Screen LibrarySingular traces : theory and applications /
This book is the first complete study and monograph dedicated to singular traces. The text mathematically formalises the study of traces in a self contained theory of functional analysis. Extensive notes will treat the historical development. The final section will contain the most complete and conc...
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| Main Authors: | |
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| Corporate Authors: | |
| Group Author: | ; |
| Published: |
De Gruyter,
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| Publisher Address: | Berlin : |
| Publication Dates: | [2012] |
| Literature type: | eBook |
| Language: | English |
| Series: |
De gruyter studies in mathematics ;
46 |
| Subjects: | |
| Online Access: |
http://dx.doi.org/10.1515/9783110262551 http://www.degruyter.com/doc/cover/9783110262551.jpg |
| Summary: |
This book is the first complete study and monograph dedicated to singular traces. The text mathematically formalises the study of traces in a self contained theory of functional analysis. Extensive notes will treat the historical development. The final section will contain the most complete and concise treatment known of the integration half of Connes' quantum calculus. Singular traces are traces on ideals of compact operators that vanish on the subideal of finite rank operators. Singular traces feature in A. Connes' interpretation of noncommutative residues. Particularly the Dixmier trace, which generalises the restricted Adler-Manin-Wodzicki residue of pseudo-differential operators and plays the role of the residue for a new catalogue of 'geometric' spaces, including Connes-Chamseddine standard models, Yang-Mills action for quantum differential forms, fractals, isospectral deformations, foliations and noncommutative index theory. The theory of singular traces has been studied after Connes' application to non-commutative geometry and physics by various authors. Recent work by Nigel Kalton and the authors has advanced the theory of singular traces. Singular traces can be equated to symmetric functionals of symmetricsequence or function spaces, residues of zeta functions and heat kernel asymptotics, and characterised by Lidksii and Fredholm formulas. The traces and formulas used in noncommutative geometry are now completely understood in this theory, with surprising new mathematical and physical consequences. For mathematical readers the text offers fundamental functional analysis results and, due to Nigel Kalton's contribution, a now complete theory of traces on compact operators. For mathematical physicists and o |
| Carrier Form: | 1 online resource (468pages). |
| ISBN: | 9783110262551（electronic bk.） |
| Index Number: | QA322 |
| CLC: | O177.2 |
| Contents: |
Frontmatter -- Preface -- Notations -- Contents -- Introduction -- Part I. Preliminary Material -- Chapter 1. What is a Singular Trace? -- Chapter 2. Preliminaries on Symmetric Operator Spaces -- Part II. General Theory -- Chapter 3. Symmetric Operator Spaces -- Chapter 4. Symmetric Functionals -- Chapter 5. Commutator Subspace -- Chapter 6. Dixmier Traces -- Part III. Traces on Lorentz Ideals -- Chapter 7. Lidskii Formulas for Dixmier Traces on Lorentz Ideals -- Chapter 8. Heat Kernel Formulas and -function Residues -- Chapter 9. Measurability in Lorentz Ideals -- Part IV. Applications to Noncommutative Geometry -- Chapter 10. Preliminaries to the Applications -- Chapter 11. Trace Theorems -- Chapter 12. Residues and Integrals in Noncommutative Geometry -- Appendix A. Operator Results -- Bibliography -- Index |