# Fourier analysis for type III representations of the noncommutative torus

## TL;DR

This paper develops Fourier analysis tools for type III representations of the noncommutative 2-torus, extending classical harmonic analysis results to a noncommutative setting with modular structures.

## Contribution

It introduces Fourier transforms for type III representations of the noncommutative torus and proves analogues of classical theorems like Riemann-Lebesgue and Hausdorff-Young in this context.

## Key findings

- Established noncommutative Riemann-Lebesgue Lemma
- Proved Hausdorff-Young inequality for the Fourier transform
- Demonstrated $L^p$ convergence of Cesaro means and Poisson kernel inversion

## Abstract

For the noncommutative 2-torus, we define and study Fourier transforms arising from representations of states with central supports in the bidual, exhibiting a possibly nontrivial modular structure (i.e. type III representations). We then prove the associated noncommutative analogous of Riemann-Lebesgue Lemma and Hausdorff-Young Theorem. In addition, the $L^p$- convergence result of the Cesaro means (i.e. the Fejer theorem), and the Abel means reproducing the Poisson kernel are also established, providing inversion formulae for the Fourier transforms in $L^p$ spaces, $p\in[1,2]$. Finally, in $L^2(M)$ we show how such Fourier transforms "diagonalise" appropriately some particular cases of modular Dirac operators, the latter being part of a one-parameter family of modular spectral triples naturally associated to the previously mentioned non type ${\rm II}_1$ representations.