The quantum Gromov-Hausdorff propinquity

Abstract

We introduce the quantum Gromov-Hausdorff propinquity, a new distance between quantum compact metric spaces, which extends the Gromov-Hausdorff distance to noncommutative geometry and strengthens Rieffel's quantum Gromov-Hausdorff distance and Rieffel's proximity by making *-isomorphism a necessary condition for distance zero, while being well adapted to Leibniz seminorms. This work offers a natural solution to the long-standing problem of finding a framework for the development of a theory of Leibniz Lip-norms over C*algebras.

Quantum metrics from traces on full matrix algebras

“…where the Jordan product is a • b = ab+ba 2 and the Lie product is {a, b} = ab−ba 2i . In [7], F. Latrémolière introduced a quantum analogue to the Gromov-Hausdorff distance [3], the quantum Gromov-Hausdorff propinquity. Indeed, the quantum Gromov-Hausdorff propinquity is a distance between quasi-Leibniz quantum compact metric spaces that preserves the C*-algebraic and metric structures and recovers the topology of the Gromov-Hausdorff distance, and thus provides an appropriate framework for the study of noncommutative metric geometry.…”

“…The convergence is studied by use of a distance on the classes of these spaces. F. Latrémolière introduced the quantum Gromov-Hausdorff propinquity [7,5,15] as a noncommutative analogue of the Gromov-Hausdorff distance [3], adapted to the theory of C*-algebras.…”

Inductive Limits of C*-Algebras and Compact Quantum Metric Spaces

“…We take the following theorem from [27] since it adapts the dual propinquity from [25] to the quasi-Leibniz case. Theorem 1.5 ([27, Definition 2.23, Theorem 2.28]).…”

The modular Gromov–Hausdorff propinquity

“…The resulting family of metrics, which we named the Gromov-Hausdorff propinquities, has been showed to enjoy various nice properties and applications. We know of several convergence and finite dimensional approximations results, such as the continuity of quantum tori and approximations of quantum tori by fuzzy tori [4,7] (later on approached using our propinquity and different techniques in [22], with potential connection to quantum information theory), continuity for families of AF algebras [13], full matrix approximations of coadjoint orbits for semi-simple Lie groups [30], continuity for conformal and other types of deformations [21]. We have a noncommutative analogue of Gromov's compactness theorem [8] and conditions for preservation of symmetries [15].…”