Gromov-Hausdorff Distance for Quantum Metric Spaces

By Marc Rieffel. 2000

206 citations recorded by the source corpus.

By a quantum metric space we mean a C^*-algebra (or more generally an order-unit space) equipped with a generalization of the Lipschitz seminorm on functions which is defined by an ordinary metric. We develop for compact quantum metric spaces a version of Gromov-Hausdorff distance. We show that the basic theorems of the classical theory have natural quantum analogues. Our main example involves the quantum tori, $A_{\th}$. We show, for consistently defined ``metrics'', that if a sequence $\{\th_n\}$ of parameters converges to a parameter $\th$, then the sequence $\{A_{\th_n}\}$ of quantum tori converges in quantum Gromov-Hausdorff distance to $A_{\th}$.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b0645b050a510d671e5a6105dc7f6868d930eae5

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Quantum automorphism groups of small metric spaces (2003, background): “See [Paschke and Sitarz 1998; Rieffel 2004] for more comments on this subject.”
On Gromov-Hausdorff convergence for operator metric spaces (2004, unknown): “ar X iv :m at h/ 04 11 15 7v 4 [ m at h.”
The Modular Gromov-Hausdorff Propinquity (2013, methodology): “This weakened form of the desired coincidence property sparked a lot of research toward the goal of strengthening Rieffel’s distance to make various stronger notions of isomorphisms necessary for distance zero [16, 24, 26, 17, 47, 22].”
Gromov-Hausdorff Distance for Quantum Metric Spaces/Matrix Algebras Converge to the Sphere for Quantum Gromov-Hausdorff Distance (2001, background): “Then the mapping σ from B to A has an adjoint operator, σ̆, from A to B.”
Spectral triples for AF C*-algebras and metrics on the cantor set (2003, background): “Rieffel has studied many aspects of the Gromov–Hausdorff distance between non commutative compact metric spaces, and published a memoir in the series of the American Mathematical Society on these matters [23].”
Hyperbolic Group C *-Algebras and Free-Product C *-Algebras as Compact Quantum Metric Spaces (2003, background): “In [ 14 ] this property was taken as the defining property for a “compact quantum metric space”.”
Matricial quantum Gromov–Hausdorff distance (2002, methodology): “O A ] 3 0 Ju l 2 1.”