The dual modular Gromov–Hausdorff propinquity and completeness

Abstract

We introduce in this paper the dual modular propinquity, a complete metric, up to full modular quantum isometry, on the class of metrized quantum vector bundles, i.e. of Hilbert modules endowed with a type of densely defined norm, called a D-norm, which generalize the operator norm given by a connection on a Riemannian manifold. The dual modular propinquity is weaker than the modular propinquity yet it is complete, which is the main purpose of its introduction. Moreover, we show that the modular propinquity can be extended to a larger class of objects which involve quantum compact metric spaces acting on metrized quantum vector bundles.