The quantum Gromov-Hausdorff Hypertopology on the class of pointed Proper Quantum Metric Spaces
Frederic Latremoliere
TL;DR
The paper constructs a hypertopology for pointed proper quantum metric spaces by introducing a local metametric built from tunnels between nonunital separable C*-algebras equipped with a Lipschitz-type seminorm and a pin state. It proves that the metametric vanishes exactly when a full quantum isometry exists, and that a relaxed triangle inequality governs tunnel composition, enabling a coherent convergence theory. A global hypertopology is defined via a quantum Gromov-Hausdorff distance (DH) that captures both state-space and operator-algebraic data, and is shown to be compatible with the quantum propinquity in the compact case while extending convergence to locally compact noncommutative spaces. The work also provides finite-dimensional approximations and noncompact examples, illustrating how locally compact noncommutative metric geometry can be developed beyond the compact setting.
Abstract
We introduce a hypertopology, induced by an inframetric up to full quantum isometry, on the class of pointed proper quantum metric spaces, which are separable, possibly non-unital, C*-algebras endowed with an analogue of the Lipschitz seminorm, with a distinguished state, and with a particular type of approximate units. Our hypertopology provides an analogue of the Gromov-Hausdorff distance on proper metric spaces, and in fact, convergence in the latter implies convergence in the former. Moreover, when restricted to the class of quantum compact metric spaces, our new topology is compatible with the topology of the Gromov-Hausdorff propinquity. We include new examples of noncompact, noncommutative pointed proper quantum metric spaces which are limits, for our new topology, of finite dimensional quantum compact metric spaces. This article thus provides a first answer to the challenging question of how to extend noncommutative metric geometry to the locally compact quantum space realm.