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Abstract

Motivated by the quest for an analytic framework to study classes of ${\rm C}^*$-algebras and associated structures as geometric objects, we introduce a metric on Hilbert modules equipped with a generalized form of a differential structure, thus extending Gromov–Hausdorff convergence theory to vector bundles and quantum vector bundles—not convergence as total space but indeed as quantum vector bundle. Our metric is new even in the classical picture, and creates a framework for the study of the moduli spaces of modules over ${\rm C}^*$-algebras from a metric perspective. We apply our construction, in particular, to the continuity of Heisenberg modules over quantum $2$-tori.