Semi-finite noncommutative geometry and some applications

These notes are a summary of talks given in Shonan, Japan in February 2008 with modifications from a later series of talks at the Hausdorff Institute for Mathematics in Bonn in July 2008 and at the Erwin Schr¨odinger Institute in October 2008. The intention is to give a short discussion of recent results in noncommutative geometry (NCG) where one extends the usual point of view of [22] by replacing the bounded operators B(H) on a Hilbert space H by certain sub-algebras; namely semi-finite von Neumann algebras. These are weakly closed subalgebras of the bounded operators on a Hilbert space that admit a faithful, normal semi-finite trace. The exposition is partly historical and intended to explain where this idea came from and how it links to other developments in index theory and NCG going back over 20 years.