Singular traces: theory and applications
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
De Gruyter
2013
|
| Series: | De Gruyter Studies in Mathematics
46 |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | XVI, 452 S. 240 mm x 170 mm |
| ISBN: | 3110262509 9783110262506 |
Staff View
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Record in the Search Index
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IMAGE 1
CONTENTS
PREFACE V
INTRODUCTION 1
I PRELIMINARY MATERIAL
1 WHAT IS A SINGULAR TRACE? 15
1.1 COMPACT OPERATORS 15
1.2 CALKIN CORRESPONDENCE N . 22
1.3 EXAMPLES O F TRACES 28
1.3.1 THE CANONICAL TRACE 29
1.3.2 THE DIXMIER TRACE 29
1.3.3 LIDSKII FORMULATION O F TRACES 33
1.4 NOTES 34
2 PRELIMINARIES ON SYMMETRIC OPERATOR SPACES 38
2.1 VON NEUMANN ALGEBRAS 38
2.2 SEMIFINITE NORMAL TRACES 42
2.3 GENERALIZED SINGULAR VALUE FUNCTION 46
2.4 CALKIN CORRESPONDENCE IN THE SEMIFINITE SETTING 53
2.5 SYMMETRIC OPERATOR SPACES 56
2.6 EXAMPLES O F SYMMETRIC OPERATOR SPACES 59
2.7 TRACES ON SYMMETRIC OPERATOR SPACES 68
2.8 NOTES 70
II GENERAL THEORY
3 SYMMETRIC OPERATOR SPACES 79
3.1 INTRODUCTION 79
3.2 SUBMAJORIZATION IN THE FINITE-DIMENSIONAL SETTING 80
3.3 HARDY-LITTLEWOOD(-POLYA) SUBMAJORIZATION 83
3.4 UNIFORM SUBMAJORIZATION 88
3.5 SYMMETRIC OPERATOR SPACES FROM SYMMETRIC FUNCTION SPACES 97
3.6 SYMMETRIC FUNCTION SPACES FROM SYMMETRIC SEQUENCE SPACES 101
3.7 NOTES 104
HTTP://D-NB.INFO/1024905985
IMAGE 2
X I V
CONTENTS
4 SYMMETRIC FUNCTIONALS 107
4.1 INTRODUCTION 107
4.2 JORDAN DECOMPOSITION O F SYMMETRIC FUNCTIONALS 109
4.3 LATTICE STRUCTURE ON THE SET O F SYMMETRIC FUNCTIONALS 114
4.4 LIFTING O F SYMMETRIC FUNCTIONALS 117
4.5 FIGIEL-KALTON THEOREM 120
4.6 EXISTENCE O F SYMMETRIC FUNCTIONALS 123
4.7 EXISTENCE O F FULLY SYMMETRIC FUNCTIONALS 130
4.8 THE SETS O F SYMMETRIC AND FULLY SYMMETRIC FUNCTIONALS ARE DIFFERENT
133
4.9 SYMMETRIC FUNCTIONALS ON SYMMETRIC OPERATOR SPACES 142
4.10 HOW LARGE IS THE SET O F SYMMETRIC FUNCTIONALS? 146
4.11 NOTES 152
5 COMMUTATOR SUBSPACE 153
5.1 INTRODUCTION 153
5.2 NORMAL OPERATORS IN THE COMMUTATOR SUBSPACE 155
5.3 NORMAL OPERATORS IN THE CLOSED COMMUTATOR SUBSPACE 162
5.4 SUBHARMONIC FUNCTIONS ON MATRIX ALGEBRAS 168
5.5 QUASI-NILPOTENT OPERATORS BELONG TO THE COMMUTATOR SUBSPACE 173
5.6 DESCRIPTION O F THE COMMUTATOR SUBSPACE 182
5.7 COMMUTATOR SUBSPACE O F THE WEAK IDEAL I OO 187
5.8 NOTES 192
6 DIXMIER TRACES 194
6.1 INTRODUCTION 194
6.2 EXTENDED LIMITS 196
6.3 DIXMIER TRACES ON LORENTZ IDEALS 198
6.4 FULLY SYMMETRIC FUNCTIONALS ON LORENTZ IDEALS ARE DIXMIER TRACES . .
203
6.5 DIXMIER TRACES ON FULLY SYMMETRIC IDEALS O F C ( H ) 206
6.6 RELATIVELY NORMAL FUNCTIONALS 209
6.7 WODZICKI REPRESENTATION O F DIXMIER TRACES 214
6.8 NOTES 217
IMAGE 3
CONTENTS XV
III TRACES ON LORENTZ IDEALS
7 LIDSKII FORMULAS FOR DIXMIER TRACES ON LORENTZ IDEALS 225
7.1 INTRODUCTION 225
7.2 DISTRIBUTION FORMULAS FOR DIXMIER TRACES 226
7.3 LIDSKII FORMULAS FOR DIXMIER TRACES 232
7.4 SPECIAL CASES AND COUNTEREXAMPLES 235
7.5 DIAGONAL FORMULAS FOR DIXMIER TRACES FAIL 241
7.6 NOTES V 242
8 HEAT KERNEL FORMULAS AND -FUNCTION RESIDUES 244
8.1 INTRODUCTION 244
8.2 HEAT KERNEL FUNCTIONALS 246
8.3 FULLY SYMMETRIC FUNCTIONALS ARE HEAT KERNEL FUNCTIONALS 252
8.4 GENERALIZED HEAT KERNEL FUNCTIONALS 256
8.5 REDUCTION O F GENERALIZED HEAT KERNEL FUNCTIONALS 258
8.6 ^-FUNCTION RESIDUES 263
8.7 NOT EVERY DIXMIER TRACE IS A ^-FUNCTION RESIDUE 268
8.8 NOTES 271
9 MEASURABILITY IN LORENTZ IDEALS 272
9.1 INTRODUCTION 272
9.2 POSITIVE DIXMIER MEASURABLE OPERATORS IN LORENTZ IDEALS 273
9.3 POSITIVE DIXMIER MEASURABLE OPERATORS IN M . \ T 0 0 2 7 6
9.4 C-INVARIANT EXTENDED LIMITS 281
9.5 POSITIVE M-MEASURABLE OPERATORS 287
9.6 ADDITIONAL IN VARIANCE O F DIXMIER TRACES 291
9.7 MEASURABLE OPERATORS IN 1 O O 297
9.8 NOTES 300
IV APPLICATIONS TO NONCOMMUTATIVE GEOMETRY
10 PRELIMINARIES TO THE APPLICATIONS 311
10.1 SUMMARY O F TRACES ON I ICO AND M \ J00 311
10.2 PSEUDO-DIFFERENTIAL OPERATORS AND THE NONCOMMUTATIVE RESIDUE 317
10.3 PSEUDO-DIFFERENTIAL OPERATORS ON MANIFOLDS 330
10.4 NOTES 334
IMAGE 4
X V I CONTENTS
11 TRACE THEOREMS 336
11.1 INTRODUCTION 336
11.2 MODULATED OPERATORS 339
11.3 LAPLACIAN MODULATED OPERATORS AND EXTENSION O F THE NONCOMMUTATIVE
RESIDUE 345
11.4 EIGENVALUES O F LAPLACIAN MODULATED OPERATORS 355
11.5 TRACE THEOREM ON 359
11.6 TRACE THEOREM ON CLOSED RIEMANNIAN MANIFOLDS 362
11.7 INTEGRATION O F FUNCTIONS 372
11.8 NOTES 380
12 RESIDUES AND INTEGRALS IN NONCOMMUTATIVE GEOMETRY 382
12.1 INTRODUCTION 382
12.2 THE NONCOMMUTATIVE RESIDUE IN NONCOMMUTATIVE GEOMETRY 385
12.3 THE INTEGRAL IN NONCOMMUTATIVE GEOMETRY 390
12.4 EXAMPLE O F ISOSPECTRAL DEFORMATIONS 396
12.5 EXAMPLE O F THE NONCOMMUTATIVE TORUS 405
12.6 CLASSICAL LIMITS 411
12.7 NOTES 416
A OPERATOR RESULTS 420
A. 1 MATRIX RESULTS 420
A.2 OPERATOR INEQUALITIES 422
BIBLIOGRAPHY 429
INDEX 445 |
| any_adam_object | 1 |
| author | Lord, Steven Sukochev, Fedor Zanin, Dmitriy |
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| author_facet | Lord, Steven Sukochev, Fedor Zanin, Dmitriy |
| author_role | aut aut aut |
| author_sort | Lord, Steven |
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| dewey-full | 515.724 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.724 |
| dewey-search | 515.724 |
| dewey-sort | 3515.724 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| language | English |
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| physical | XVI, 452 S. 240 mm x 170 mm |
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| spelling | Lord, Steven Verfasser (DE-588)1029362149 aut Singular traces theory and applications Fedor Sukochev ; Steven Lord ; Dmitriy Zanin Berlin [u.a.] De Gruyter 2013 XVI, 452 S. 240 mm x 170 mm txt rdacontent n rdamedia nc rdacarrier De Gruyter Studies in Mathematics 46 Nichtkommutative Geometrie (DE-588)4171742-9 gnd rswk-swf Operatorenideal (DE-588)4284995-0 gnd rswk-swf Operatorraum (DE-588)4591231-2 gnd rswk-swf Spur Mathematik (DE-588)4202272-1 gnd rswk-swf Spur Mathematik (DE-588)4202272-1 s Operatorraum (DE-588)4591231-2 s Operatorenideal (DE-588)4284995-0 s Nichtkommutative Geometrie (DE-588)4171742-9 s DE-604 Sukochev, Fedor Verfasser (DE-588)1029362300 aut Zanin, Dmitriy Verfasser (DE-588)1029362211 aut Erscheint auch als Online-Ausgabe 978-3-11-026255-1 De Gruyter Studies in Mathematics 46 (DE-604)BV000005407 46 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=4097204&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025432088&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Lord, Steven Sukochev, Fedor Zanin, Dmitriy Singular traces theory and applications De Gruyter Studies in Mathematics Nichtkommutative Geometrie (DE-588)4171742-9 gnd Operatorenideal (DE-588)4284995-0 gnd Operatorraum (DE-588)4591231-2 gnd Spur Mathematik (DE-588)4202272-1 gnd |
| subject_GND | (DE-588)4171742-9 (DE-588)4284995-0 (DE-588)4591231-2 (DE-588)4202272-1 |
| title | Singular traces theory and applications |
| title_auth | Singular traces theory and applications |
| title_exact_search | Singular traces theory and applications |
| title_full | Singular traces theory and applications Fedor Sukochev ; Steven Lord ; Dmitriy Zanin |
| title_fullStr | Singular traces theory and applications Fedor Sukochev ; Steven Lord ; Dmitriy Zanin |
| title_full_unstemmed | Singular traces theory and applications Fedor Sukochev ; Steven Lord ; Dmitriy Zanin |
| title_short | Singular traces |
| title_sort | singular traces theory and applications |
| title_sub | theory and applications |
| topic | Nichtkommutative Geometrie (DE-588)4171742-9 gnd Operatorenideal (DE-588)4284995-0 gnd Operatorraum (DE-588)4591231-2 gnd Spur Mathematik (DE-588)4202272-1 gnd |
| topic_facet | Nichtkommutative Geometrie Operatorenideal Operatorraum Spur Mathematik |
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