# Index Theory for Locally Compact Noncommutative Geometries

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*A. L. Carey; V. Gayral; A. Rennie; F. A. Sukochev*

Spectral triples for nonunital algebras model locally compact
spaces in noncommutative geometry. In the present text, the authors
prove the local index formula for spectral triples over nonunital
algebras, without the assumption of local units in our algebra. This
formula has been successfully used to calculate index pairings in
numerous noncommutative examples. The absence of any other effective
method of investigating index problems in geometries that are
genuinely noncommutative, particularly in the nonunital situation, was
a primary motivation for this study and the authors illustrate this
point with two examples in the text.

In order to understand
what is new in their approach in the commutative setting the authors
prove an analogue of the Gromov-Lawson relative index formula (for
Dirac type operators) for even dimensional manifolds with bounded
geometry, without invoking compact supports. For odd dimensional
manifolds their index formula appears to be completely new.

#### Table of Contents

# Table of Contents

## Index Theory for Locally Compact Noncommutative Geometries

- Cover Cover11 free
- Title page i2 free
- Introduction 18 free
- Chapter 1. Pseudodifferential Calculus and Summability 714 free
- Chapter 2. Index Pairings for Semifinite Spectral Triples 3340
- 2.1. Basic definitions for spectral triples 3340
- 2.2. The Kasparov class and Fredholm module of a spectral triple 3441
- 2.3. The numerical index pairing 3845
- 2.4. Smoothness and summability for spectral triples 4148
- 2.5. Some cyclic theory 4855
- 2.6. The Kasparov product, numerical index and Chern character 4956
- 2.7. Digression on the odd index pairing for nonunital algebras 5259

- Chapter 3. The Local Index Formula for Semifinite Spectral Triples 5562
- 3.1. The resolvent and residue cocycles and other cochains 5562
- 3.2. The resolvent cocycle and variations 5764
- 3.3. The double construction, invertibility and reduced cochains 5966
- 3.4. Algebraic properties of the expectations 6168
- 3.5. Continuity of the resolvent cochain 6471
- 3.6. Cocyclicity of the resolvent and residue cocycles 6774
- 3.7. The homotopy to the Chern character 6976
- 3.8. Removing the invertibility of 𝒟 7986
- 3.9. The local index formula 8390
- 3.10. A nonunital McKean-Singer formula 8491
- 3.11. A classical example with weaker integrability properties 8693

- Chapter 4. Applications to Index Theorems on Open Manifolds 8996
- Chapter 5. Noncommutative Examples 103110
- Appendix A. Estimates and Technical Lemmas 117124
- Bibliography 125132
- Index 129136 free
- Back Cover Back Cover1142