Fourier Analysis for Type III Representations of the Noncommutative Torus

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-LebesgueLemmaand 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 is an element of [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 II1 representations.