The canonical trace and the noncommutative residue on the noncommutative torus

TL;DR

This paper constructs a canonical trace and residue on pseudodifferential operators on noncommutative tori, linking them to classical results and deriving implications for regularized traces and conformal invariance.

Contribution

It introduces a canonical trace and residue on noncommutative tori, extending classical concepts and characterizing these traces uniquely in the noncommutative setting.

Findings

Canonical trace reduces to Kontsevich-Vishik trace in the commutative case.

Characterization of the noncommutative residue as a unique trace vanishing on trace-class operators.

Derived defect formulas for regularized traces and established conformal invariance of the zeta function at zero.

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) non--integer 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.