#### Document Type

Article

#### Publication Date

2-23-2014

#### Keywords

Noncommutative metric geometry, Gromov-Hausdorff convergence, Monge-Kantorovich distance, Quantum Metric Spaces, Lip-norms.

#### 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, which resolves several important issues raised by recent research in noncommutative metric geometry: our new metric makes *-isomorphism a necessary condition for distance zero, 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 new tool for noncommutative metric geometry which offers a solution to several important problems in the field and is designed to allow for the generalizations of techniques from metric geometry to C*-algebra theory.

#### Recommended Citation

Latrémolière, Frédéric, "The Dual Gromov-Hausdorff Propinquity" (2014). *Mathematics Preprint Series.* Paper 20.

http://digitalcommons.du.edu/math_preprints/20