Given a unital $C^*$-algebra and a faithful trace, we prove that the topology on the associated density space induced by the $C^*$-norm is finer than the Bures metric topology. We also provide an example when this containment is strict. Next, we provide a metric on the density space induced by a quantum metric in the sense of Rieffel and prove that the induced topology is the same as the topology induced by the Bures metric and $C^*$-norm when the $C^*$-algebra is assumed to be finite dimensional. Finally, we provide an example where the Bures metric and induced quantum metric are not metric equivalent. Thus, we provide a bridge between these aspects of quantum information theory and noncommutative metric geometry.

We provide a detailed study of actions of the integers on compact quantum metric spaces, which includes general criteria ensuring that the associated crossed product algebra is again a compact quantum metric space in a natural way. Moreover, we provide a flexible set of assumptions ensuring that a continuous family of $\ast$-automorphisms of a compact quantum metric space yields a field of crossed product algebras which varies continuously in Rieffel’s quantum Gromov–Hausdorff distance. Finally, we show how our results apply to continuous families of Lip-isometric actions on compact quantum metric spaces and to families of diffeomorphisms of compact Riemannian manifolds which vary continuously in the Whitney $C^{1}$-topology.

Given a unital inductive limit of C*-algebras for which each C*-algebra of the inductive sequence comes equipped with a Rieffel compact quantum metric, we produce sufficient conditions to build a compact quantum metric on the inductive limit from the quantum metrics on the inductive sequence by utilizing the completeness of the dual Gromov–Hausdorff propinquity of Latrémolière on compact quantum metric spaces. This allows us to place new quantum metrics on all unital approximately finite-dimensional (AF) algebras that extend our previous work with Latrémolière on unital AF algebras with faithful tracial state. As a consequence, we produce a continuous image of the entire Fell topology on the ideal space of any unital AF algebra in the dual Gromov–Hausdorff propinquity topology.