C * -algebras:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English French |
| Published: |
Amsterdam [u.a.]
North-Holland
1977
|
| Series: | North-Holland mathematical library
15 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIII, 492 S. |
| ISBN: | 0720407621 |
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Record in the Search Index
| _version_ | 1804136095710969856 |
|---|---|
| adam_text | CONTENTS
Introduction xi
Part I. C* ALGEBRAS
Chapter 1. Normed involutive algebras 3
1.1. Involutive algebras 3
1.2. Normed involutive algebras 6
1.3. C* algebras 8
1.4. Commutative C* algebras 11
1.5. Functional calculus in C* algebras 12
1.6. Positive elements in C* algebras 15
1.7. Approximate identities in C* algebras 18
1.8. Quotient of a C* algebra 20
1.9. Addenda 22
Chapter 2. Positive forms and representations .... 26
2.1. Positive forms 26
2.2. Representations 31
2.3. Topologically irreducible representations 34
2.4. Positive forms and representations 37
2.5. Pure forms and irreducible representations .... 41
2.6. Existence of representations of C* algebras .... 45
2.7. The enveloping C* algebra of an involutive Banach
algebra 47
2.8. A theorem on transitivity 50
2.9. Ideals in C* algebras 54
2.10. Extension of representations of C* algebras .... 58
2.11. Passage to an ideal and to a quotient algebra ... 60
2.12. Addenda 64
vi CONTENTS
Chapter 3. The spectrum of a C* algebra 69
3.1. The Jacobson topology 69
3.2. The spectrum of an ideal and of a quotient algebra . 72
3.3. Norm and topology 74
3.4. Second definition of the topology on the spectrum . . 76
3.5. Third definition of the topology on the spectrum . . 80 »
3.6. Finite dimensional representations 84
3.7. More about the spaces Repn(A) 86
3.8. The Mackey Borel structure 89
3.9. Addenda 90
Chapter 4. Liminal C* algebras 94 ]i
4.1. The algebra of compact operators 94 ^
4.2. Liminal C* algebras 98
4.3. Postliminal C* algebras 99
4.4. The spectrum of a postliminal C* algebra 102 4
4.5. C* algebras with continuous trace 104
4.6. Borel structure on the spectrum of a postliminal C*
algebra 108
4.7. Addenda 109
Chapter 5. The type of a representation 114
5.1. Comparison of representations and comparison of
projections 114
5.2. Disjunction 115 *
5.3. Quasi equivalence 118
5.4. Representations of type I 121 j
5.5. Involutive algebras of type I 126 I
5.6. Representations of types II and III 127
5.7. Addenda 128 !
Chapter 6. Traces and representations 131
6.1. Traces 131
6.2. Bitraces 133
6.3. Maximal bitraces 135
6.4. Relations between traces and bitraces 137
6.5. The sum of two traces 139
6.6. Traces and representations 141
6.7. Characters and traceable factor representations . . 143
CONTENTS vii
6.8. Finite traces 146
6.9. Addenda 149
Chapter 7. The quasi spectrum 151
7.1. The space of factor representations 151
7.2. Definition of the quasi spectrum 154
7.3. Relations between the spectrum and the quasi spectrum 155
7.4. The finite part of the quasi spectrum 159
7.5. Addenda 160
Chapter 8. Integration and disintegration of
representations 162
8.1. Integration of representations 162
8.2. Equivalence of two direct integrals of representations 165
8.3. Disintegration of representations 167
8.4. Central disintegration 169
8.5. Disintegration into irreducible representations . . . 173
8.6. The case of postliminal C* algebras 174
8.7. An interlude 178
8.8. Disintegration of a positive form and of a trace . . 180
8.9. Addenda 188
Chapter 9. C* algebras of type I 190
9.1. Statement of the theorem, start of the proof ... 190
9.2. Preliminaries concerning systems of matrix units . . 191
9.3. Some lemmas 197
9.4. The proof of the theorem, concluded 206
9.5. Addenda 209
Chapter 10. Continuous fields of C* algebras .... 211
10.1. Continuous fields of Banach spaces 211
10.2. Total subsets 215
10.3. Continuous fields of C* algebras 218
10.4. The C* algebra defined by a continuous field of C*
algebras 219
10.5. The continuous field of C* algebras defined by certain
C* algebras 223
10.6. Some remarks concerning elementary C* algebras . 227
viii CONTENTS
10.7. The continuous field of elementary C* algebras defined
by a continuous field of Hilbert spaces 230 ,
10.8. Locally trivial fields of elementary C* algebras . . . 237
10.9. Application to C* algebras with continuous trace . . 245
10.10. Addenda 246
Chapter 11. Extension to C* algebras of the Stone
Weierstrass theorem 250
11.1. The case of postliminal C* algebras 250
11.2. Abundance of pure states in certain C* algebras . . 253
11.3. Statement of the theorem 255
11.4. Several lemmas 256
11.5. Proof of the theorem 261
11.6. Addenda 263
Chapter 12. The enveloping von neumann algebra of a
c* algebra 264
12.1. The second dual of a C* algebra 264
12.2. Polar decomposition of a linear from 267
12.3. Decomposition of an hermitian form into positive and
negative parts 271 ;
12.4. The positive part of an ideal in a C* algebra . . 273
12.5. Addenda 275
!
Part II. APPLICATIONS TO GROUP REPRESENTATIONS
Chapter 13. Unitary representations of locally
compact groups 279 ;
13.1. Elementary definitions concerning representations . . 279
13.2. The involutive algebra Ll(G) 282
13.3. Representations of G and representations of L G) . 283
13.4. Positive forms on L G) and positive definite functions 286 ,
13.5. Weak* convergence and compact convergence of con¬
tinuous positive definite functions 291
13.6. Pure positive definite functions 292
13.7. Positive definite measures 295
13.8. Square integrable positive definite functions .... 299
13.9. The C* algebra of a locally compact group .... 303 ,
CONTENTS ix
13.10. The Hilbert algebra of a unimodular locally compact
group 304
13.11. Addenda 305
Chapter 14. Square integrable irreducible
representations 309
14.1. Definition of square integrable representations . . . 309
14.2. Square integrable representations and minimal biinvar¬
iant subspaces of L2(G) 310
14.3. Coefficients of square integrable representations . . 311
14.4. Formal dimension and trace 315
14.5. Integrable representations 316
14.6. Addenda 317
Chapter 15. Representations of compact groups . . . 319
15.1. Complete reducibility 319
15.2. Irreducible representations of a compact group . . . 321
15.3. Characters of compact groups 322
15.4. Representations of finite groups 327
15.5. Use of compact subgroups of arbitrary groups . . . 329
15.6. Addenda 331
Chapter 16. Almost periodic functions 333
16.1. The compact group associated with a topological group 333
16.2. Almost periodic functions 335
16.3. The mean of an almost periodic function 337
16.4. Groups injectable in a compact group 338
16.5. Addenda 341
Chapter 17. Characters of a locally compact group . 343
17.1. Definitions 343
17.2. The character defined by a measure, and by a dis¬
tribution 344
17.3. Characters of finite type 348
17.4. Addenda 351
Chapter 18. The dual of a locally compact group . . 353
18.1. Definition of the dual 353
18.2. The Fourier transformation 356
18.3. The reduced dual 357
X CONTENTS
18.4. The reduced dual and integrable representations . . 362
18.5. The Mackey Borel structure 363
18.6. The quasi dual 364
18.7. Integration and disintegration of representations . . 364
18.8. The Plancherel measure 367
18.9. Addenda 370
Appendix A. von Neumann algebras 374
Appendix B. Miscellaneous results 393
Index of notation 401
Index of terminology 404
References 408
Supplementary bibliography 486
|
| adam_txt |
CONTENTS
Introduction xi
Part I. C* ALGEBRAS
Chapter 1. Normed involutive algebras 3
1.1. Involutive algebras 3
1.2. Normed involutive algebras 6
1.3. C* algebras 8
1.4. Commutative C* algebras 11
1.5. Functional calculus in C* algebras 12
1.6. Positive elements in C* algebras 15
1.7. Approximate identities in C* algebras 18
1.8. Quotient of a C* algebra 20
1.9. Addenda 22
Chapter 2. Positive forms and representations . 26
2.1. Positive forms 26
2.2. Representations 31
2.3. Topologically irreducible representations 34
2.4. Positive forms and representations 37
2.5. Pure forms and irreducible representations . 41
2.6. Existence of representations of C* algebras . 45
2.7. The enveloping C* algebra of an involutive Banach
algebra 47
2.8. A theorem on transitivity 50
2.9. Ideals in C* algebras 54
2.10. Extension of representations of C* algebras . 58
2.11. Passage to an ideal and to a quotient algebra . 60
2.12. Addenda 64
vi CONTENTS
Chapter 3. The spectrum of a C* algebra 69
3.1. The Jacobson topology 69
3.2. The spectrum of an ideal and of a quotient algebra . 72
3.3. Norm and topology 74
3.4. Second definition of the topology on the spectrum . . 76
3.5. Third definition of the topology on the spectrum . . 80 »
3.6. Finite dimensional representations 84
3.7. More about the spaces Repn(A) 86
3.8. The Mackey Borel structure 89
3.9. Addenda 90
Chapter 4. Liminal C* algebras 94 ]i
4.1. The algebra of compact operators 94 ^
4.2. Liminal C* algebras 98
4.3. Postliminal C* algebras 99
4.4. The spectrum of a postliminal C* algebra 102 4
4.5. C* algebras with continuous trace 104
4.6. Borel structure on the spectrum of a postliminal C*
algebra 108
4.7. Addenda 109
Chapter 5. The type of a representation 114
5.1. Comparison of representations and comparison of
projections 114
5.2. Disjunction 115 *
5.3. Quasi equivalence 118 \
5.4. Representations of type I 121 j
5.5. Involutive algebras of type I 126 I
5.6. Representations of types II and III 127
5.7. Addenda 128 !
Chapter 6. Traces and representations 131
6.1. Traces 131
6.2. Bitraces 133
6.3. Maximal bitraces 135
6.4. Relations between traces and bitraces 137
6.5. The sum of two traces 139
6.6. Traces and representations 141
6.7. Characters and traceable factor representations . . 143
CONTENTS vii
6.8. Finite traces 146
6.9. Addenda 149
Chapter 7. The quasi spectrum 151
7.1. The space of factor representations 151
7.2. Definition of the quasi spectrum 154
7.3. Relations between the spectrum and the quasi spectrum 155
7.4. The finite part of the quasi spectrum 159
7.5. Addenda 160
Chapter 8. Integration and disintegration of
representations 162
8.1. Integration of representations 162
8.2. Equivalence of two direct integrals of representations 165
8.3. Disintegration of representations 167
8.4. Central disintegration 169
8.5. Disintegration into irreducible representations . . . 173
8.6. The case of postliminal C* algebras 174
8.7. An interlude 178
8.8. Disintegration of a positive form and of a trace . . 180
8.9. Addenda 188
Chapter 9. C* algebras of type I 190
9.1. Statement of the theorem, start of the proof . 190
9.2. Preliminaries concerning systems of matrix units . . 191
9.3. Some lemmas 197
9.4. The proof of the theorem, concluded 206
9.5. Addenda 209
Chapter 10. Continuous fields of C* algebras . 211
10.1. Continuous fields of Banach spaces 211
10.2. Total subsets 215
10.3. Continuous fields of C* algebras 218
10.4. The C* algebra defined by a continuous field of C*
algebras 219
10.5. The continuous field of C* algebras defined by certain
C* algebras 223
10.6. Some remarks concerning elementary C* algebras . 227
viii CONTENTS
10.7. The continuous field of elementary C* algebras defined
by a continuous field of Hilbert spaces 230 ,
10.8. Locally trivial fields of elementary C* algebras . . . 237
10.9. Application to C* algebras with continuous trace . . 245
10.10. Addenda 246
Chapter 11. Extension to C* algebras of the Stone
Weierstrass theorem 250
11.1. The case of postliminal C* algebras 250
11.2. Abundance of pure states in certain C* algebras . . 253
11.3. Statement of the theorem 255
11.4. Several lemmas 256 '
11.5. Proof of the theorem 261
11.6. Addenda 263
Chapter 12. The enveloping von neumann algebra of a
c* algebra 264
12.1. The second dual of a C* algebra 264
12.2. Polar decomposition of a linear from 267
12.3. Decomposition of an hermitian form into positive and
negative parts 271 ;
12.4. The positive part of an ideal in a C* algebra . . 273
12.5. Addenda 275
!
Part II. APPLICATIONS TO GROUP REPRESENTATIONS
Chapter 13. Unitary representations of locally
compact groups 279 ;
13.1. Elementary definitions concerning representations . . 279 '
13.2. The involutive algebra Ll(G) 282
13.3. Representations of G and representations of L\G) . 283
13.4. Positive forms on L\G) and positive definite functions 286 ,
13.5. Weak* convergence and compact convergence of con¬
tinuous positive definite functions 291
13.6. Pure positive definite functions 292
13.7. Positive definite measures 295
13.8. Square integrable positive definite functions . 299
13.9. The C* algebra of a locally compact group . 303 ,
CONTENTS ix
13.10. The Hilbert algebra of a unimodular locally compact
group 304
13.11. Addenda 305
Chapter 14. Square integrable irreducible
representations 309
14.1. Definition of square integrable representations . . . 309
14.2. Square integrable representations and minimal biinvar¬
iant subspaces of L2(G) 310
14.3. Coefficients of square integrable representations . . 311
14.4. Formal dimension and trace 315
14.5. Integrable representations 316
14.6. Addenda 317
Chapter 15. Representations of compact groups . . . 319
15.1. Complete reducibility 319
15.2. Irreducible representations of a compact group . . . 321
15.3. Characters of compact groups 322
15.4. Representations of finite groups 327
15.5. Use of compact subgroups of arbitrary groups . . . 329
15.6. Addenda 331
Chapter 16. Almost periodic functions 333
16.1. The compact group associated with a topological group 333
16.2. Almost periodic functions 335
16.3. The mean of an almost periodic function 337
16.4. Groups injectable in a compact group 338
16.5. Addenda 341
Chapter 17. Characters of a locally compact group . 343
17.1. Definitions 343
17.2. The character defined by a measure, and by a dis¬
tribution 344
17.3. Characters of finite type 348
17.4. Addenda 351
Chapter 18. The dual of a locally compact group . . 353
18.1. Definition of the dual 353
18.2. The Fourier transformation 356
18.3. The reduced dual 357
X CONTENTS
18.4. The reduced dual and integrable representations . . 362
18.5. The Mackey Borel structure 363
18.6. The quasi dual 364
18.7. Integration and disintegration of representations . . 364
18.8. The Plancherel measure 367
18.9. Addenda 370
Appendix A. von Neumann algebras 374
Appendix B. Miscellaneous results 393
Index of notation 401
Index of terminology 404
References 408
Supplementary bibliography 486 |
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| illustrated | Not Illustrated |
| index_date | 2024-07-02T16:16:18Z |
| indexdate | 2024-07-09T20:50:57Z |
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| open_access_boolean | |
| owner | DE-706 DE-20 |
| owner_facet | DE-706 DE-20 |
| physical | XIII, 492 S. |
| publishDate | 1977 |
| publishDateSearch | 1977 |
| publishDateSort | 1977 |
| publisher | North-Holland |
| record_format | marc |
| series | North-Holland mathematical library |
| series2 | North-Holland mathematical library |
| spelling | Dixmier, Jacques Verfasser aut Les C * -algébres et leurs représentations C * -algebras Jacques Dixmier Amsterdam [u.a.] North-Holland 1977 XIII, 492 S. txt rdacontent n rdamedia nc rdacarrier North-Holland mathematical library 15 C*-algebra's gtt C*-algèbres ram Représentations d'algèbres ram Représentations de groupes ram C*-algebras Representations of algebras Representations of groups Darstellung Mathematik (DE-588)4128289-9 gnd rswk-swf Darstellung (DE-588)4200624-7 gnd rswk-swf C-Stern-Algebra (DE-588)4136693-1 gnd rswk-swf Darstellung Mathematik (DE-588)4128289-9 s DE-604 C-Stern-Algebra (DE-588)4136693-1 s Darstellung (DE-588)4200624-7 s 1\p DE-604 North-Holland mathematical library 15 (DE-604)BV000005206 15 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015336675&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Dixmier, Jacques C * -algebras North-Holland mathematical library C*-algebra's gtt C*-algèbres ram Représentations d'algèbres ram Représentations de groupes ram C*-algebras Representations of algebras Representations of groups Darstellung Mathematik (DE-588)4128289-9 gnd Darstellung (DE-588)4200624-7 gnd C-Stern-Algebra (DE-588)4136693-1 gnd |
| subject_GND | (DE-588)4128289-9 (DE-588)4200624-7 (DE-588)4136693-1 |
| title | C * -algebras |
| title_alt | Les C * -algébres et leurs représentations |
| title_auth | C * -algebras |
| title_exact_search | C * -algebras |
| title_exact_search_txtP | C * -algebras |
| title_full | C * -algebras Jacques Dixmier |
| title_fullStr | C * -algebras Jacques Dixmier |
| title_full_unstemmed | C * -algebras Jacques Dixmier |
| title_short | C * -algebras |
| title_sort | c algebras |
| topic | C*-algebra's gtt C*-algèbres ram Représentations d'algèbres ram Représentations de groupes ram C*-algebras Representations of algebras Representations of groups Darstellung Mathematik (DE-588)4128289-9 gnd Darstellung (DE-588)4200624-7 gnd C-Stern-Algebra (DE-588)4136693-1 gnd |
| topic_facet | C*-algebra's C*-algèbres Représentations d'algèbres Représentations de groupes C*-algebras Representations of algebras Representations of groups Darstellung Mathematik Darstellung C-Stern-Algebra |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015336675&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000005206 |
| work_keys_str_mv | AT dixmierjacques lescalgebresetleursrepresentations AT dixmierjacques calgebras |