eBook ISBN:  9781470403942 
Product Code:  MEMO/168/796.E 
List Price:  $63.00 
MAA Member Price:  $56.70 
AMS Member Price:  $37.80 
eBook ISBN:  9781470403942 
Product Code:  MEMO/168/796.E 
List Price:  $63.00 
MAA Member Price:  $56.70 
AMS Member Price:  $37.80 

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 orderunit 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.

Table of Contents

Chapters

GromovHausdorff distance for quantum metric spaces

1. Introduction

2. Compact quantum metric spaces

3. Quotients (= “subsets”)

4. Quantum GromovHausdorff distance

5. Bridges

6. Isometries

7. Distance zero

8. Actions of compact groups

9. Quantum tori

10. Continuous fields of orderunit spaces

11. Continuous fields of lipnorms

12. Completeness

13. Finite approximation and compactness

Matrix algebras converge to the sphere for quantum GromovHausdorff 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


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By a quantum metric space we mean a \(C^*\)algebra (or more generally an orderunit 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

GromovHausdorff distance for quantum metric spaces

1. Introduction

2. Compact quantum metric spaces

3. Quotients (= “subsets”)

4. Quantum GromovHausdorff distance

5. Bridges

6. Isometries

7. Distance zero

8. Actions of compact groups

9. Quantum tori

10. Continuous fields of orderunit spaces

11. Continuous fields of lipnorms

12. Completeness

13. Finite approximation and compactness

Matrix algebras converge to the sphere for quantum GromovHausdorff 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