A Noncommutative Residue on Tori and a Semiclassical Limit

TL;DR

This paper introduces a noncommutative residue for pseudodifferential operators on tori, proving its uniqueness as a trace and connecting it to noncommutative geometry via semiclassical limits.

Contribution

It defines a new noncommutative residue on tori and establishes its uniqueness and relation to noncommutative tori in the semiclassical limit.

Findings

The noncommutative residue is the unique trace up to a constant.

It extends to arbitrary-dimensional tori.

In the two-torus case, it relates to noncommutative geometry.

Abstract

We define a noncommutative residue for classical Euclidean pseudodifferential operators on a torus of arbitrary dimension. We prove that, up to multiplication by a constant, it is the unique trace on the algebra of classical pseudodifferential operators modulo infinitely smoothing operators. In the case of the two torus, we show that the noncommutative residue is the semiclassical limit of a noncommutative residue defined on classical pseudodifferential operators on noncommutative two tori.