Measured quantum groupoids in action:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Montreuil
Gauthier-Villars
2008 [erschienen] 2009
|
| Series: | Mémoire de la Société Mathématique de France
114 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Physical Description: | 150 S. |
| ISBN: | 9782856292655 |
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MARC
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| 100 | 1 | |a Enock, Michel |e Verfasser |0 (DE-588)113258658 |4 aut | |
| 245 | 1 | 0 | |a Measured quantum groupoids in action |c Michel Enock |
| 264 | 1 | |a Montreuil |b Gauthier-Villars |c 2008 [erschienen] 2009 | |
| 300 | |a 150 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Mémoire de la Société Mathématique de France |v 114 | |
| 650 | 7 | |a Groupes localement compacts |2 ram | |
| 650 | 7 | |a Groupoïdes quantiques |2 ram | |
| 650 | 7 | |a Von Neumann, Algèbres de |2 ram | |
| 650 | 4 | |a Locally compact groups | |
| 650 | 4 | |a Quantum groupoids | |
| 650 | 4 | |a Von Neumann algebras |x Crossed products | |
| 650 | 0 | 7 | |a Quantengruppe |0 (DE-588)4252437-4 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Quantengruppe |0 (DE-588)4252437-4 |D s |
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Record in the Search Index
| _version_ | 1813056763303297024 |
|---|---|
| adam_text |
CONTENTS
1.
Introduction
. 1
1.1. 1
1.2. 1
1-3. 2
1.4. 2
1.5. 2
1.6. 2
2.
Preliminaries
. 5
2.1.
Spatial theory
[5], [31], [34] . 5
2.2.
Jones' basic construction and operator-valued weights; depth
2
inclusions
8
2.2.1.
Proposition
. 9
2.2.2.
Lemma
. 10
2.3.
Relative tensor product
[5], [31], [34] . 10
2.4.
Fiber product
[35], [15] . 14
2.5.
Slice maps
[8] . 15
2.6.
Vaes' Radon-Nikodym theorem
. 17
3.
Measured quantum groupoids
. 19
3.1.
Definition
. 19
3.2.
Definition
. 21
3.3.
Algebras and Hopf-bimodules associated to a pseudo-multiplicative
unitary
. 22
3.4.
Fundamental example
. 23
3.5.
Definitions
([20], [21]) . 24
3.6.
Theorem([20],
[21]) . 24
3.7.
Definitions
. 25
3.8.
Theorem
([21], [11]) . 26
3.9.
Notations
. 28
3.10.
Theorem
([21]) . 28
3.11.
Theorem
([21]) . 30
3.12.
Theorem([21])
. 31
3.13.
Example
. 32
3.14.
Example
. 32
CONTENTS
3.15. Theorem([15], [8] 8.2 and 8.3) . 33
4.
Left
invariance
revisited
. 35
4.1.
Definitions and notations
. 35
4.2.
Lemma
. 36
4.3.
Lemma
. 36
4.4.
Definitions and lemma
. 37
4.5.
Proposition
. 38
4.6.
Proposition
. 39
4.7.
Proposition
. 41
4.8.
Theorem
. 42
4.9.
Proposition
. 42
4.10.
Proposition
. 43
4.11.
Proposition
. 44
4.12.
Theorem
. 46
5.
Corepresentations of measured
quantum groupoids
. 47
5.1.
Definition
. 47
5.2.
Theorem
. 48
5.3.
Corollary
. 49
5.4.
Proposition
. 49
5.5.
Proposition
. 51
5.6.
Example
. 51
5.7.
Proposition
. 52
5.8.
Proposition
. 52
5.9.
Proposition
. 53
5.10.
Theorem
. 53
5.11.
Theorem
. 54
6.
Actions of measured quantum groupoids
. 55
6.1.
Definition
. 55
6.2.
Example
. 56
6.3.
Example
. 56
6.4.
Example
. 56
6.5.
Example
. 56
6.6.
Proposition
. 57
6.7.
Theorem
. 57
6.8.
Proposition
. 58
6.9.
Definition
. 59
6.10.
Example
. 59
6.11.
Definition
. 59
6.12.
Proposition
. 59
6.13.
Proposition
. 60
MÉMOIRES DE LA
SMF
114
CONTENTS
6.14. Definition . 60
7.
Some technical properties of actions
. 61
7.1.
Definition
. 61
7.2.
Proposition
. 61
7.3.
Definition
. 62
7.4.
Lemma
. 62
7.5.
Lemma
. 63
7.6.
Proposition
. 64
7.7.
Proposition
. 64
7.8.
Corollary
. 65
8.
The standard implementation of an action: the case of a ¿-invariant
weight
. 67
8.1.
Definition
. 67
8.2.
Example
. 68
8.3.
Lemma
. 68
8.4.
Proposition
. 69
8.5.
Proposition
. 71
8.6.
Theorem
. 73
8.7.
Lemma
. 74
8.8.
Theorem
. 75
8.9.
Corollary
. 78
8.10.
Corollary
. 79
8.11.
Corollary
. 79
9.
Crossed-product and dual actions
. 81
9.1.
Definition
. 81
9.2.
Example
. 81
9.3.
Example
. 81
9.4.
Theorem
. 82
9.5.
Example
. 83
9.6.
Example
. 83
9.7.
Proposition
. 83
9.8.
Theorem
. 84
9.9.
Definition
. 84
9.10.
Lemma
. 85
9.11.
Proposition
. 85
9.12.
Lemma
. 86
10.
An auxilliary weight on the crossed-product
. 89
10.1.
Proposition
. 89
10.2.
Proposition
. 90
10.3.
Proposition
. 91
SOCIÉTÉ MATHÉMATIQUE DE
FRANCE
200β
CONTENTS
10.4. Lemma . 92
10.5. Proposition . 93
10.6.
Corollary
. 94
10.7. Theorem . 95
10.8. Theorem . 95
10.9. Proposition . 96
10.10. Definition . 96
10.11. Proposition . 96
10.12.
Corollary
. 96
11. Biduality . 97
11.1. Lemma . 97
11.2. Proposition . 98
11.3. Proposition .100
11.4. Proposition .101
11.5. Theorem .102
11.6. Theorem .103
11.7. Theorem .104
11.8. Theorem .105
11.9. Theorem .105
11.10.
Remark
.105
12.
Characterization of crossed-products
.107
12.1.
Notations
.107
12.2.
Lemma
.107
12.3.
Theorem
.109
12.4.
Corollary
.
Ill
12.5.
Corollary
.112
13.
Dual weight; bidual weight; depth
2
inclusion associated to an
action
.113
13.1.
Definition
.113
13.2.
Example
.113
13.3.
Theorem
.114
13.4.
Theorem
.114
13.5.
Proposition
.115
13.6.
Lemma
.116
13.7.
Theorem
.116
13.8.
Theorem
.118
13.9.
Theorem
.119
13.10.
Theorem
.120
13.11.
Remark
. .121
MÉMOIRES DE LA
SMF
114
CONTENTS
14.
The measured quantum groupoid
associated to an action
.123
14.1.
Theorem
.123
14.2.
Theorem
.124
14.3.
Theorem
.124
14.4.
Theorem
.125
14.5.
Corollary
.126
14.6.
Corollary
.126
14.7.
Proposition
.127
14.8.
Example
.127
14.9.
Remark
.128
Appendix
.129
A. Coinverse
and scaling group
of a measured quantum groupoid
.131
A.I. Lemma
.131
A.2. Definition
.132
A.3. Proposition
.132
A.4. Proposition
.133
A.5. Theorem
.133
A.6. Theorem
.134
A.7. Theorem
.134
A.8. Definition
.135
A.9. Theorem
.135
A.10. Proposition
.136
B. Automorphism groups on the basis of a measured quantum
groupoid
.137
B.I. Lemma
.137
B.2. Proposition
.138
B.3. Proposition
.138
B.4. Corollary
.140
B.5. Lemma
.141
B.6. Lemma
.142
B.7. Proposition
.143
B.8. Theorem
.146
Bibliography
.147
SOCIÉTÉ MATHÉMATIQUE DE
FRANCE
2008
Franek Lesieur
had introduced in his thesis (now published in an expended
and revised version in the
Mémoires
de
la
SMF
(2007))
a notion
of measured
quantum groupoid, in the setting of
von
Neumann algebras and a simplification
of Lesieur's axioms is presented in an appendix of this article. We here de¬
velop the notions of actions, crossed-product, and obtain a biduality theorem,
following what had been done by Stefaan Vaes for locally compact quantum
groups. Moreover, we prove that the inclusion of the initial algebra into its
crossed-product is depth
2,
which gives a converse of a result proved by Jean-
Michel
Vallin
and the author. More precisely, to any action of a measured
quantum groupoid, we associate another measured quantum groupoid. In par¬
ticular, starting from an action of a locally compact quantum group, we obtain
a measured quantum groupoid canonically associated to this action; when the
action is outer, this measured quantum groupoid is the initial locally compact
quantum group.
Frank Lesieur
a introduit dans sa thèse (maintenant publiée dans une version
révisée et complétée dans les Mémoires de la
SMF
(2007))
une notion de grou¬
poïde quantique mesuré, dans le cadre des
algebres
de
von Neumann, et
une
simplification des axiomes de Lesieur est placée en appendice de cet article.
Nous développons ici les notions d'action d'un groupoïde quantique mesuré,
de produit-croisé et un théorème de bidualité est démontré, en s'inspirant lar¬
gement de ce qui a été fait par Stefaan Vaes pour les groupes quantiques
localement compacts. Ainsi, nous prouvons que l'inclusion de l'algèbre initiale
dans son produit croisé est de profondeur
2,
ce qui fournit une réciproque
à un résultat démontré par Jean-Michel
Vallin
et l'auteur. De plus, à toute
action d'un groupoïde quantique mesuré, on associe un autre groupoïde quan¬
tique mesuré
;
ainsi, en particulier, on construit un groupoïde quantique mesuré
associé canoniquement à toute action d'un groupe quantique localement com¬
pact
;
quand cette action est extérieure, ce groupoïde quantique mesuré est le
groupe quantique initial. |
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| id | DE-604.BV035675231 |
| illustrated | Not Illustrated |
| indexdate | 2024-10-16T08:01:09Z |
| institution | BVB |
| isbn | 9782856292655 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-017729443 |
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| owner_facet | DE-19 DE-BY-UBM DE-703 DE-355 DE-BY-UBR DE-824 DE-29T |
| physical | 150 S. |
| publishDate | 2008 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | Gauthier-Villars |
| record_format | marc |
| series2 | Mémoire de la Société Mathématique de France |
| spelling | Enock, Michel Verfasser (DE-588)113258658 aut Measured quantum groupoids in action Michel Enock Montreuil Gauthier-Villars 2008 [erschienen] 2009 150 S. txt rdacontent n rdamedia nc rdacarrier Mémoire de la Société Mathématique de France 114 Groupes localement compacts ram Groupoïdes quantiques ram Von Neumann, Algèbres de ram Locally compact groups Quantum groupoids Von Neumann algebras Crossed products Quantengruppe (DE-588)4252437-4 gnd rswk-swf Quantengruppe (DE-588)4252437-4 s DE-604 Société Mathématique de France Mémoire de la Société Mathématique de France 114 (DE-604)BV000000921 114 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017729443&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017729443&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Enock, Michel Measured quantum groupoids in action Groupes localement compacts ram Groupoïdes quantiques ram Von Neumann, Algèbres de ram Locally compact groups Quantum groupoids Von Neumann algebras Crossed products Quantengruppe (DE-588)4252437-4 gnd |
| subject_GND | (DE-588)4252437-4 |
| title | Measured quantum groupoids in action |
| title_auth | Measured quantum groupoids in action |
| title_exact_search | Measured quantum groupoids in action |
| title_full | Measured quantum groupoids in action Michel Enock |
| title_fullStr | Measured quantum groupoids in action Michel Enock |
| title_full_unstemmed | Measured quantum groupoids in action Michel Enock |
| title_short | Measured quantum groupoids in action |
| title_sort | measured quantum groupoids in action |
| topic | Groupes localement compacts ram Groupoïdes quantiques ram Von Neumann, Algèbres de ram Locally compact groups Quantum groupoids Von Neumann algebras Crossed products Quantengruppe (DE-588)4252437-4 gnd |
| topic_facet | Groupes localement compacts Groupoïdes quantiques Von Neumann, Algèbres de Locally compact groups Quantum groupoids Von Neumann algebras Crossed products Quantengruppe |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017729443&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017729443&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000921 |
| work_keys_str_mv | AT enockmichel measuredquantumgroupoidsinaction |