Standard homogeneous C*-algebras as compact quantum metric spaces

TL;DR

This paper introduces a family of seminorms on continuous functions from a compact metric space to a unital C*-algebra, establishing conditions for these to form compact quantum metric spaces and extending convergence results in metric geometry.

Contribution

It defines seminorms on C(X, A) that produce compact quantum metrics when A is finite-dimensional and extends Gromov-Hausdorff convergence to matrix-valued function spaces.

Findings

Seminorms induce compact quantum metrics for finite-dimensional A

Isometric embedding of X into the state space of C(X,A)

Extension of Gromov-Hausdorff convergence to matrix function spaces

Abstract

Given a compact metric space X and a unital C*-algebra A, we introduce a family of seminorms on the C*-algebra of continuous functions from X to A, denoted C(X, A), induced by classical Lipschitz seminorms that produce compact quantum metrics in the sense of Rieffel if and only if A is finite-dimensional. As a consequence, we are able isometrically embed X into the state space of C(X,A). Furthermore, we are able to extend convergence of compact metric spaces in the Gromov-Hausdorff distance to convergence of spaces of matrices over continuous functions on the associated compact metric spaces in Latremoliere's Gromov-Hausdorff propinquity.