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Extending Rieffel’s precompactness to quantum Riemannian 1-spaces

Determine how to extend Rieffel’s precompactness theorem for compact quantum metric spaces to the class of quantum Riemannian 1-spaces defined via semigroups and graph amplitudes, thereby accommodating noncommutative geometric data beyond C*-algebraic quantum metric spaces.

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Background

Rieffel introduced a notion of compact quantum metric space with a Gromov–Hausdorff-type topology and proved a precompactness theorem in that setting.

Quantum Riemannian 1-spaces in this paper are built from amplitude assignments to metrized graphs and semigroup generators; adapting Rieffel’s methods to this different framework is unclear and unresolved.

References

Although the precompactness theorem was formulated and proved by Rieffel for compact quantum metric spaces, it is not clear how to extend his approach to the case of quantum Riemannian $1$-spaces discussed in this paper.

— Moduli space of Conformal Field Theories and non-commutative Riemannian geometry (2506.00896 - Soibelman, 1 Jun 2025) in Subsection 1.3, Spectral triples, Bakry calculus and Wasserstein spaces (footnote)