The Dual Gromov-Hausdorff Propinquity
Abstract
Motivated by the quest for an analogue of the Gromov-Hausdorff distance in noncommutative geometry which is well-behaved with respect to C*-algebraic structures, we propose a complete metric on the class of Leibniz quantum compact metric spaces, named the dual Gromov-Hausdorff propinquity. This metric resolves several important issues raised by recent research in noncommutative metric geometry: it makes *-isomorphism a necessary condition for distance zero, it is well-adapted to Leibniz seminorms, and --- very importantly --- is complete, unlike the quantum propinquity which we introduced earlier. Thus our new metric provides a natural tool for noncommutative metric geometry, designed to allow for the generalizations of techniques from metric geometry to C*-algebra theory.
- Publication:
-
arXiv e-prints
- Pub Date:
- November 2013
- DOI:
- 10.48550/arXiv.1311.0104
- arXiv:
- arXiv:1311.0104
- Bibcode:
- 2013arXiv1311.0104L
- Keywords:
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- Mathematics - Operator Algebras;
- Mathematics - Metric Geometry;
- Primary: 46L89;
- 46L30;
- 58B34
- E-Print:
- 42 pages in elsarticle 3p format. This third version has many small typos corrections and small clarifications included. Intended form for publication