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


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.