The modular 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. Resume: Motives par la quete d’une metrique analogue a la distance de Gromov-Hausdorff pour la geometrie noncommutative et adaptee aux C*-algebres, nous proposons une distance complete sur la classe des espaces metriques compacts quantiques de Leibniz. Cette nouvelle distance, que nous appelons la proximite duale de Gromov-Hausdorff, resout plusieurs problemes importants que la recherche courante en geometrie metrique noncommutative a reveles. En particulier, il est necessaire pour les C*-algebres d’etre isomorphes pour avoir distance zero, et tous les espaces quantiques compacts impliques dans le calcul de la proximite duale sont de type Leibniz. En outre, notre distance est complete. Notre proximite duale de Gromov-Hausdorff est donc un nouvel outil naturel pour le developpement de la geometrie metrique noncommutative.