We discuss the explicit construction of non type \(II_1\) representations and relative modular spectral triples of the noncommutative 2-torus \(A_\alpha\), provided α is a (special kind of) Liouville number, where the nontrivial modular

structure plays a crucial role.

For such representations, we briefly discuss the appropriate Fourier analysis, by proving the analogous of many of the classical known theorems in Harmonic Analysis such as the Riemann-Lebesgue Lemma, the Hausdorff-Young Theorem, and the \(L^p\)-convergence results such as these associated to the Cesaro means (i.e. the Fejer theorem) and the Abel means reproducing the Poisson kernel. We show how those Fourier transforms ”diagonalise” appropriately some examples of the Dirac operators associated to the previous mentioned spectral triples.

Finally, we outline the description (which is contained in a work in progress) of a deformed generalisation of ”Fredholm module”, i.e. a suitably deformed commutator of the ”phase” of the involved Dirac operator with elements of a subset which, depending on the cases under consideration, is either a dense *-algebra or an essential operator system, of the (representation of the) \(C^*\)-algebra associated to the quantum manifold under consideration. This definition of deformed Fredholm module is new even in the usual cases associated to a trace and could provide other, hopefully interesting, applications.

The present talk is based on the following papers:

(1) F. Fidaleo and L. Suriano: Type III representations and modular spectral triples for the noncommutative torus, J. Funct. Anal. 275 (2018), 1484-1531.

(2) F. Fidaleo: Fourier analysis for type III representations of the noncommutative torus, J. Fourier Anal. Appl. 25 (201), 2801-2835.

(3) F. Ciolli and F. Fidaleo: Type III modular spectral triples and deformed Fredholm modules, work in preparation.