2 A. L. CAREY, V. GAYRAL, A. RENNIE, and F. A. SUKOCHEV
The noncommutative results. The index theorems we prove rely on a gen-
eral nonunital noncommutative integration theory and the index theory developed
in detail in Chapters 1 and 2.
Chapter 1 presents an integration theory for weights which is compatible with
Connes and Moscovici’s approach to the pseudodifferential calculus for spectral
triples. This integration theory is the key technical innovation, and allows us to
treat the unital and nonunital cases on the same footing.
An important feature of our approach is that we can eliminate the need to
assume the existence of ‘local units’ which mimic the notion of compact support,
[27,49,50]. The difficulty with the local unit approach is that there are no general
results guaranteeing their existence. Instead we identify subalgebras of integrable
and square integrable elements of our algebra, without the need to control ‘sup-
ports’.
In Chapter 2 we introduce a triple (A, H, D) where H is a Hilbert space, A is
a (nonunital) ∗-algebra of operators represented in a semifinite von Neumann sub-
algebra of B(H), and D is a self-adjoint unbounded operator on H whose resolvent
need not be compact, not even in the sense of semifinite von Neumann algebras.
Instead we ask that the product a(1 +
D2)−1/2
is compact, and it is the need to
control this product that produces much of the technical difficulty.
We remark that there are good cohomological reasons for taking the effort to
prove our results in the setting of semifinite noncommutative geometry, and that
these arguments are explained in [24]. In particular, [24, Th´ eor` eme 15] identifies
a class of cyclic cocycles on a given algebra which have a natural representation as
Chern characters, provided one allows semifinite Fredholm modules.
We refer to the case when D does not have compact resolvent as the ‘nonunital
case’, and justify this terminology in Lemma 2.2. Instead of requiring that D be
Fredholm we show that a spectral triple (A, H, D), in the sense of Chapter 2, defines
an associated semifinite Fredholm module and a KK-class for A.
This is an important point. It is essential in the nonunital version of the theory
to have an appropriate definition of the index which we are computing. Since the
operator D of a general spectral triple need not be Fredholm, this is accomplished
by following [35] to produce a KK-class. Then the index pairing can be defined
via the Kasparov product.
The role of the additional smoothness and summability assumptions on the
spectral triple is to produce the local index formula for computing the index pairing.
Our smoothness and summability conditions are defined using the smooth version
of the integration theory in Chapter 1. This approach is justified by Propositions
2.16 and 2.17, which compare our definition with a more standard definition of
finite summability.
Having identified workable definitions of smoothness and summability, the main
technical obstacle we have to overcome in Chapter 2 is to find a suitable Fr´echet
completion of A stable under the holomorphic functional calculus. The integration
theory of Chapter 1 provides such an algebra, and in the unital case it reduces to
previous solutions of this problem, [49, Lemma
16]1.
In Chapter 3 we establish our local index formula in the sense of Connes-
Moscovici. The underlying idea here is that Connes’ Chern character, which defines
1Despite
being about nonunital spectral triples, [49, Lemma 16] produces a Fr´ echet comple-
tion which only takes smoothness, not integrability, into account.
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