Noncommutative Residue and Canonical Trace on Noncommutative Tori. Uniqueness Results

TL;DR

This paper proves the uniqueness of the noncommutative residue and canonical trace on pseudodifferential operators on noncommutative tori, extending previous results and covering arbitrary dimensions.

Contribution

It establishes the first comprehensive uniqueness theorems for these traces on noncommutative tori of any dimension, improving earlier partial results.

Findings

Uniqueness of the noncommutative residue up to constant multiple.

Uniqueness of the canonical trace up to constant multiple.

Extension of results to arbitrary-dimensional noncommutative tori.

Abstract

In this paper we establish uniqueness theorems for the noncommutative residue and the canonical trace on pseudodifferential operators on noncommutative tori of arbitrary dimension. The former is the unique trace up to constant multiple on integer order pseudodifferential operators.The latter is the unique trace up to constant multiple on non-integer order pseudodifferential operators. This improves previous uniqueness results by Fathizadeh-Khalkhali, Fathizadeh-Wong, and L\'evy-Neira-Paycha.