The Modular Gromov-Hausdorff Propinquity

Abstract

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 C*-algebras from a metric perspective. We apply our construction, in particular, to the continuity of Heisenberg modules over quantum $2$-tori.Comment: 64 Pages. Contain the first section of ArXiv:1608.04881; split due to paper length. Sections 7 and 8 reworke