Some applications of conditional expectations to convergence for the quantum Gromov-Hausdorff propinquity

TL;DR

This paper explores the convergence of compact metric spaces to matrix algebras within the quantum Gromov-Hausdorff framework, using conditional expectations to analyze group actions and fixed point subalgebras.

Contribution

It demonstrates that all compact metric spaces can be approximated by matrix algebras in the quantum propinquity and establishes the continuity of fixed point subalgebras under group actions.

Findings

Compact metric spaces are in the closure of full matrix algebras for the quantum Gromov-Hausdorff propinquity.

The fixed point subalgebra map is continuous with respect to the Hausdorff and propinquity topologies.

Conditional expectations are effectively used to analyze convergence and continuity in quantum metric spaces.

Abstract

We prove that all the compact metric spaces are in the closure of the class of full matrix algebras for the quantum Gromov-Hausdorff propinquity. We also show that given an action of a compact metrizable group G on a quasi-Leibniz compact quantum metric space (A,Lip), the function associating any closed subgroup of G group to its fixed point C*-subalgebra in A is continuous from the topology of the Hausdorff distance to the topology induced by the propinquity. Our techniques are inspired from our work on AF algebras as quantum metric spaces, as they are based on the use of various types of conditional expectations.