Gromov-Hausdorff convergence of state spaces for spectral truncations

TL;DR

This paper investigates the convergence of metric spaces derived from spectral truncations of geometric objects, establishing conditions for Gromov-Hausdorff convergence and illustrating with examples like the circle and sphere.

Contribution

It provides general conditions for Gromov-Hausdorff convergence of state spaces in spectral truncations, extending understanding of noncommutative geometric approximations.

Findings

Convergence conditions for spectral truncations

Examples include circle and sphere approximations

Spectral triples can approximate classical geometries

Abstract

We study the convergence aspects of the metric on spectral truncations of geometry. We find general conditions on sequences of operator system spectral triples that allows one to prove a result on Gromov-Hausdorff convergence of the corresponding state spaces when equipped with Connes' distance formula. We exemplify this result for spectral truncations of the circle, Fourier series on the circle with a finite number of Fourier modes, and matrix algebras that converge to the sphere.