Gromov-Hausdorff Distance for Quantum Metric Spaces

Abstract

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, $A_{þ}$. We show, for consistently defined ``metrics'', that if a sequence $\{{þ}_n\}$ of parameters converges to a parameter $þ$, then the sequence $\{A_{{þ}_n}\}$ of quantum tori converges in quantum Gromov-Hausdorff distance to $A_{þ}$.

81 pages. Several minor improvements and several references added. To appear Memoirs Amer. Math. Soc

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Cited by in corpus (7)

• Dimension and dynamical entropy for metrized C*-algebras
• Representations of Lie groups and random matrices
• Metrics On The Quantum Heisenberg Manifold
• Finiteness of metrics on state spaces
• Quantum metrics on noncommutative spaces
• From C*algebra extensions to CQMS, ${SU}_q(2)$, Podles sphere and other examples
• Vietoris topology on hyperspaces associated to a noncommutative compact space