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# Title: 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.
 Comments: 64 Pages. Contain the first section of ArXiv:1608.04881; split due to paper length. Sections 7 and 8 reworked Subjects: Operator Algebras (math.OA); Functional Analysis (math.FA) MSC classes: 46L89, 46L30, 58B34 Cite as: arXiv:1608.04881 [math.OA] (or arXiv:1608.04881v5 [math.OA] for this version)

## Submission history

From: Frederic Latremoliere [view email]
[v1] Wed, 17 Aug 2016 07:21:08 GMT (93kb)
[v2] Fri, 2 Sep 2016 05:44:10 GMT (99kb)
[v3] Tue, 21 Mar 2017 07:00:05 GMT (58kb)
[v4] Mon, 28 Aug 2017 14:58:31 GMT (58kb)
[v5] Sun, 18 Feb 2018 04:05:06 GMT (58kb)