Fourier Analysis for Type III Representations of the Noncommutative Torus

  • Francesco FidaleoEmail author
Research Article


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.


Noncommutative harmonic analysis Noncommutative geometry Noncommutative torus Type III representations Noncommutative measure theory Modular spectral triples 

Mathematics Subject Classification

43A99 46L36 46L51 46L65 46L87 58B34 



The author acknowledges the financial support of Italian INDAM-GNAMPA. The present project is part of “MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006”. He is grateful to the referees for a very careful reading of the manuscript, and for several suggestions which contribute to improve the presentation of the paper.


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Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di Roma Tor VergataRomeItaly

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