Index Theory for Locally Compact Noncommutative Geometries.

Cover -- Title page -- Introduction -- Chapter 1. Pseudodifferential Calculus and Summability -- 1.1. Square-summability from weight domains -- 1.2. Summability from weight domains -- 1.3. Smoothness and summability -- 1.4. The pseudodifferential calculus -- 1.5. Schatten norm estimates for tame pseudodifferential operators -- Chapter 2. Index Pairings for Semifinite Spectral Triples -- 2.1. Basic definitions for spectral triples -- 2.2. The Kasparov class and Fredholm module of a spectral triple -- 2.3. The numerical index pairing -- 2.4. Smoothness and summability for spectral triples -- 2.5. Some cyclic theory -- 2.6. The Kasparov product, numerical index and Chern character -- 2.7. Digression on the odd index pairing for nonunital algebras -- Chapter 3. The Local Index Formula for Semifinite Spectral Triples -- 3.1. The resolvent and residue cocycles and other cochains -- 3.2. The resolvent cocycle and variations -- 3.3. The double construction, invertibility and reduced cochains -- 3.4. Algebraic properties of the expectations -- 3.5. Continuity of the resolvent cochain -- 3.6. Cocyclicity of the resolvent and residue cocycles -- 3.7. The homotopy to the Chern character -- 3.8. Removing the invertibility of -- 3.9. The local index formula -- 3.10. A nonunital McKean-Singer formula -- 3.11. A classical example with weaker integrability properties -- Chapter 4. Applications to Index Theorems on Open Manifolds -- 4.1. A spectral triple for manifolds of bounded geometry -- 4.2. An index formula for manifolds of bounded geometry -- 4.3. An ²-index theorem for manifolds of bounded geometry -- Chapter 5. Noncommutative Examples -- 5.1. Torus actions on *-algebras -- 5.2. Moyal plane -- Appendix A. Estimates and Technical Lemmas -- A.1. Background material on the pseudodifferential expansion -- A.2. Estimates for Chapter 2 -- Bibliography -- Index.

Back Cover.

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.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.