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Index Theory for Locally Compact Noncommutative Geometries
A. L. Carey, Mathematical Sciences Institute, Australian National University, Canberra, Australia, V. Gayral, Université de Reims, France, A. Rennie, University of Wollongong, Australia, and F. A. Sukochev, University of New South Wales, Kensington, Australia
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Memoirs of the American Mathematical Society
2014; 130 pp; softcover
Volume: 231
ISBN-10: 0-8218-9838-8
ISBN-13: 978-0-8218-9838-3
List Price: US$76
Individual Members: US$45.60
Institutional Members: US$60.80
Order Code: MEMO/231/1085
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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

  • Introduction
  • Pseudodifferential calculus and summability
  • Index pairings for semifinite spectral triples
  • The local index formula for semifinite spectral triples
  • Applications to index theorems on open manifolds
  • Noncommutative examples
  • Appendix A. Estimates and technical lemmas
  • Bibliography
  • Index
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