In my opening lecture, I explained how the classical notions of 1) measure theory ; 2) topology of locally compact spaces and 3) de Rham current theory are replaced in the non-commutative case by 1) von Neumann algebras ; 2) C*-algebras ; 3) cyclic cohomology. I followed a similar plan in my lecture, and for lack of time I was only able to deal with 1) and 2). I therefore started with the modular theory of von Neumann algebras. As this was not an audience of specialists, my aim in this part of the lecture was to deal entirely with modular theory, with all demonstrations, in the case where the von Neumann algebra M is constructed from a geometric situation : action of a group by diffeomorphisms or a laminated variety. In this case, we can explicitly describe the generic element of M, as well as all faithful normal states on M, so that we can directly calculate all modular automorphism groups and verify all results of modular theory. In the second part of the lecture I showed how the C* algebra associated with a leafing is introduced in a necessary way when studying the spectral properties of differential operators on noncompact varieties leaves of a compact variety leafing. I then introduced the longitudinal Euler class of an arbitrary leaflet as an element of K0 (C* (V, F)) and showed its invariance by longitudinal homotopy. The lecture ended with the longitudinal index theorem [with values in K0 (C* (V, F))] and its corollary for laminations with transverse measure.
Lecture