorem computes (and constructs) the index of an equivariant elliptic operator on such covering spaces. The canonical trace on a discrete group defines a natural equivariant $KK$ class with real coefficients. Using such a class we will rephrase Atiyah’s $L^2$-index theorem for coverings and generalise it to a property of group actions on $C^*$-algebras, which can be called “$K$-theoretical free and properness of the action”. In case of such \emph{good} actions secondary classes, attached to unitary representations of the discrete group can be constructed in this broader setting. We will review these constructions which are based on joint work with Sara Azzali and Georges Skandalis.