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.