Fourier Analysis for Type III Representations of the Noncommutative Torus

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 $\mathrm{{III}}$ 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 $\mathrm{{II_1}}$ representations.