TY  - JOUR
A1  - Levy, Cyril
A1  - Jimenez, Carolina Neira
A1  - Paycha, Sylvie
T1  - THE CANONICAL TRACE AND THE NONCOMMUTATIVE RESIDUE ON THE NONCOMMUTATIVE TORUS
JF  - Transactions of the American Mathematical Society
N2  - 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.
Y1  - 2016
U6  - https://doi.org/10.1090/tran/6369
SN  - 0002-9947
SN  - 1088-6850
VL  - 368
SP  - 1051
EP  - 1095
PB  - American Mathematical Soc.
CY  - Providence
ER  -