Quantum ultrametrics on AF algebras and the Gromov-Hausdorff propinquity (Q2787157)
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scientific article; zbMATH DE number 6545414
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Quantum ultrametrics on AF algebras and the Gromov-Hausdorff propinquity |
scientific article; zbMATH DE number 6545414 |
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24 February 2016
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noncommutative metric geometry
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Gromov-Hausdorff convergence
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Monge-Kantorovich distance
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quantum metric spaces
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Lip-norm
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AF algebras
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\(C^*\)-algebra
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Quantum ultrametrics on AF algebras and the Gromov-Hausdorff propinquity (English)
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This work transfers AF algebras into the realm of noncommutative metric geometry in the sense now explained. The authors construct quantum metric structures on unital AF algebras with a faithful tracial state. They prove that for such metrics, AF algebras are limits of their defining inductive sequences of finite-dimensional \(C^*\)-algebras for the quantum propinquity. After this, for the quantum propinquity, they study the geometry of three natural classes of AF algebras equipped with their quantum metrics. In addition, they present several compact classes of AF algebras for the quantum propinquity, and they show continuity of their family of Lip-norms on a fixed AF algebra.
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