The noncommutative geometry of the Landau Hamiltonian: Differential aspects

We develop the differential aspects of a noncommutative geometry for the Quantum Hall Effect in the continuous, with the ambition of proving Kubos formula. Taking inspiration from the ideas developed by Bellissard during the 80s we build a Fredholm module for the $C^*$-algebra of continuous magnetic operators, based on a Dirac operator closely related to the quantum harmonic oscillator. An important piece of Bellissards theory (the so-called second Connes formula) is proved. This work provides the continuation of the recent article [DS].