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THE CANONICAL TRACE AND THE NONCOMMUTATIVE RESIDUE ON THE NONCOMMUTATIVE TORUS

  • 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 zeroUsing 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.show moreshow less

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Metadaten
Author:Cyril Levy, Carolina Neira Jimenez, Sylvie Paycha
DOI:https://doi.org/10.1090/tran/6369
ISSN:0002-9947
ISSN:1088-6850
Parent Title (English):Transactions of the American Mathematical Society
Publisher:American Mathematical Soc.
Place of publication:Providence
Document Type:Article
Language:English
Year of first Publication:2016
Year of Completion:2016
Release Date:2020/03/22
Volume:368
Pagenumber:45
First Page:1051
Last Page:1095
Organizational units:Mathematisch-Naturwissenschaftliche Fakultät / Institut für Mathematik
Peer Review:Referiert