Connes Trace Theorem for Curved Noncommutative Tori. Application to Scalar Curvature

TL;DR

This paper extends Connes' trace theorem to curved noncommutative tori of any dimension, linking noncommutative integrals with Riemannian geometry and defining scalar curvature in this setting.

Contribution

It generalizes Connes' trace theorem to curved noncommutative tori and introduces a scalar curvature concept within this framework.

Findings

Proved a version of Connes' trace theorem for all dimensions n≥2.

Reproduced and improved earlier results for n=2 and n=4.

Established a link between noncommutative integrals and Riemannian densities.

Abstract

In this paper we prove a version of Connes' trace theorem for noncommutative tori of any dimension~$n\geq 2$. This allows us to recover and improve earlier versions of this result in dimension $n=2$ and $n=4$ by Fathizadeh-Khalkhali. We also recover the Connes integration formula for flat noncommutative tori of McDonald-Sukochev-Zanin. As a further application we prove a curved version of this integration formula in terms of the Laplace-Beltrami operator defined by an arbitrary Riemannian metric. For the class of so-called self-compatible Riemannian metrics (including the conformally flat metrics of Connes-Tretkoff) this shows that Connes' noncommutative integral allows us to recover the Riemannian density. This exhibits a neat link between this notion of noncommutative integral and noncommutative measure theory in the sense of operator algebras. As an application of these results, we setup a natural notion of scalar curvature for curved noncommutative tori.