C*-algebraic quantum Gromov-Hausdorff distance

Abstract

We introduce a new quantum Gromov-Hausdorff distance between C*-algebraic compact quantum metric spaces. Because it is able to distinguish algebraic structures, this new distance fixes a weakness of Rieffel's quantum distance. We show that this new quantum distance has properties analogous to the basic properties of the classical Gromov-Hausdorff distance, and we give criteria for when a parameterized family of C*-algebraic compact quantum metric spaces is continuous with respect to this new distance.

27 pages. Small modifications

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• On Gromov-Hausdorff convergence for operator metric spaces
• The Podles spheres converge to the sphere
• Curved Noncommutative Tori as Leibniz Quantum Compact Metric Spaces
• The problem of completeness for Gromov-Hausdorff metrics on C*-algebras
• A Gromov-Hausdorff Distance between von Neumann Algebras and an Application to Free Quantum Fields
• Topographic Gromov-Hausdorff quantum Hypertopology for Quantum Proper Metric Spaces