The noncommutative geometry of the Landau Hamiltonian: differential aspects

In this work we study the differential aspects of noncommutative geometry for magnetic $C^*$-algebra which is a 2-cocycle deformation group $\mathbb{R}^2$. This algebra intimately related to Quantum Hall Effect in continuous, and our results aim provide new geometric interpretation Kubo's formula. Taking inspiration from ideas developed by Bellissard during 80's, build an appropriate Fredholm module based on Dirac operator square root (\`a la Dirac) quantum harmonic oscillator. Our main result consist establishing important piece Bellissard's theory, so-called second Connes' order do so, establish equality three cyclic 2-cocycles defined dense subalgebra $C^*$-algebra. Two these are literature quantized calculus, with use Dixmier trace operator.