INTRODUCTION 5 We were also motivated to consider Atiyah’s L2-index Theorem in this setting. Because we prove our index formula in the general framework of operators affiliated to semifinite von Neumann algebras we are able, with some additional effort, to obtain at the same time a version of the L2-index Theorem of Atiyah for Dirac type operators on the universal cover of M (whether M is closed or not). We are able to reduce our proof in this L2-setting to known results about the local asymptotics at small time of heat kernels on covering spaces. The key point here is that our residue cocycle formula gives a uniform approach to all of these ‘classical’ index theorems. Summary of the exposition. Chapter 1 begins by introducing the integra- tion theory we employ, which is a refinement of the ideas introduced in [10]. Then we examine the interaction of our integration theory with various notions of smooth- ness for spectral triples. In particular, we follow Higson, [32], and [15] in extending the Connes-Moscovici pseudodifferential calculus to the nonunital setting. Finally we prove some trace estimates that play a key role in the subsequent technical parts of the discussion. All these generalisations are required for the proof of our main result in Chapter 3. Chapter 2 explains how our definition of semifinite spectral triple results in an index pairing from Kasparov’s point of view. In other words, while our spectral triple does not a priori involve (possibly unbounded) Fredholm operators, there is an associated index problem for bounded Fredholm operators in the setting of Kasparov’s KK-theory. We then show that by modifying our original spectral triple we may obtain an index problem for unbounded Fredholm operators without changing the Kasparov class in the bounded picture. This modification of our unbounded spectral triple proves to be essential, in two ways, for us to obtain our residue formula in Chapter 3. The method we use in Chapter 3 to prove the existence of a formula of Connes- Moscovici type for the index pairing of our K-homology class with the K-theory of the nonunital algebra A is a modification of the argument in [17]. This argument is in turn closely related to the approach of Higson [32] to the Connes-Moscovici formula. The idea is to start with the resolvent cocycle of [15–17] and show that it is well defined in the nonunital setting. We then show that there is an extension of the results in [17] that gives a homotopy of the resolvent cocycle to the Chern character for the Fredholm module associated to the spectral triple. The residue cocycle can then be derived from the resolvent cocycle in the nonunital case by much the same argument as in [15,16]. In order to avoid cluttering our exposition with proofs of nonunital modifi- cations of the estimates of these earlier papers, we relegate much detail to the Appendix. Modulo these technicalities we are able to show, essentially as in [17], that the residue cocycle and the resolvent cocycle are index cocycles in the class of the Chern character. Then Theorem 3.33 in Chapter 3 is the main result of this memoir. It gives a residue formula for the numerical index defined in Chapter 2 for spectral triples. We conclude Chapter 3 with a nonunital McKean-Singer formula and an exam- ple showing that the integrability hypotheses can be weakened still further, though we do not pursue the issue of finding the weakest conditions for our local index formula to hold in this text.
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