Fourier analysis for type III representations of the noncommutative\n torus

2019/03/15 by Francesco Fidaleo, Fidaleo, Francesco
Mathematics · #43A99 #46L36 #46L51 #46L65 #46L87 #58B34 #81R60 #Advanced Algebra and Geometry #Advanced Operator Algebra Research #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #Operator Algebras (math.OA) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1903.06710

openalex publication_date 2019/03/15 · openalex created_date 2022/07/29 · openalex updated_date 2026/07/28

Abstract

For the noncommutative 2-torus, we define and study Fourier transforms\narising from representations of states with central supports in the bidual,\nexhibiting a possibly nontrivial modular structure (i.e. type III\nrepresentations).\n We then prove the associated noncommutative analogous of Riemann-Lebesgue\nLemma and Hausdorff-Young Theorem. In addition, the L^p- convergence result\nof the Cesaro means (i.e. the Fejer theorem), and the Abel means reproducing\nthe Poisson kernel are also established, providing inversion formulae for the\nFourier transforms in L^p spaces, p\∈[1,2].\n Finally, in L²(M) we show how such Fourier transforms "diagonalise"\nappropriately some particular cases of modular Dirac operators, the latter\nbeing part of a one-parameter family of modular spectral triples naturally\nassociated to the previously mentioned non type rm II₁ representations.\n

Related

• type III representations and modular spectral triples for the noncommutative torus
• Solvmanifolds and noncommutative tori with real multiplication
• Bargmann-Fock realization of the noncommutative torus
• Fourier transform for D-algebras
• Spectral Geometry