| Résume | Principal symbols of pseudodifferential operators can be approached in a C*-algebraic manner, through which classical notions like the cosphere bundle and geodesic flow can be generalised to noncommutative geometry. Based on joint work with Edward McDonald, I will show how this can be connected to Quantum Ergodicity via spectral truncations of spectral triples - connecting to work by Connes and Van Suijlekom. Finally, I will consider the case of non-compact manifolds, where the pseudodifferential calculus of scattering operators by Melrose provides a good framework for an extension of Connes' trace theorem. This latter result is a work in progress in collaboration with Galina Levitina, Edward McDonald, Fedor Sukochev and Dmitriy Zanin. |
