The Bures metric and the quantum metric on the density space of a C*-algebra: the non-unital case

TL;DR

This paper extends the Bures and quantum metrics to non-unital C*-algebras with faithful traces, analyzing their topological properties and providing examples of non-compactness in both commutative and noncommutative cases.

Contribution

It generalizes the definitions of the Bures and quantum metrics to non-unital C*-algebras and explores their topological and compactness properties, including new classes of quantum metric spaces.

Findings

Bures metric remains a metric in the non-unital case with weaker topology than the C*-norm

Density space with Bures metric is non-compact if and only if the algebra is infinite dimensional

Quantum metric topology is weaker than the C*-norm topology, with examples showing non-compactness

Abstract

Building off work of Farenick and Rahaman, we extend the definition of the density space and the Bures metric to the setting of non-unital C*-algebras equipped with a faithful trace and prove that the Bures metric is also a metric in this case and show that its topology is weaker than the topology induced by the C*-norm. Furthermore, we prove a Heine-Borel type theorem for C*-algebras and the density space. In particular, we prove that for any C*-algebra (unital or non-unital) equipped with a faithful trace, the density space equipped with the Bures metric topology is not compact if and only if the C*-algebra is infinite dimensional. We also exhibit several examples of sequences that have no converging sequence in the unital and non-unital case including both commutative and noncommutative C*-algebras. Next, building off work from some of the authors, we extend the definition of the quantum metric on the density space to the non-unital C*-algebra case by introducing the notion of a quantum Lipschitz triple, which form a subclass of quantum locally compact metric spaces of Latr\'emoli\`ere that utilize Rieffel's notion of a quantum metric (we also introduce new classes of quantum locally compact metric spaces that include certain noncommutative homogeneous C*-algebras). Furthermore, we prove that this quantum metric topology is weaker than the topology of the one induced by the C*-norm and finish the article with an analysis of matrix-valued functions on the quantized interval, which provides commutative and noncommuataive examples where the quantum metric topology on the density space is not compact and is not uniformly equivalent to both the Bures metric and the metric induced by the C*-norm.