Noncommutative Geometry: Lectures given at the C.I.M.E. Summer School held in Martina Franca, Italy, September 3-9, 2000

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Noncommutative Geometry is one of the most deep and vital research subjects of present-day Mathematics. Its development, mainly due to Alain Connes, is providing an increasing number of applications and deeper insights for instance in Foliations, K-Theory, Index Theory, Number Theory but also in Quantum Physics of elementary particles. The purpose of the Summer School in Martina Franca was to offer a fresh invitation to the subject and closely related topics; the contributions in this volume include the four main lectures, cover advanced developments and are delivered by prominent specialists.

 

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Contents

1 Introduction
2
2 Cyclic Cohomology
6
3 Calculus and Infinitesimals
12
4 Spectral Triples
16
5 Operator Theoretic Local Index Formula
21
Case q 0
23
7 The Local Index Formula for SU_q2 q 0
32
8 The etaCochain
47
2 Coarse Equivalence QuasiIsometries and Uniform Embeddings
254
3 Exact Groups
256
4 Exactness and the BaumConnes and Novikov Conjectures
258
5 Gromov Groups and Expanders
259
6 Final Remarks
260
References
261
1 Introduction
263
2 Algebraic Quantum Field Theory
264

9 PseudoDifferential Calculus and the Cosphere Bundle on SU_q2 q in 0 1
50
10 Dimension Spectrum and Residues for SU_q2 q in 0 1
55
11 The Local Index Formula for SU_q2 q in 0 1
57
12 Quantum Groups and Invariant Cyclic Cohomology
63
1 Introduction
72
2 Some Examples of Algebras
76
3 Locally Convex Algebras
78
4 Standard Extensions of a Given Algebra
83
5 Preliminaries on Homological Algebra
86
6 Definition of Cyclic HomologyCohomology Using the Cyclic Bicomplex and the Connes Complex
88
7 The Algebra OmegaA of Abstract Differential Forms over A and Its Operators
93
8 Periodic Cyclic Homology and the Bivariant Theory
96
9 Mixed Complexes
99
10 The XComplex Description of Cyclic Homology
100
11 Cyclic Homology as Noncommutative de Rham Theory
106
12 Homotopy Invariance for Cyclic Theory
108
13 Morita Invariance for Periodic Cyclic Theory
110
14 Morita Invariance for the Nonperiodic Theory
111
15 Excision for Periodic Cyclic Theory
112
16 Excision for the Nonperiodic Theory
113
18 The Chern Character for KTheory Classes Given by Idempotents and Invertibles
114
19 Cyclic Cocycles Associated with Fredholm Modules
116
20 Bivariant KTheory for Locally Convex Algebras
118
21 The Bivariant ChernConnes Character
122
22 Entire Cyclic Cohomology
124
23 Local Cyclic Cohomology
130
References
134
1 KTheory
138
2 Bivariant KTheory
169
3 Groups with the Haagerup Property
197
4 Injectivity Arguments
220
5 Counterexamples
233
References
248
1 Introduction
252
3 Quantum Fields and Local Observables
265
4 Quantum Field Theory
269
5 Spacetime and Its Symmetries
271
6 Local Observables
274
7 Additivity
277
8 Local Normality
281
9 Inclusions of von Neumann Algebras
283
10 Standard Split Inclusions
287
11 Some Properties of Nets
291
12 Duality
296
13 Intertwiners
299
14 States of Relevance
300
15 Charges in Particle Physics
301
16 The Selection Criterion I
302
17 Charges of Electromagnetic Type
304
18 Solitonic Sectors
305
19 Scattering Theory
306
20 Modular Theory
307
21 Conformal Field Theory
308
22 Curved Spacetime
309
23 Partially Ordered Sets
310
24 Representations and Duality
318
25 The Selection Criterion II
320
26 The Cohomological Interpretation
321
27 Tensor Structure
324
28 Localized Endomorphisms
327
29 Left Inverses
330
30 Change of Index Set
333
References
339
List of Participants
343
LIST OF CIME SEMINARS
344
2004 COURSES LIST
348
Lecture Notes in Mathematics
350
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