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.
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