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.