INTRODUCTION 3

an element of the cyclic cohomology of A, computes the index pairing defined

by a Fredholm module. Any cocycle in the same cohomology class as the Chern

character will therefore also compute the index pairing. In this memoir we define

several cocycles that represent the Chern character and which are expressed in

terms of the unbounded operator D. These cocycles generalise those found in

[15–17] (where semifinite versions of the local index formula were first proved) to

the nonunital case. We have to prove that these additional cocycles, including the

residue cocycle, are in the class of the Chern character in the (b, B)-complex.

Our main result (stated in Theorem 3.33 of Chapter 3) is then an expression for

the index pairing using a nonunital version of the semifinite local index formula of

[15,16], which is in turn a generalisation to the setting of semifinite von Neumann

algebras of the original Connes-Moscovici [25] formula. Our noncommutative index

formula is given by a sum of residues of zeta functions and is easily recognisable

as a direct generalisation of the unital formulas of [15,16,25]. We emphasise that

even for the standard B(H) case our local index formula is new.

One of the main diﬃculties that we have to overcome is that while there is a well

understood theory of Fredholm (or Kasparov) modules for nonunital algebras, the

‘right framework’ for working with unbounded representatives of these K-homology

classes has proved elusive. We believe that we have found the appropriate formalism

and the resulting residue index formula provides evidence that the approach to

spectral triples over nonunital algebras initiated in [10] is fundamentally sound

and leads to interesting applications. Related ideas on the K-homology point of

view for relative index theorems are to be found in [52], [9] and [19], and further

references in these texts.

We also discuss some fully noncommutative applications in Chapter 5, including

the type I spectral triple of the Moyal plane constructed in [27] and semifinite spec-

tral triples arising from torus actions on

C∗-algebras,

but leave other applications,

such as those to the results in [44], [46] and [60], to elsewhere.

To explain how we arrived at the technical framework described here, consider

the simplest possible classical case, where H = L2(R), D =

d

idx

and A is a certain

∗-subalgebra of the algebra of smooth functions on R. Let P = χ[0,∞)(D) be the

projection defined using the functional calculus and the characteristic function of

the half-line and let u be a unitary in A such that u − 1 converges to zero at ±∞

‘suﬃciently rapidly’. Then the classical Gohberg-Krein theory gives a formula for

the index of the Fredholm operator PMuP where Mu is the operator of multiplica-

tion by u on L2(R). In proving this theorem for general symbols u, one confronts

the classical question (studied in depth in [56]) of when an operator of the form

(Mu − 1)(1 + D2)−s/2, s 0, is trace class. In the general noncommutative setting

of this article, this question and generalisations must still be confronted and this is

done in Chapter 1.

The results for manifolds. In the case of closed manifolds, the local index

formula in noncommutative geometry (due to Connes-Moscovici [25]) can serve as a

starting point to derive the Atiyah-Singer index theorem for Dirac type operators.

This proceeds by a Getzler type argument enunciated in this setting by Ponge,

[47], though similar arguments have been used previously with the JLO cocycle

as a starting point in [7, 23]. While there is already a version of this Connes-

Moscovici formula that applies in the noncompact case [50], it relies heavily on the

use of compact support assumptions.