The canonical trace and the noncommutative residue on the noncommutative torus
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- by Cyril Lévy, Carolina Neira Jiménez and Sylvie Paycha PDF
- Trans. Amer. Math. Soc. 368 (2016), 1051-1095 Request permission
Abstract:
Using a global symbol calculus for pseudodifferential operators on tori, we build a canonical trace on classical pseudodifferential operators on noncommutative tori in terms of a canonical discrete sum on the underlying toroidal symbols. We characterise the canonical trace on operators on the noncommutative torus as well as its underlying canonical discrete sum on symbols of fixed (resp. any) noninteger order. On the grounds of this uniqueness result, we prove that in the commutative setup, this canonical trace on the noncommutative torus reduces to Kontsevich and Vishik’s canonical trace which is thereby identified with a discrete sum. A similar characterisation for the noncommutative residue on noncommutative tori as the unique trace which vanishes on trace–class operators generalises Fathizadeh and Wong’s characterisation in so far as it includes the case of operators of fixed integer order. By means of the canonical trace, we derive defect formulae for regularized traces. The conformal invariance of the $\zeta$–function at zero of the Laplacian on the noncommutative torus is then a straightforward consequence.References
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Additional Information
- Cyril Lévy
- Affiliation: Institut für Mathematik, Am Neuen Palais 10, Universität Potsdam, 14469 Potsdam, Germany
- Address at time of publication: Centre Universitaire Jean-François Champollion, Place Verdun 81000 Albi, France – and – Institut de Mathématiques de Toulouse, 118 route de Narbonne 31062 Toulouse Cedex 9, France
- Email: levy@math.uni-potsdam.de, cyril.levy@univ-jfc.fr
- Carolina Neira Jiménez
- Affiliation: Fakultät für Mathematik, Universitätsstrasse 31, University of Regensburg, 92040 Regensburg, Germany
- Address at time of publication: Departamento de Matemáticas, Universidad Nacional de Colombia, Carrera 30 #45 - 03, Bogotá, Colombia
- Email: Carolina.neira-jimenez@mathematik.uni-regensburg.de, cneiraj@unal.edu.co
- Sylvie Paycha
- Affiliation: Institut für Mathematik, Am Neuen Palais 10, Universität Potsdam, 14469 Potsdam, Germany
- MR Author ID: 137200
- Email: paycha@math.uni-potsdam.de
- Received by editor(s): March 12, 2013
- Received by editor(s) in revised form: July 20, 2013, July 22, 2013, and December 14, 2013
- Published electronically: July 9, 2015
- © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 1051-1095
- MSC (2010): Primary 58B34, 58J42
- DOI: https://doi.org/10.1090/tran/6369
- MathSciNet review: 3430358