Sylvie Paycha

Traces on the noncommutative torus

The global symbol calculus for pseudodifferential operators on tori can be generalised to noncommutative tori. In this global approach, the quantisation map is invertible and traces are discrete sums. On the noncommutative torus, Fathizadeh and Wong had characterised the Wodzicki residue as the unique trace which vanishes on trace-class operators. In contrast, we build and characterise the canonical trace on classical pseudodifferential operators on a noncommutative torus, which extends the ordinary trace on trace-class operators. It can be written as a canonical discrete sum on the underlying toroidal symbols. 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. By means of the canonical trace, we derive defect formulae for regularised traces on noncommutative toris. The conformal invariance of the zeta-function at zero of the Laplacian on the noncommutative torus is then a straightforward consequence.

This is based on joint work with Cyril Lévy and Carolina Neira Jiménez.