Noncommutative geometry:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English French |
| Published: |
San Diego [u.a.]
Academic Press
1994
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Hier auch später erschienene, unveränderte Nachdrucke |
| Physical Description: | XIII, 661 S. graph. Darst. |
| ISBN: | 012185860X 9780121858605 |
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MARC
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| 084 | |a MAT 500f |2 stub | ||
| 100 | 1 | |a Connes, Alain |d 1947- |e Verfasser |0 (DE-588)112614760 |4 aut | |
| 240 | 1 | 0 | |a Géométrie non commutative |
| 245 | 1 | 0 | |a Noncommutative geometry |c Alain Connes |
| 264 | 1 | |a San Diego [u.a.] |b Academic Press |c 1994 | |
| 300 | |a XIII, 661 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Hier auch später erschienene, unveränderte Nachdrucke | ||
| 650 | 7 | |a Algebraïsche meetkunde |2 gtt | |
| 650 | 4 | |a Anneaux non commutatifs | |
| 650 | 7 | |a Functionaalanalyse |2 gtt | |
| 650 | 7 | |a Globale analyse |2 gtt | |
| 650 | 4 | |a Géométrie algébrique | |
| 650 | 7 | |a Mathematische fysica |2 gtt | |
| 650 | 7 | |a Niet-commutatieve structuren |2 gtt | |
| 650 | 7 | |a Operatoren |2 gtt | |
| 650 | 4 | |a Geometry, Algebraic | |
| 650 | 4 | |a Noncommutative rings | |
| 650 | 0 | 7 | |a Operatoralgebra |0 (DE-588)4129366-6 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Nichtkommutative Geometrie |0 (DE-588)4171742-9 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Nichtkommutative Differentialgeometrie |0 (DE-588)4311174-9 |2 gnd |9 rswk-swf |
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| 689 | 2 | 1 | |a Nichtkommutative Differentialgeometrie |0 (DE-588)4311174-9 |D s |
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Record in the Search Index
| _version_ | 1821976831464570880 |
|---|---|
| adam_text |
TABLE
OF
CONTENTS
PREFACE
.
хш
INTRODUCTION
. 1
NONCOMMUTATIVE SPACES AND MEASURE THEORY
.33
1. Heisenberg
and the Noncommutative Algebra of Physical
Quantities Associated to a Microscopic System
.33
2.
Statistical State of a Macroscopic System and Quantum
Statistical Mechanics
.39
3.
Modular Theory and the Classification of Factors
.43
4.
Geometric Examples of
von
Neumann Algebras: Measure
Theory of Noncommutative Spaces
.45
a. Classical Lebesgue measure theory
. 46
β.
Foliations
.49
y. The
von
Neumann algebra of a foliation
.50
5.
The Index Theorem for Measured Foliations
.59
a. Transverse measures for foliations
.60
β.
The
Ruelle-Sullivan
cycle and the
Euler
number of a
measured foliation
.64
y. The index theorem for measured foliations
.69
A. Appendix: Transverse Measures and Averaging Sequences
.72
B. Appendix: Abstract Transverse Measure Theory
.72
C. Appendix: Noncommutative Spaces and Set Theory
.74
v
vi
TABLE OF
CONTENTS
П.
TOPOLOGY AND K-THEORY
.79
1.
C*-algebras and their ^-theory
.80
2.
Elementary Examples of Quotient Spaces
.85
a. Open covers of manifolds
.86
β.
The dual of the infinite dihedral group
Γ
= 1x112.87
3.
The Space X of Penrose Tilings
.88
4.
Duals of Discrete Groups and the Novikov Conjecture
.94
5.
The Tangent Groupoid of a Manifold
.99
6.
Wrong-way Functoriality in K-theory as a Deformation
.107
а. The index groupoid of a linear map
.107
β.
Construction of/!
є
E{T*M
©
f*TN,N)
.108
y.
jř-orientations
of vector bundles and maps
.109
б.
Wrong-way functoriality for K-oriented maps
.110
7.
The Orbit Space of a Group Action
.
Ill
8.
The Leaf Space of a Foliation
.118
a. Construction of C?(V,F)
.118
β.
Closed transversals and idempotents of C?(V,F)
.120
y. The analytic assembly map
μ
:
K^T{BG)
-
ÍT(C*(V,F))
.125
9.
The Longitudinal Index Theorem for Foliations
.129
a. Construction of Ind(D)
є
K0(J)
.130
β.
Significance of the C*-algebra index
.132
y. The longitudinal index theorem
.133
10.
The Analytic Assembly Map and Lie Groups
.136
а. Geometric cycles for smooth groupoids
.136
β.
Lie groups and deformations
.140
y. The G-equivariant index of elliptic operators on
homogeneous spaces of lie groups
.142
б.
The K-theory K(C*(G)) for lie groups
.148
ε.
The general conjecture for smooth groupoids
.151
A. Appendix: C*-modules and Strong Morita Equivalence
.152
B. Appendix: f-theory and Deformations of Algebras
.158
а. Defformations of C*-algebras and asymptotic
morphisms
.159
β.
Composition of asymptotic morphisms
.161
y. Asymptotic morphisms and exact sequences of
C*-algebras
.163
б.
The cone of a map and half-exactness
.165
ε.
f
-theory
.168
C.
Appendix: Crossed Products of C*-algebras and the Thom
Isomorphism
.171
D. Appendix: Penrose Tilings
.175
TABLE
OF
CONTENTS vü
Ш.
CYCLIC COHOMOLOGY AND DIFFERENTIAL GEOMETRY
.179
1.
Cyclic Cohomology
.182
a. Characters of cycles and the cup product in HC*
.183
β.
Cobordisms of cycles and the operator
В
.194
у.
The exact couple relating
ЯС*(Л)
to
Hochschild
cohomology
.199
2.
Examples
.207
а. Я
=
CiV), V a compact smooth manifold
.207
β.
The cyclic cohomology of the
noncommutative
torus
Л = Ле,вєЛ/2
.212
у.
The cyclic cohomology of the group ring
СГ
for
Γ
a discrete
group
.213
б.
Cyclic cohomology of C?(V
хГ).
216
3.
Pairing of Cyclic Cohomology with K-Theory
.223
4.
The Higher Index Theorem for Covering Spaces
.233
«.The smooth groupoid of a covering space
.233
β.
The group ring
31Γ
.234
y. The index theorem
.237
5.
The Novikov Conjecture for Hyperbolic Groups
.238
o«. Word hyperbolic groups
.238
β.
The Haagerup inequality
.241
y. Extension to Cf
(Γ)
of
Aľ-theory
invariants
.242
6.
Factors of Type
Ш,
Cyclic Cohomology and the Godbillon-Vey
Invariant
.244
or. Extension of densely defined cyclic cocycles on Banach
algebras
.246
β.
The Bott-Thurston cocycle and the equality
GV = is£t[V/Fl
.257
у.
Invariant measures on the flow of weights
.259
7.
The Transverse Fundamental Class for Foliations and
Geometric Corollaries
.263
а. The transverse fundamental class
.263
β.
Geometric corollaries
.268
y. Index formula for longitudinal elliptic operators
.271
A. Appendix: The Cyclic Category
Λ
.2 74
α.
The simplicial category
Δ
.275
β.
The cyclic category
Λ
.276
у.
The
Л
-module
Л"
associated to an algebra
Ά
.280
б.
Cyclic spaces and S1 spaces
.282
B. Appendix: Locally Convex Algebras
.283
С
Appendix: Stability under Holomorphic Functional
Calculus
.285
vüi
TABLE OF CONTENTS
IV. QUANTIZED CALCULUS
.287
1.
Quantized Differential Calculus and Cyclic Cohomology
.292
а. The cycle associated to
a Fredholm
module
.292
β.
The periodicity operator
5
and the Chern character
.294
y. Pairing with K-theory and index formula
.296
2.
The Dixmier Trace and the Hochschild Class of the
Character
.299
ix.
General properties of interpolation ideals L{pA)
.299
β.
The Dixmier trace
.303
y. The residue formula for the Hochschild class of the
character of
Fredholm
modules
.308
б.
Growth of algebras and degree of summability of
It-cycles
.310
3.
Quantized Calculus in One Variable and Fractal Sets
.313
a. Quantized calculus in one variable
.314
β.
The class of df in
£"·"/.£&·".317
y. The Dixmier trace of
ƒ
(Z)\dZ\p
.321
Ő.
The harmonic measure and non-normality of the Dixmier
trace
.326
ε.
Cantor sets, Dixmier trace and Minkowski measure
.327
4.
Conformai
Manifolds
.331
<x. Quantized calculus on
conformai
manifolds
.331
β.
Perturbation of
Fredhohn
modules by the
commutant
von
Neumann algebra
.335
y. The 4-dimensional analogue of the Polyakov action
.338
5.
Fredholm
Modules and Rank-One Discrete Groups
.340
6.
Elliptic Theory on the
Noncommutative
Torus l\ and the
Quantum Hall Effect
.347
<x. Elliptic theory on T|
.348
β.
The quantum Hall effect
.355
y. The work of J. Bellissard on the integrality of
σΗ
.357
7.
Entire Cyclic Cohomology
.366
а. Entire cyclic cohomology of Banach algebras
.367
β.
Infinite-dimensional cycles
.371
y. Traces on (¿A and
ΈΑ
.374
б.
Pairing with KQ(A)
.378
ε.
Entire cyclic cohomology of S1
.381
8.
The Chern Character of 0-summable
Fredholm
Modules
.390
а. Fredholm
modules and
íT-cycles
.
.^
.391
β.
The supergroup
R1·1
and the convolution algebra
£
of
operator-valued distributions on
[0, +00[ .395
y. The Chern character of X-cycles
.399
б.
The index formula
.403
ε.
The
ДО
cocycle
.405
TABLE
OF
CONTENTS ix
9. 0-summable
íf-cycles,
Discrete Groups and
Quantum
Field
Theory
.407
Oř.
Discrete subgroups of lie groups
.407
ß. Supersymmetric
quantum field theory
.415
A. Appendix: Kasparov's Bivariant Theory
.428
B. Appendix: Real and Complex Interpolation of Banach
Spaces
.436
C. Appendix: Normed Ideals of Compact Operators
.439
D. Appendix: The Chern Character of Deformations of
Algebras
.443
V. OPERATOR ALGEBRAS
.447
1.
The Papers of Murray and
von
Neumann
.448
а. Examples of
von
Neumann algebras
.449
β.
Reduction theory
.452
y. Comparison of subrepresentations, comparison of
projections and the relative dimension function
.453
б.
Algebraic isomorphism and spatial isomorphism
.455
ε.
The first two examples of type
Πχ
factors, the hyperfinite
factor and the property
Γ
.456
2.
Representations of C*-algebras
.457
3.
The Algebraic Framework for
Noncommutative
Integration
and the Theory of Weights
.460
4.
The Factors of Powers,
Araki
and Woods, and of
Krieger.463
5.
The
Radon-Nikodým
Theorem and Factors of Type
ΙΠΑ
.469
a. The
Radon-Nikodým
theorem
.469
β.
The factors of type
Шл
.472
6.
Noncommutative Ergodic
Theory
.475
а. Rokhlin's theorem
.476
β.
Entropy
.477
y. Approximately inner automorphisms
.482
б.
Centrally trivial automorphisms
.482
f. The obstruction
γ{θ)
.484
ζ.
The list of automorphisms of JR up to outer conjugacy
.485
η.
Automorphisms of the AraM-Woods factor Rotl of type
Π«
.'.487
7.
Amenable
von
Neumann Algebras
.488
a. Approximation by finite-dimensional algebras
.488
β.
The properties
Ρ
of Schwartz,
E
of Hakeda and Tomiyama,
and
injecüvity
.489
у.
Semidiscrete
von Neumann
algebras .
490
8.
The Flow of Weights: mod{M)
.493
a. The discrete decomposition of factors of type
Шо
.493
β.
Continuous decomposition of type
Ш
factors
.494
y. Functorial definition of the flow of weights
.495
TABLE
OF
CONTENTS
б.
Virtual
groups and the flow of weights as modular
spectrum
.497
9.
The Classification of Amenable Factors
.499
а. Factors of type
Пх
.499
β.
Factors of type
Π«,
.501
y. Factors of type
ША, А є
]0,1[
.502
б.
Factors of type
Шо
.503
ε.
Factors of type
ΠΙχ
.504
10.
Subfactors of Type
Щ
Factors
.505
a. Index of subfactors
.505
β.
Positive Markov traces on
Hecke
algebras
.508
11. Hecke
Algebras, Type HI Factors and Statistical Theory of
Prime Numbers
.510
a. Description of the system and its phase transition
.510
β.
Bosonic second quantization and prime numbers as a
subset of
R
.515
у.
Products of trees and the
noncommutative
Hecke
algebra
.518
A. Appendix: Crossed Products of
von
Neumann Algebras
.524
B. Appendix: Correspondences
.526
а. Half densities and the identity correspondence
.527
β.
Correspondences and ♦-homomorphisms
.529
y. Coefficients of correspondences and completely positive
maps
.531
б.
Composition of correspondences
.533
ε.
Correspondences, hyperfiniteness and property
Τ
.536
VI. THE METRIC ASPECT OF NONCOMMUTATTVE GEOMETRY
. 539
1.
Riemannian Manifolds and the Dirac Operator
.543
2.
Positivity
in
Hochschild Cohomology
and Inequalities for the
Yang-Mills Action
.556
3.
Product of the Continuum by the Discrete and the Symmetry
Breaking Mechanism
.561
4.
The Notion of Manifold in
Noncommutative
Geometry
.585
с«.
The classical notion of manifold
.585
β.
Bivariant K-theory and
Poincaré
duality
.587
y.
Poincaré
duality and cyclic cohomology
.591
б.
Bivector potentials on an
(Jł,S)-bimodule
(Sj,D,y)
.594
5.
The Standard
ІД1)
x
51/(2)
x
SUO)
Model
.595
α.
The dictionary
.596
β.
The standard model
.598
y. Geometric structure of the finite space
F
.601
б.
Geometric structure of the standard model
.604
£.
Unimodularity condition and hypercharges
.609
TABLE
OF
CONTENTS xi
BIBLIOGRAPHY
.613
NOTATION AND CONVENTIONS.645
INDEX.649 |
| any_adam_object | 1 |
| author | Connes, Alain 1947- |
| author_GND | (DE-588)112614760 |
| author_facet | Connes, Alain 1947- |
| author_role | aut |
| author_sort | Connes, Alain 1947- |
| author_variant | a c ac |
| building | Verbundindex |
| bvnumber | BV010060863 |
| callnumber-first | Q - Science |
| callnumber-label | QA564 |
| callnumber-raw | QA564 |
| callnumber-search | QA564 |
| callnumber-sort | QA 3564 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 240 SK 620 |
| classification_tum | MAT 149f MAT 110f MAT 500f |
| ctrlnum | (OCoLC)30738985 (DE-599)BVBBV010060863 |
| dewey-full | 516.3/5 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.3/5 |
| dewey-search | 516.3/5 |
| dewey-sort | 3516.3 15 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV010060863 |
| illustrated | Illustrated |
| indexdate | 2025-01-22T19:01:48Z |
| institution | BVB |
| isbn | 012185860X 9780121858605 |
| language | English French |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-006675067 |
| oclc_num | 30738985 |
| open_access_boolean | |
| owner | DE-91 DE-BY-TUM DE-91G DE-BY-TUM DE-20 DE-355 DE-BY-UBR DE-703 DE-384 DE-19 DE-BY-UBM DE-29T DE-83 DE-188 DE-11 DE-824 |
| owner_facet | DE-91 DE-BY-TUM DE-91G DE-BY-TUM DE-20 DE-355 DE-BY-UBR DE-703 DE-384 DE-19 DE-BY-UBM DE-29T DE-83 DE-188 DE-11 DE-824 |
| physical | XIII, 661 S. graph. Darst. |
| publishDate | 1994 |
| publishDateSearch | 1994 |
| publishDateSort | 1994 |
| publisher | Academic Press |
| record_format | marc |
| spelling | Connes, Alain 1947- Verfasser (DE-588)112614760 aut Géométrie non commutative Noncommutative geometry Alain Connes San Diego [u.a.] Academic Press 1994 XIII, 661 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Hier auch später erschienene, unveränderte Nachdrucke Algebraïsche meetkunde gtt Anneaux non commutatifs Functionaalanalyse gtt Globale analyse gtt Géométrie algébrique Mathematische fysica gtt Niet-commutatieve structuren gtt Operatoren gtt Geometry, Algebraic Noncommutative rings Operatoralgebra (DE-588)4129366-6 gnd rswk-swf Nichtkommutative Geometrie (DE-588)4171742-9 gnd rswk-swf Nichtkommutative Differentialgeometrie (DE-588)4311174-9 gnd rswk-swf Algebraische Geometrie (DE-588)4001161-6 gnd rswk-swf Nichtkommutative Geometrie (DE-588)4171742-9 s DE-604 Algebraische Geometrie (DE-588)4001161-6 s Operatoralgebra (DE-588)4129366-6 s Nichtkommutative Differentialgeometrie (DE-588)4311174-9 s 1\p DE-604 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006675067&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 | Connes, Alain 1947- Noncommutative geometry Algebraïsche meetkunde gtt Anneaux non commutatifs Functionaalanalyse gtt Globale analyse gtt Géométrie algébrique Mathematische fysica gtt Niet-commutatieve structuren gtt Operatoren gtt Geometry, Algebraic Noncommutative rings Operatoralgebra (DE-588)4129366-6 gnd Nichtkommutative Geometrie (DE-588)4171742-9 gnd Nichtkommutative Differentialgeometrie (DE-588)4311174-9 gnd Algebraische Geometrie (DE-588)4001161-6 gnd |
| subject_GND | (DE-588)4129366-6 (DE-588)4171742-9 (DE-588)4311174-9 (DE-588)4001161-6 |
| title | Noncommutative geometry |
| title_alt | Géométrie non commutative |
| title_auth | Noncommutative geometry |
| title_exact_search | Noncommutative geometry |
| title_full | Noncommutative geometry Alain Connes |
| title_fullStr | Noncommutative geometry Alain Connes |
| title_full_unstemmed | Noncommutative geometry Alain Connes |
| title_short | Noncommutative geometry |
| title_sort | noncommutative geometry |
| topic | Algebraïsche meetkunde gtt Anneaux non commutatifs Functionaalanalyse gtt Globale analyse gtt Géométrie algébrique Mathematische fysica gtt Niet-commutatieve structuren gtt Operatoren gtt Geometry, Algebraic Noncommutative rings Operatoralgebra (DE-588)4129366-6 gnd Nichtkommutative Geometrie (DE-588)4171742-9 gnd Nichtkommutative Differentialgeometrie (DE-588)4311174-9 gnd Algebraische Geometrie (DE-588)4001161-6 gnd |
| topic_facet | Algebraïsche meetkunde Anneaux non commutatifs Functionaalanalyse Globale analyse Géométrie algébrique Mathematische fysica Niet-commutatieve structuren Operatoren Geometry, Algebraic Noncommutative rings Operatoralgebra Nichtkommutative Geometrie Nichtkommutative Differentialgeometrie Algebraische Geometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006675067&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT connesalain geometrienoncommutative AT connesalain noncommutativegeometry |