Introduction Our objective in writing this memoir is to establish a unified framework to deal with index theory on locally compact spaces, both commutative and noncom- mutative. In the commutative situation this entails index theory on noncompact manifolds where Dirac-type operators, for example, typically have noncompact re- solvent, are not Fredholm, and so do not have a well-defined index. In initiating this study we were also interested in understanding previous approaches to this problem such as those of Gromov-Lawson [29] and Roe [51] from a new viewpoint: that of noncommutative geometry. In this latter setting the main tool, the Connes- Moscovici local index formula, is not adapted to nonunital examples. Thus our primary objective here is to extend that theorem to this broader context. Index theory provided one of the main motivations for noncommutative geom- etry. In [20,21] it is explained how to express index pairings between the K-theory and K-homology of noncommutative algebras using Connes’ Chern character for- mula. In examples this formula can be diﬃcult to compute. A more tractable analytic formula is established by Connes and Moscovici in [23] using a repre- sentative of the Chern character that arises from unbounded Kasparov modules or ‘spectral triples’ as they have come to be known. Their resulting ‘local index formula’ is an analytic cohomological expression for index pairings that has been exploited by many authors in calculations in fully noncommutative settings. In previous work [15–17] some of the present authors found a new proof of the formula that applied for unital spectral triples in semifinite von Neumann alge- bras. However for some time the understanding of the Connes-Moscovici formula in nonunital situations has remained unsatisfactory. The main result of this article is a residue formula of Connes-Moscovici type for calculating the index pairing between the K-homology of nonunital algebras and their K-theory. This latter view of index theory, as generalised by Kasparov’s bivariant KK functor, is central to our approach and we follow the general philos- ophy enunciated by Higson and Roe, [33]. One of our main advances is to avoid ad hoc assumptions on our algebras (such as the existence of local units, see below). To illustrate our main result in practice we present two examples in Chapter 5. Elsewhere we will explain how a version of the example of nonunital Toeplitz theory in [46] can be derived from our local index formula. To understand what is new about our theorem in the commutative case we apply our residue formula to manifolds of bounded geometry, obtaining a cohomological formula of Atiyah- Singer type for the index pairing. We also prove an L2-index theorem for coverings of such manifolds. We now explain in some detail these and our other results. 1

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