Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Request Permissions   Purchase Content 
 

 

The canonical trace and the noncommutative residue on the noncommutative torus


Authors: Cyril Lévy, Carolina Neira Jiménez and Sylvie Paycha
Journal: Trans. Amer. Math. Soc. 368 (2016), 1051-1095
MSC (2010): Primary 58B34, 58J42
DOI: https://doi.org/10.1090/tran/6369
Published electronically: July 9, 2015
MathSciNet review: 3430358
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 58B34, 58J42

Retrieve articles in all journals with MSC (2010): 58B34, 58J42


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
Email: paycha@math.uni-potsdam.de

DOI: https://doi.org/10.1090/tran/6369
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
Article copyright: © Copyright 2015 American Mathematical Society