Contents
Introduction 1
Chapter 1. Pseudodifferential Calculus and Summability 7
1.1. Square-summability from weight domains 7
1.2. Summability from weight domains 11
1.3. Smoothness and summability 20
1.4. The pseudodifferential calculus 22
1.5. Schatten norm estimates for tame pseudodifferential operators 29
Chapter 2. Index Pairings for Semifinite Spectral Triples 33
2.1. Basic definitions for spectral triples 33
2.2. The Kasparov class and Fredholm module of a spectral triple 34
2.3. The numerical index pairing 38
2.4. Smoothness and summability for spectral triples 41
2.5. Some cyclic theory 48
2.6. The Kasparov product, numerical index and Chern character 49
2.7. Digression on the odd index pairing for nonunital algebras 52
Chapter 3. The Local Index Formula for Semifinite Spectral Triples 55
3.1. The resolvent and residue cocycles and other cochains 55
3.2. The resolvent cocycle and variations 57
3.3. The double construction, invertibility and reduced cochains 59
3.4. Algebraic properties of the expectations 61
3.5. Continuity of the resolvent cochain 64
3.6. Cocyclicity of the resolvent and residue cocycles 67
3.7. The homotopy to the Chern character 69
3.8. Removing the invertibility of D 79
3.9. The local index formula 83
3.10. A nonunital McKean-Singer formula 84
3.11. A classical example with weaker integrability properties 86
Chapter 4. Applications to Index Theorems on Open Manifolds 89
4.1. A spectral triple for manifolds of bounded geometry 89
4.2. An index formula for manifolds of bounded geometry 96
4.3. An L2-index theorem for manifolds of bounded geometry 98
Chapter 5. Noncommutative Examples 103
5.1. Torus actions on
C∗-algebras
103
5.2. Moyal plane 110
Appendix A. Estimates and Technical Lemmas 117
iii
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