A Gromov-Hausdorff Distance between von Neumann Algebras and an Application to Free Quantum Fields

TL;DR

This paper introduces a new metric for von Neumann algebras based on a norm inducing the $w^*$-topology, connecting it with existing topologies and applying it to free quantum field algebras to analyze their continuity properties.

Contribution

It proposes a novel Gromov-Hausdorff type distance for von Neumann algebras and demonstrates its application to free quantum fields, revealing continuity with respect to the mass parameter.

Findings

The new distance relates to Gromov-Hausdorff and Effros-Marechal topologies.

It shows local algebras of free quantum fields are continuous in the mass parameter.

The construction links quantum metric spaces with quantum field theory.

Abstract

A distance between von Neumann algebras is introduced, depending on a further norm inducing the $w^*$-topology on bounded sets. Such notion is related both with the Gromov-Hausdorff distance for quantum metric spaces of Rieffel and with the Effros-Marechal topology on the von Neumann algebras acting on a Hilbert space. This construction is tested on the local algebras of free quantum fields endowed with norms related with the Buchholz-Wichmann nuclearity condition, showing the continuity of such algebras w.r.t. the mass parameter.