The canonical trace and the noncommutative residue on the noncommutative torus

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 ζfunction at zero of the Laplacian on the noncommutative torus is then a straightforward consequence.

Connes’s trace theorem for curved noncommutative tori: Application to scalar curvature

“…Remark 5.4. Under the equivalence between our classes of ΨDOs and the classes of toroidal ΨDOs (see [31]) the above noncommutative residue agrees with the noncommutative residue for toroidal ΨDOs introduced in [38].…”

Scalar curvature for the noncommutative two torus

“…defines a trace on the algebra of pseudodifferential operators [7,19], it follows from the uniqueness of traces on the algebra of pseudodifferential operators [18,15,25] that it coincides with the noncommutative residue defined in [15]. Therefore there exists a constant c such that for any P…”

“…For noncommutative four tori T 4 Θ , the scalar curvature is computed in [15] and it is shown that flat metrics are the critical points of the analog of the Einstein-Hilbert action. Also noncommutative residues for noncommutative tori were studied in [18,25,15] (see also [30]). We refer to [31,22] and [24] for detailed discussions on noncommutative residues for classical manifolds.…”

Wodzicki residue and minimal operators on a noncommutative 4-dimensional torus

“…As it has been shown [12] and more generally in [15] Wodzicki residue exists also in the case of the pseudodifferential calculus over noncommutative tori.…”

“…The symbol calculus defined in [4] and developed further in [3] (see also [15]) is easily generalised to the d-dimensional case and to the operators defined above. We shall briefly review the basic definitions and methods used further in the note.…”