
eBook ISBN: | 978-1-4704-0394-2 |
Product Code: | MEMO/168/796.E |
List Price: | $63.00 |
MAA Member Price: | $56.70 |
AMS Member Price: | $37.80 |

eBook ISBN: | 978-1-4704-0394-2 |
Product Code: | MEMO/168/796.E |
List Price: | $63.00 |
MAA Member Price: | $56.70 |
AMS Member Price: | $37.80 |
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 168; 2004; 91 ppMSC: Primary 46; Secondary 53; 58; 60
By a quantum metric space we mean a \(C^*\)-algebra (or more generally an order-unit space) equipped with a generalization of the usual Lipschitz seminorm on functions which one associates to 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, \(A_\theta\). We show, for consistently defined “metrics”, that if a sequence \(\{\theta_n\}\) of parameters converges to a parameter \(\theta\), then the sequence \(\{A_{\theta_n}\}\) of quantum tori converges in quantum Gromov–Hausdorff distance to \(A_\theta\).
ReadershipGraduate students and research mathematicians interested in functional analysis.
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Table of Contents
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Chapters
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Gromov-Hausdorff distance for quantum metric spaces
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1. Introduction
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2. Compact quantum metric spaces
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3. Quotients (= “subsets”)
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4. Quantum Gromov-Hausdorff distance
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5. Bridges
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6. Isometries
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7. Distance zero
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8. Actions of compact groups
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9. Quantum tori
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10. Continuous fields of order-unit spaces
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11. Continuous fields of lip-norms
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12. Completeness
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13. Finite approximation and compactness
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Matrix algebras converge to the sphere for quantum Gromov-Hausdorff distance
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0. Introduction
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1. The quantum metric spaces
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2. Choosing the bridge constant $\gamma $
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3. Compact semisimple Lie groups
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4. Covariant symbols
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5. Contravariant symbols
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6. Conclusion of the proof of Theorem 3.2
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By a quantum metric space we mean a \(C^*\)-algebra (or more generally an order-unit space) equipped with a generalization of the usual Lipschitz seminorm on functions which one associates to 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, \(A_\theta\). We show, for consistently defined “metrics”, that if a sequence \(\{\theta_n\}\) of parameters converges to a parameter \(\theta\), then the sequence \(\{A_{\theta_n}\}\) of quantum tori converges in quantum Gromov–Hausdorff distance to \(A_\theta\).
Graduate students and research mathematicians interested in functional analysis.
-
Chapters
-
Gromov-Hausdorff distance for quantum metric spaces
-
1. Introduction
-
2. Compact quantum metric spaces
-
3. Quotients (= “subsets”)
-
4. Quantum Gromov-Hausdorff distance
-
5. Bridges
-
6. Isometries
-
7. Distance zero
-
8. Actions of compact groups
-
9. Quantum tori
-
10. Continuous fields of order-unit spaces
-
11. Continuous fields of lip-norms
-
12. Completeness
-
13. Finite approximation and compactness
-
Matrix algebras converge to the sphere for quantum Gromov-Hausdorff distance
-
0. Introduction
-
1. The quantum metric spaces
-
2. Choosing the bridge constant $\gamma $
-
3. Compact semisimple Lie groups
-
4. Covariant symbols
-
5. Contravariant symbols
-
6. Conclusion of the proof of Theorem 3.2