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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