Non-commutative analysis:
Gespeichert in:
| Beteiligte Personen: | , |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
New Jersey
World Scientific
[2017]
|
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029875611&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | xxviii, 533 Seiten Illustrationen, Diagramme |
| ISBN: | 9789813202122 9789813202115 |
Internformat
MARC
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| 245 | 1 | 0 | |a Non-commutative analysis |c Palle Jorgensen, Feng Tian |
| 264 | 1 | |a New Jersey |b World Scientific |c [2017] | |
| 300 | |a xxviii, 533 Seiten |b Illustrationen, Diagramme | ||
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| 700 | 1 | |a Tian, Feng |e Verfasser |0 (DE-588)1108833748 |4 aut | |
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Datensatz im Suchindex
| _version_ | 1819285270146580480 |
|---|---|
| adam_text | Contents
Foreword vii
Preface ix
Acknowledgments xv
Abbreviations, Notation, Some Core Theorems xxiii
I Introduction and Motivation 1
1 Subjects and User’s Guide 3
1.1 Motivation ............................................... 3
1.2 Key Themes in the Book: A Bird’s-eye Preview ............. 5
1.2.1 Operators in Hilbert Space.......................... 5
1.2.2 Multivariable Spectral Theory ...................... 7
1.2.3 Noncommutative Analysis............................. 8
1.2.4 Probability......................................... 9
1.2.5 Other Neighboring Areas............................ 12
1.2.6 Unitary Representations............................ 13
1.3 Note on Cited Books and Papers.......................... 13
1.4 Reader Guide............................................. 15
1.5 A Word About the Exercises............................... 16
1.6 List of Applications..................................... 17
1.7 Groups and Physics ..................................... 18
II Topics from Functional Analysis and Operators in
Hilbert Space: A Selection 21
2 Elementary Facts 23
2.1 A Sample of Topics....................................... 25
2.2 Duality ................................................. 27
2.2.1 Duality and Measures............................... 33
2.2.2 Other Spaces in Duality............................ 36
XVII
xviii NON-COMMUTATIVE ANALYSIS
2.3 Transfinite Induction (Zorn and All That)........................ 37
2.4 Basics of Hilbert Space Theory................................... 39
2.4.1 Positive Definite Functions ............................... 42
2.4.2 Orthonormal Bases.......................................... 46
2.4.3 Bounded Operators in Hilbert Space......................... 50
2.4.4 The Gram-Schmidt Process and Applications.................. 55
2.5 Dirac’s Notation................................................. 63
2.5.1 Three Norm-Completions..................................... 67
2.5.2 Connection to Quantum Mechanics............................ 70
2.5.3 Probabilistic Interpretation of Parseval in Hilbert Space . 75
2.6 The Lattice Structure of Projections............................. 77
2.7 Multiplication Operators......................................... 83
2.A Hahn-Banach Theorems............................................. 85
2. B Banach-Limit..................................................... 86
3 Unbounded Operators in Hilbert Space 89
3.1 Domain, Graph, and Adjoints...................................... 90
3.2 Characteristic Matrix ........................................... 97
3.2.1 Commutants.................................................101
3.3 Unbounded Operators Between Different
Hilbert Spaces ..................................................102
3.3.1 An application to the Malliavin derivative.................109
3.4 Normal Operators............................................... Ill
3.5 Polar Decomposition..............................................113
3. A Stone’s Theorem..................................................114
4 Spectral Theory 119
4.1 An Overview.................................................... 120
4.2 Multiplication Operator Version..................................125
4.2.1 Transformation of Measures.................................128
4.2.2 Direct Integral Representation.............................131
4.2.3 Proof of Theorem 4.1 continued:............................132
4.3 Project ion-Valued Measure (PVM).................................136
4.4 Convert M p to a PVM (projection-valued measure) ................140
4.5 The Spectral Theorem for Compact Operators ......................144
4.5.1 Preliminaries..............................................144
4.5.2 Integral operators ..................................150
III Applications 153
5 GNS and Representations 155
5.1 Definitions and Facts: An Overview ..............................158
5.2 The GNS Construction.............................................163
5.3 States, Dual and Pre-dual........................................168
5.4 New Hilbert Spaces From “old”....................................174
CONTENTS
xix
5.4.1 GNS ....................................................174
5.4.2 Direct sum ®a ..........................................175
5.4.3 Hilbert-Schmidt operators (continuing the discussion in 1) 175
5.4.4 Tensor-Product g ^2...................................176
5.4.5 Contractive Inclusion ..................................176
5.4.6 Inflation (Dilation)....................................176
5.4.7 Quantum Information.....................................177
5.4.8 Reflection Positivity (or renormalization) . . . 179
5.5 A Second Duality Principle: A Metric on the Set of Probability
Measures ......................................................184
5.6 Abelian C*-algebras............................................187
5.7 States and Representations.....................................189
5.7.1 Normal States ..........................................196
5.7.2 A Dictionary of operator theory and quantum mechanics 197
5.8 Krein-Milman, Choquet, Decomposition of States . ..............198
5.8.1 Noncommutative Radon-Nikodym Derivative.................202
5.8.2 Examples of Disintegration..............................202
5.9 Examples of C*-algebras........................................203
5.10 Examples of Representations....................................211
5.11 Beginning of Multiplicity Theory............................. . 214
5.A The Fock-state, and Representation of CCR, Realized as Malli-
avin Calculus..................................................220
6 Completely Positive Maps 223
6.1 Motivation ................................................... 224
6.2 CP v.s. GNS....................................................226
6.3 Stinespring’s Theorem..........................................228
6.4 Applications...................................................233
6.5 Factorization..................................................239
6.6 Endomorphisms, Representations of and Numerical Range . 241
7 Brownian Motion 249
7.1 Introduction, Applications, and Context for Path-space Analysis 250
7.2 The Path Space............................................... 259
7.3 Decomposition of Brownian Motion...............................266
7.4 The Spectral Theorem, and Karhunen-Loeve Decomposition . . . 270
7.5 Large Matrices Revisited...................................... 271
8 Lie Groups, and their Unitary Representations 273
8.1 Motivation ................................................. 276
8.2 Unitary One-Parameter Groups...................................280
8.3 Group - Algebra - Representations..............................281
8.3.1 Example - ax + b group..................................286
8.4 Induced Representations........................................288
8.4.1 Integral operators and induced representations..........297
8.5 Example - Heisenberg group.....................................302
XX
NON-COMMUTATIVE ANALYSIS
8.5.1 ax + b group..........................................305
8.6 Co-adjoint Orbits............................................306
8.6.1 Review of some Lie theory.............................306
8.7 Garding Space................................................310
8.8 Decomposition of Representations ............................315
8.9 Summary of Induced Representations, the Example of d/dx . . . 318
8.9.1 Equivalence and imprimitivity for induced representations 319
8.10 Connections to Nelson’s Spectral Theory......................322
8.11 Multiplicity Revisited.......................................326
8.A The Stone-von Neumann Uniqueness Theorem .......329
9 The Kadison-Singer Problem 333
9.1 Statement of the Problem.....................................334
9.2 The Dixmier Trace............................................340
9.3 Frames in Hilbert Space......................................341
IV Extension of Operators 347
10 Selfadjoint Extensions 349
10.1 Extensions of Hermitian Operators............................350
10.2 Cayley Transform.............................................362
10.3 Boundary Triple..............................................364
10.4 The Friedrichs Extension ....................................371
10.5 Rigged Hilbert Space.........................................376
11 Unbounded Graph-Laplacians 385
11.1 Basic Setting................................................387
11.1.1 Infinite Path Space ..................................389
11.2 The Energy Hilbert Spaces J%e ...............................389
11.3 The Graph-Laplacian..........................................392
11.4 The Friedrichs Extension of A, the Graph Laplacian...........394
11.5 A ID Example.................................................395
12 Reproducing Kernel Hilbert Space 403
12.1 Fundamentals.................................................403
12.2 Application to Optimization..................................407
12.2.1 Application: Least square-optimization ...............409
12.3 A Digression: Stochastic Processes...........................413
12.4 Two Extension Problems.......................................415
12.5 The Reproducing Kernel Hilbert Space J%f.....................416
12.6 Type I v.s. Type II Extensions...............................425
12.7 The Case of e~K M 1........................................426
12.7.1 The Selfadjoint Extensions Aq D —iDF .................427
12.7.2 The Spectra of the s.a. Extensions Aq D ~iDF........431
CONTENTS
xxi
V Appendix 439
A An Overview of Functional Analysis Books (Cast of Charac-
ters) 441
B Terminology from Neighboring Areas 451
Classical Wiener measure/space
Hilbert’s sixth problem
Infinite-dimensional analysis
Monte Carlo (MC) simulation
Multiresolution analysis (MRA)
Quantum field theory (QFT)
Quantum Information (QI)
Quantum mechanics(QM)
Quantum probability (QP)
Signal processing (SP)
Stochastic processes (SP)
Uncertainty quantification (UQ)
Unitary representations (UR)
Wavelets
C Often Cited 459
D Prizes and Fame 475
Quotes: Index of Credits 477
E List of Exercises 479
F Definitions of Frequently Occurring Terms 485
G List of Figures 489
List of Tables 493
Bibliography 495
Index
525
|
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| author | Jørgensen, Palle E. T. 1947- Tian, Feng |
| author_GND | (DE-588)124805515 (DE-588)1108833748 |
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| bvnumber | BV044475290 |
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| id | DE-604.BV044475290 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T18:04:07Z |
| institution | BVB |
| isbn | 9789813202122 9789813202115 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-029875611 |
| oclc_num | 989061648 |
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| owner_facet | DE-188 DE-384 DE-19 DE-BY-UBM DE-739 DE-29T DE-20 |
| physical | xxviii, 533 Seiten Illustrationen, Diagramme |
| publishDate | 2017 |
| publishDateSearch | 2017 |
| publishDateSort | 2017 |
| publisher | World Scientific |
| record_format | marc |
| spellingShingle | Jørgensen, Palle E. T. 1947- Tian, Feng Non-commutative analysis Operatortheorie (DE-588)4075665-8 gnd Hilbert-Raum (DE-588)4159850-7 gnd Funktionalanalysis (DE-588)4018916-8 gnd |
| subject_GND | (DE-588)4075665-8 (DE-588)4159850-7 (DE-588)4018916-8 |
| title | Non-commutative analysis |
| title_auth | Non-commutative analysis |
| title_exact_search | Non-commutative analysis |
| title_full | Non-commutative analysis Palle Jorgensen, Feng Tian |
| title_fullStr | Non-commutative analysis Palle Jorgensen, Feng Tian |
| title_full_unstemmed | Non-commutative analysis Palle Jorgensen, Feng Tian |
| title_short | Non-commutative analysis |
| title_sort | non commutative analysis |
| topic | Operatortheorie (DE-588)4075665-8 gnd Hilbert-Raum (DE-588)4159850-7 gnd Funktionalanalysis (DE-588)4018916-8 gnd |
| topic_facet | Operatortheorie Hilbert-Raum Funktionalanalysis |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029875611&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT jørgensenpalleet noncommutativeanalysis AT tianfeng noncommutativeanalysis |