27 jan 2016 -- 16:00
Aula D'Antoni, Dip.Matematica, Università "Tor Vergata", Roma
Abstract.
A construction of Atiyah, Patodi and Singer (APS) associates an odd K-theory class with coefficients in $\mathbb{R}/\mathbb{Z}$ to any unitary flat bundle on a compact manifold. The pairing of this K-class with the K-homology class represented by a selfadjoint differential operator is computed by the Index theorem for flat bundles as the $\mathbb{R}/\mathbb{Z}$ class of the rho invariant, a spectral invariant of the operator. In this seminar we illustrate how von Neumann algebras may be used to construct a model of K-theory with coefficients (in $\mathbb{R}$ and $\mathbb{R}/\mathbb{Z}$) for $C^*$ algebras, we give a canonical operator algebraic construction of the APS class and we compute its paring with elliptic operators as a Kasparov product. If time permits we will extend these results to study actions of discrete groups on noncommutative $C^*$ algebras.