type III representations and modular spectral triples for the noncommutative torus

TL;DR

This paper constructs and analyzes type III representations and modular spectral triples for the noncommutative torus, especially for Liouville numbers, extending the understanding of their von Neumann algebra types and spectral geometry.

Contribution

It demonstrates the existence of type III and related factor representations with modular spectral triples for noncommutative tori at Liouville numbers, generalizing to CCR algebras.

Findings

Existence of type III and II_infinity representations for certain irrational rotations.

Construction of modular spectral triples linked to the Tomita modular operator.

Method applicable to CCR algebras with symplectic structures.

Abstract

It is well known that for any irrational rotation number $\a$, the noncommutative torus $\ba_\a$ must have representations $\pi$ such that the generated von Neumann algebra $\pi(\ba_\a)"$ is of type $\ty{III}$. Therefore, it could be of interest to exhibit and investigate such kind of representations, together with the associated spectral triples whose twist of the Dirac operator and the corresponding derivation arises from the Tomita modular operator. In the present paper, we show that this program can be carried out, at least when $\a$ is a Liouville number satisfying a faster approximation property by rationals. In this case, we exhibit several type $\ty{II_\infty}$ and $\ty{III_\l}$, $\l\in[0,1]$, factor representations and modular spectral triples. The method developed in the present paper can be generalised to CCR algebras based on a locally compact abelian group equipped with a symplectic form.