External products of spectral metric spaces

TL;DR

This paper characterizes compact quantum metric spaces via finite dimensional approximations, introduces matrix analogues, and demonstrates their stability under tensor products and external products in spectral triples.

Contribution

It introduces matrix compact quantum metric spaces and proves their stability under minimal tensor and external product operations.

Findings

Matrix compact quantum metric spaces are stable under tensor products.

Matrix spectral metric spaces are stable under external products.

Several noncommutative examples of matrix compact quantum metric spaces are provided.

Abstract

In this paper, we present a characterization of compact quantum metric spaces in terms of finite dimensional approximations. This characterization naturally leads to the introduction of a matrix analogue of a compact quantum metric space. As an application, we show that matrix compact quantum metric spaces are stable under minimal tensor products and more specifically that matrix spectral metric spaces are stable under the external product operation on unital spectral triples. We present several noncommutative examples of matrix compact quantum metric spaces.