Spectral Distances: Results for Moyal Plane and Noncommutative Torus

TL;DR

This paper investigates spectral distances in noncommutative geometries, specifically Moyal planes and tori, providing explicit formulas and analyzing topological properties of the state space.

Contribution

It derives explicit spectral distance formulas for specific states in noncommutative Moyal planes and tori, highlighting topological differences from classical geometries.

Findings

Explicit distance formulas for pure states in Moyal planes

Existence of states at infinite spectral distance

Topological differences from classical spaces

Abstract

The spectral distance for noncommutative Moyal planes is considered in the framework of a non compact spectral triple recently proposed as a possible noncommutative analog of non compact Riemannian spin manifold. An explicit formula for the distance between any two elements of a particular class of pure states can be determined. The corresponding result is discussed. The existence of some pure states at infinite distance signals that the topology of the spectral distance on the space of states is not the weak * topology. The case of the noncommutative torus is also considered and a formula for the spectral distance between some states is also obtained.