Memoirs of the American Mathematical Society 2014; 130 pp; softcover Volume: 231 ISBN10: 0821898388 ISBN13: 9780821898383 List Price: US$76 Individual Members: US$45.60 Institutional Members: US$60.80 Order Code: MEMO/231/1085
 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 GromovLawson 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
