Quantum Metric Spaces and the Gromov-Hausdorff Propinquity

Abstract

We present a survey of the dual Gromov-Hausdorff propinquity, a noncommutative analogue of the Gromov-Hausdorff distance which we introduced to provide a framework for the study of the noncommutative metric properties of C*-algebras. We first review the notions of quantum locally compact metric spaces, and present various examples of such structures. We then explain the construction of the dual Gromov-Hausdorff propinquity, first in the context of quasi-Leibniz quantum compact metric spaces, and then in the context of pointed proper quantum metric spaces. We include a few new result concerning perturbations of the metrics on Leibniz quantum compact metric spaces in relation with the dual Gromov-Hausdorff propinquity.

Harmonic Analysis Approach to Gromov–Hausdorff Convergence for Noncommutative Tori

“…In this section we work in the framework of Gromov-Hausdorff propinquity developed by Latrémolière recently in [Lat15a,Lat16,Lat15b]. We will show that our previous results on convergence of matrix algebras to rotation algebras actually hold in the strong sense of (1) For all a ∈ A such that |||a||| A ≤ 1, there exists b ∈ B such that |||b||| B ≤ 1 and…”

“…Θ n (n) in Proposition 7.5. By Definition 2.21 in [Lat15a] and the existence of a derivation δ as defined in (3.4), (A 2d Θ , |||.|||) and (A 2d Θ n (n), |||.|||) are Leibniz pairs. Indeed, the conditions in the definition were proved in [JM10,JMP14]; see also [Zen14] for more remarks on the Lip-norms.…”

“…It follows that |||·||| is a lower semicontinuous Lip-norm. Therefore, by Definition 2.44 in [Lat15a] and by the choice of F , A 2d Θ n (n) and A 2d Θ are F -quasi-Leibniz quantum compact metric spaces. Here, in fact they are Leibniz quantum compact metric spaces.…”

“…A very similar criterion is used in Latrémolière [Lat15a] to show convergence in the propinquity sense. Starting from I) and II) it is very easy to construct an operator space X which witnesses the fact that the Gromov-Hausdorff distance between S(A Θ ) and S(B n ) is ≤ ε.…”

Metrized Quantum Vector Bundles over Quantum Tori Built from Riemannian Metrics and Rosenberg's Levi-Civita Connections

“…where the object on the right-hand side is a · A Θ -compact subset of A Θ (see [5,Remark 2.46]). Hence, as X ∈ D 1 g is arbitrary, we have shown that there exists a single · A Θ -compact subset of A Θ that contains the A Θ -coefficients of all elements of D 1 g , which implies that D 1 g is · st -precompact in χ Θ .…”