Leibniz seminorms for "Matrix algebras converge to the sphere"

TL;DR

This paper constructs Leibniz seminorms for matrix algebras converging to the sphere, enabling rigorous analysis of non-commutative vector bundles in quantum geometry.

Contribution

It introduces a method to build Leibniz seminorms satisfying strong properties for matrix algebras approaching the sphere, connecting non-commutative geometry with physics.

Findings

Constructed seminorms satisfying strong Leibniz property for matrix algebras

Utilized coherent states and Berezin symbols as key tools

Facilitated rigorous analysis of vector bundles in quantum geometry

Abstract

In an earlier paper of mine relating vector bundles and Gromov-Hausdorff distance for ordinary compact metric spaces, it was crucial that the Lipschitz seminorms from the metrics satisfy a strong Leibniz property. In the present paper, for the now non-commutative situation of matrix algebras converging to the sphere (or to other spaces) for quantum Gromov-Hausdorff distance, we show how to construct suitable seminorms that also satisfy the strong Leibniz property. This is in preparation for making precise certain statements in the literature of high-energy physics concerning "vector bundles" over matrix algebras that "correspond" to monopole bundles over the sphere. We show that a fairly general source of seminorms that satisfy the strong Leibniz property consists of derivations into normed bimodules. For matrix algebras our main technical tools are coherent states and Berezin symbols.