45671
2016
2016
eng
1051
1095
45
368
article
American Mathematical Soc.
Providence
1
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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 zero of the Laplacian on the noncommutative torus is then a straightforward consequence.
Transactions of the American Mathematical Society
10.1090/tran/6369
0002-9947
1088-6850
wos2016:2019
WOS:000366330100010
Levy, C (reprint author), Ctr Univ Jean Francois Champollion, Pl Verdun, F-81000 Albi, France., levy@math.uni-potsdam.de; cyril.levy@univ-jfc.fr; paycha@math.uni-potsdam.de
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2020-03-22T20:29:01+00:00
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73f596a307ee4360636407e08d878f50
Cyril Levy
Carolina Neira Jimenez
Sylvie Paycha
Institut für Mathematik
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