The Local Index Formula in Noncommutative...

The Local Index Formula in NoncommutativeGeometry

Nigel Higson�

Pennsylvania State University, University Park, Pennsylvania, USA

(Dedicated to H. Bass on the occasion of his 70th birthday)

Lectures given at theSchool on Algebraic

�-Theory and its Applications

Trieste, 8 - 19 July 2002

LNS0315006

�[email protected]

Abstract

These notes present a partial account of the local index theorem in non-commutative geometry discovered by Alain Connes and Henri Moscovici.

Contents

Preface 447

1 Elliptic Partial Differential Operators 4491.1 Laplace Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 4491.2 Unbounded Operators . . . . . . . . . . . . . . . . . . . . . . . . 4511.3 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 4521.4 Compact Resolvent . . . . . . . . . . . . . . . . . . . . . . . . . 4551.5 Weyl’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 4551.6 Elliptic Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 4591.7 Basic Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . 4601.8 Proof of the Basic Estimate . . . . . . . . . . . . . . . . . . . . . 462

2 Zeta Functions 4642.1 Outline of the Proof . . . . . . . . . . . . . . . . . . . . . . . . . 4652.2 Remark on Orders of Differential Operators . . . . . . . . . . . . 4672.3 The Actual Proof . . . . . . . . . . . . . . . . . . . . . . . . . . 467

3 Abstract Differential Operators 4763.1 Algebras of Differential Operators . . . . . . . . . . . . . . . . . 4763.2 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4773.3 Pseudodifferential Operators . . . . . . . . . . . . . . . . . . . . 4803.4 Spectral Triples . . . . . . . . . . . . . . . . . . . . . . . . . . . 4853.5 Dimension Spectrum . . . . . . . . . . . . . . . . . . . . . . . . 489

4 Computation of Residues 4904.1 Computation of the Leading Residue . . . . . . . . . . . . . . . . 4904.2 The Lower Residues . . . . . . . . . . . . . . . . . . . . . . . . 493

5 The Index Problem 4965.1 Index of Elliptic Operators . . . . . . . . . . . . . . . . . . . . . 4965.2 Square Root of the Laplacian . . . . . . . . . . . . . . . . . . . . 4985.3 Cyclic Cohomology Theory . . . . . . . . . . . . . . . . . . . . 4995.4 Chern Character and Pairings with K-Theory . . . . . . . . . . . 5025.5 Zeta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 5045.6 Formulation of the Local Index Theorem . . . . . . . . . . . . . . 506

6 The Residue Cocycle 5086.1 Improper Cocycle . . . . . . . . . . . . . . . . . . . . . . . . . . 5086.2 Residue Cocycle . . . . . . . . . . . . . . . . . . . . . . . . . . 5106.3 Complex Powers in a Differential Algebra . . . . . . . . . . . . . 511

7 Comparison with the Chern Character 5177.1 Homotopy Invariance and Index Formula . . . . . . . . . . . . . 522

8 The General Case 525

A Appendix: Compact and Trace-Class Operators 528

B Appendix: Fourier Theory 531

References 534

Local Index Formula in Noncommutative Geometry 447

Preface

Several years ago Alain Connes and Henri Moscovici discovered a quite general‘local’ index formula in noncommutative geometry [10]. The formula was origi-nally studied in relation to the transverse geometry of foliations, but more recentlyConnes has drawn attention to other possible areas of application, for example com-pact quantum groups [6] and deformations of homogeneous manifolds [8]. More-over elaborate structures in homological algebra have been devised in the course ofstudying the formula [8], and these have found application in quantum field theory[7] and elsewhere [11].

These notes provide an introduction to the local index formula. They emphasizethe basic, analytic aspects of the subject. This is in part because the analysis mustbe dealt with first, before more purely cohomological issues are tackled, and inpart because the later issues are already quite well covered in survey articles byConnes and others (see for example [5]). Moreover, on the cohomological side, thefinal and definitive results have yet to be thoroughly investigated. I hope that thereader will be able to use these notes to introduce himself to these issues of currentresearch interest.

The notes begin with a rapid account of the spectral theory of linear ellipticoperators on manifolds, which is the launching point for the local index formula.They begin right at the beginning, and I hope that they might be accessible to stu-dents with a very modest background in analysis. Two appendices deal with stillmore basic issues in Hilbert space operator theory and Fourier theory.

The first result which goes beyond the totally standard canon (but which is stillclassical) is the theorem that the zeta functions Trace �� associated to ellipticoperators admit meromorphic continuations to � � . I shall present a proof whichis more algebraic than the usual ones, and which seems to me to well adapted toConnes’ noncommutative geometric point of view.

Following that, manifolds are replaced by Connes’ ‘noncommutative geometricspaces’, and basic tools such as differential operator theory and pseudodifferentialoperator theory are developed in this context.

After the subject of cyclic cohomology theory is rapidly introduced, it becomespossible to formulate the basic index problem, which is the main topic of the notes.The final sections of the paper (from 6 to 8) prove the index formula.

The notes correspond very roughly to the first four of the six lectures I gaveat the Trieste meeting. The remaining two lectures dealt with cyclic cohomologyfor Hopf algebras. The interested reader can look at the overhead transparenciesfrom those lectures [20] to figure out more precisely what has been omitted and

448 N. Higson

what has been added (note that the division of the present notes into sections doesnot correspond to the original division into lectures). The notes borrow (in placesverbatim) from several preprints of mine [17, 18, 19] which will be published else-where. But of course they rely most of all on the work of Connes and Moscovici. Ifa result is given in the notes without attribution, the reader should not assume thatit is original in any way. Most likely the result is due, in one form or another, tothese authors.

I would like to thank Max Karoubi, Aderemi Kuku, and Claudio Pedrini for theinvitation to speak at the Trieste meeting. Many friends and colleagues helped mealong the way as I learned the topics presented here. In this regard I especially wantto thank Raphael Ponge and John Roe, along with all the members of the GeometricFunctional Analysis Seminar at Penn State.

The writing of these notes was supported in part by a grant from the US Na-tional Science Foundation, and also of course by the ICTP.


1 Elliptic Partial Differential Operators

We are going to develop the spectral theory of elliptic linear partial differentialoperators on smooth, closed manifolds. We shall approach the subject from thedirection of Hilbert space theory, which is particularly well suited for the task. Infact Hilbert space theory was invented for just this purpose.

1.1 Laplace Operators

Let � be a smooth, closed, manifold of dimension � . A linear operator � mappingthe vector space of smooth, complex-valued functions on � to itself is local if, forevery smooth function � , the support of �� is contained within the support of � .If � is local, then the value of �� at a point � �� depends only on the valuesof � near � , and as a result it makes sense to seek a local coordinate descriptionof � .

1.1 Definition. A linear partial differential operator is a local operator which inevery coordinate chart may be written

(1.1) ��

� ��

where the � � are �� -functions. Here � is a non-negative integer and the sum isover non-negative integer multi-indices �� ! �"�"� #�%$ � for which & ��&'�(��)*"*"* )+�%$+,.- . The order of � is the least � required to so represent � (in anycoordinate chart).

To begin with we are mainly interested in one example. This is the Laplaceoperator � , also known as the Laplace-Beltrami operator on a closed Riemannianmanifold. It is given by the compact formula �/�/0 � 0 , where 0 is the gradientoperator from functions to tangent vector fields, and 0 �

is its adjoint, also calledthe divergence operator (up to a sign, these are direct generalizations to manifoldsof the objects of the same name in vector calculus). In local coordinates the Laplaceoperator has the form

��21$354 687 �:9

3�6�� <;� � 3 � � 6 ) order one operator �

The order one term is a bit complicated (the exact formula is of no concern to us)but at the origin of a geodesic coordinate system all the coefficients of the order

450 N. Higson

one term vanish, and we get

��21$ 3 7 � �<;� � ;3

at the origin which is the familiar formula from ordinary vector calculus.

Our goal in Section 1 is to prove the following fundamental fact.

1.2 Theorem. Let � be the Laplace operator on a closed Riemannian manifold.There is an orthonormal basis

� � 6�� for the Hilbert space � ; �� consisting ofsmooth functions � 6 which are eigenfunctions for � :

� � 6 �� 6 � 6 for some scalar � 6 .The eigenvalues � 6 are non-negative and they tend to infinity as � tends to infinity.

It is possible to say a bit more. Since the functions � 6 constitute an orthonormalbasis for � ; �� , every function � in � ; �� can be expanded as a series

� � � 687� � 6 � 6

where the sequence of coefficients�� 6�� is square-summable. It turns out that �

� ; �� is a smooth function if and only if the sequence�� 6�� is of rapid decay, which

means that if - � then

sup6 � � & � 6 &�� This should call to mind a basic fact in the theory of Fourier series: a function onthe circle is smooth if and ony if its Fourier coefficient sequence is of rapid decay.Note that the basic functions in Fourier theory, the exponentials �

3 $�� , constitute anorthonormal basis for � ; �� consisting of eigenfunctions for the Laplace operatoron the circle, which is just 1��

. So in some sense Theorem 1.2 establishes thefirst principles of Fourier theory on any closed Riemannian manifold.

The proof of Theorem 1.2 is more or less a resume of a first course in functionalanalysis. In view of what we have said it will not surprise the reader to learn thatthe argument relies on one or two crucial computations in Fourier theory. But weshall also need to review various ideas from Hilbert space operator theory.


1.2 Unbounded Operators

An unbounded operator on a Hilbert space, � , is a linear transformation froma dense linear subspace of � into � . No continuity is assumed. When dealingwith unbounded operators it is important to keep track of domains. Unboundedoperators with different domains can’t generally be added together in a reasonableway. Unbounded operators can’t generally be composed in a reasonable way unlessthe range of the first is contained within the domain of the second.

An unbounded operator � is symmetric if

� �� for all �� dom � . An unbounded operator is self-adjoint if it is symmetric andif in addition

Range �� In finite dimensions every symmetric operator is self-adjoint. In infinite dimensionsself-adjointness is precisely the condition needed to get spectral theory going. Ob-serve that if � is symmetric then

� �� ; � � �� ; ) � � � ; which implies that if � is self-adjoint then the operators �� map dom � one-to-one and onto � , so that they have well-defined inverses (which we regard asoperators from � to itself).

1.3 Theorem. Let � be a self-adjoint operator. There is a (unique) homomorphismfrom the algebra of bounded, continuous functions on � into � �� (the algebra ofbounded operators on � ) such that

��

This is one version of the Spectral Theorem. It is proved by noting that theoperators �� generate a commutative � �

-subalgebra of � �� , and by thenapplying the basic theory of commutative � �

-algebras.Note that once we have the Spectral Theorem we can define ‘wave operators’

�3"!$#

, ‘heat operators’ � �!%#� , and so on. Thus the result is conceptually very pow-

erful.Self-adjoint operators are hard to come by in nature. Typically the natural do-

main of an unbounded operator (e.g. the smooth, compactly supported functions in

452 N. Higson

the case of a differential operator) must be enlarged, and the operator extended tothis larger domain, so as to obtain a self-adjoint operator. Here is one procedure,due to Friedrichs, which we’ll illustrate using the Laplace operator.

Let � be the Laplace operator on a Riemannian manifold � . The manifoldneed not be compact, or complete; it might have a boundary. Think of � as an un-bounded operator on � � � ; �� whose domain is the space of smooth, compactlysupported functions (on the interior of � , if � has boundary).

Observe that if � dom � then

� � � #� �� 0 � 0 � � �� Let us exploit this to define a new inner product on dom � by the formula

� � �� ) � � � �� Denote by � � the Hilbert space completion of dom � in this inner product. It is,among other things, a dense subspace of � (more about it later). Now denote by� ;

� � � the space of all � for which there exists a vector � �� (which will be� � ) � � � ) such that � � �� ! �� 1.4 Theorem (Friedrichs). The operator �%) � is self-adjoint on � ; .

The proof is a really good exercise. To get a self-adjoint extension of � , justsubtract � from � ) � .

1.3 Sobolev Spaces

We are now going to investigate in a bit more detail the Hilbert space � � whichappeared above. It appears as the space � � in the sequence of Sobolev spaces� � �� ; �"�"� associated to a closed manifold (and as it happens the Hilbertspace � ; is the same as the space � ; that we defined in the last section, at least fora closed manifold, although the proof of that fact will be postponed for a while).

Although we are interested in function spaces associated to a manifold � , weshall begin not with � but with open sets in Euclidean space.

1.5 Definition. Let be an open subset of � $ and let - be a non-negative integer.Denote by � � �� the completion of � �� in the norm

� � � ;��

�� ; � � ��


Thus the � � -norm combines the � ; -norms of all the partial derivatives of � oforder - or less.

1.6 Definition. Let � be a manifold and let�

be a compact subset of (the interiorof) � . Define a Hilbertian space1 � � �� & � � as follows.

Case 1.�

is contained in a coordinate ball. Fix a diffeomorphism from aneighbourhood of

�to an open set � � $ , use the diffeomorphism to transfer

the norm on � � �� to the smooth functions on � which are compactly supportedwithin

�, and then complete.

Case 2.�

is any compact set. Choose smooth, compactly supported functions� � �"�"� � � on � , each supported in a coordinate ball, with

� � 6 �� on�

, andlet

� 6 � supp � 6 . Let � � �� & � � be the completion of the smooth functions on �which are compactly supported in

�, in the norm

� � � ;�� 6� � 6 � � ;��

In either case, the norms depend on coordinate choices, etc, but the underlyingHilbertian spaces do not. If we fix a smooth measure on � then all the spaces� � �� & � � can be thought of as linear subspaces of � ; �� (they are dense, if

� �� ).The spaces � � �� & � � have the following invariance property: if � is a diffeo-

morphism carrying � onto an open subset of � � , and if � maps�

to� � , then �

carries � � �� & � � � isomorphically onto � � �� & � � . Moreover pointwise multipli-cation by a smooth function is a bounded operator on each � � �� & � � . Differentialoperators of order � map � ��

� �� & � � continuously into � � �� & � � .If

� � � , then we’ll write � � �� in place of � � �� & � � . In this case (where� is compact) we can give an alternate, more concise, definition of the Sobolevspaces. The set of all order - , or less, differential operators is a finitely gener-ated module over the ring of smooth functions on � . If

� � � �"�"� � � � is a finitegenerating set then

� � � �� ) *"*"* )� � � � � � � ��

(the symbol � denotes equivalence of norms).Recall that a bounded Hilbert space operator is compact if it carries the closed

unit ball into a compact set (see Appendix A for a quick review of compact operatortheory and related matters).

1A Hilbertian space is a vector space with an equivalence class of Hilbert space norms, two norms��and

�� being equivalent if there is a constant �� such that ��

� �� .

454 N. Higson

1.7 Rellich Lemma. If -�� then the inclusion of � � �� & � � into � ; �� is a

compact operator.

Proof. Fix a partition of unity� � 6�� as in Definition 1.6, with each � 6 supported in a

compact set� 6 within a coordinate neighbourood � 6 . The inclusion

� � �� & � � // � ; �� can be broken down as a composition of maps

� � �� & � ��

� � �� & � � �� *"*"* � � � �� & � � � // � ; �� *"*"* � � ; ��

� ; �� where the first vertical map is multiplication by � 6 in component � and the othermaps are induced from the obvious inclusions. It clearly suffices to show that theinclusions � � �� & � 6 � � � ; �� 6 � are compact operators. But if we embed � 6 as anopen set in a torus � 6 then in view of the commuting diagram

� � �� & � 6 ��

// � ; �� 6 �

� � �� 6 � // � ; �� 6 � OO

where the downward map is inclusion and the upward one is restriction to � 6 � � 6 ,we see that it suffices to prove that the inclusion

� � �� 6 � // � ; �� 6 �is a compact operator. This is easily accomplished by using Fourier theory — seeAppendix B.

1.8 Remark. The same argument shows that if is a bounded open set in � $ thenthe inclusion � � ��

� � ; �� is compact for all -�� .

1.9 Lemma. If � and - are non-negative integers, and if -�� ) $ ; then � � �� & � � �� & � � . As a result, � � � �� & � � �+� � �� & � � �


Proof. To prove that a function � � � �� & � � is in �� & � � it suffices to showthat each � 6 � � � ��

6 & � 6 � belongs to � � �� 6 & � 6 � (we are using the same notationas in the previous proof). After embedding � 6 as an open set in a torus � 6 , itsuffices to show that � � �� 6 � � � � �� 6 � . Once again, this is easily proved usingFourier series — see Appendix B.

1.4 Compact Resolvent

Let’s return to the Laplace operator and its self-adjoint extension. Assume that themanifold � is closed. Recall that the ‘intermediate’ Hilbert space � � we con-structed on the way to finding the Friedrichs extension of � was the completion of�� in the norm

� � � ; � � � � � ) � � � #� �� ; � � )�� ; � � �

From this it is easy to see that � � � �� . As a result, it follows from theRellich lemma that

1.10 Theorem. The bounded operator � �%) � � � � on � ; �� is compact.

Now remember from functional analysis that every compact positive-definiteoperator (such as � � ) � � � � ) has an orthonormal eigenbasis, whose correspond-ing eigenvalues constitute a sequence of positive numbers converging to zero (seeAppendix A). Hence:

1.11 Theorem. Let � be the self-adjoint operator on � ; �� obtained by the Friedrichsextension procedure from the Laplace operator on � . There is an orthonormal ba-sis for � ; �� consisting of functions � 6 dom � which are eigenfunctions for� . The corresponding eigenvalues constitute a sequence of non-negative numbersconverging to � .

1.12 Remark. We haven’t yet shown that the � 6 are smooth functions, but at anyrate we have that � � 6 � � 6 � 6 in the sense of distributions.

1.5 Weyl’s Theorem

The solution to the problem of finding an orthonormal basis for � ; �� consistingof eigenfunctions of � was first great triumph of Hilbert space theory (in fact thisis the problem which began Hilbert space theory — see [27]). Before we developthe theory any further, let us pause to prove the following very famous theorem ofWeyl.

456 N. Higson

1.13 Theorem. Let � be the (Friedrichs extension of the) Laplace operator on aclosed Riemannian � -manifold or a smooth, bounded domain in � $ . Let � � � � bethe number of eigenvalues of � (multiplicities counted) less than � . Then

lim�� Vol �� $ ; ) ��

�This is a little bit of a detour away from our main objectives, but it began a

sequence of developments which ultimately led to the local index theory we shallbe describing in these notes. For this and other reasons, Weyl’s theorem is in somesense the first theorem of noncommutative geometry.

We’ll deal with the case of domains in � $ (the case of manifolds is just a tinybit harder), and to keep things as clear as possible we’ll consider the dimension

case (although the case of general � is really no different). Thus for a smooth,bounded domain in � ; we aim to prove that

lim�� Vol ��

The first step is to check the result for some basic regions, namely rectangles.2

This, incidentally, will fix the constant �� .

1.14 Lemma. Weyl’s Theorem holds for rectangular domains.

Proof. Let us work with the rectangle of width � and height � whose bottom leftcorner is the origin in the �� -plane. For this domain an eigenbasis for theLaplace operator can be explicitly computed. The eigenfunctions are�� $ �� sin � �� sin �8��'� �and the eigenvalues are � � $ �� ; � � �� ) $ �� , where �� 5� � �

. It follows that

� � � � � # � �8� � � �� ;� ; )� ;� ; ,

�� ;

! �� Area of Ellipse " ;� ;)$# ;� ; ,

�� ; �� Thus � � � �

�! Area ��

as required.2Weyl’s Theorem holds for various non-smooth domains—as will become clear.


The proof of Weyl’s Theorem is an eigenvalue comparison argument, based onthe following simple observation.

1.15 Lemma. Let � and � be compact and positive operators on a Hilbert space� , and denote by

� � 6 �� and� � 6 �� the eigenvalue sequences of � and � . If

� � �� for all � � , then � 6 �� 6 �� , for all � .

Proof. This follows from Weyl’s formula

� 6 �� '� infdim ��

7:6� � sup��

� � � �� which is described in Appendix A, and the fact that if a bounded operator � ispositive then �

�� sup� � � 7 �

��

The main step in the proof of Weyl’s Theorem is now this:

1.16 Proposition. Suppose that � and � are bounded open sets in the plane,and that � � � . Then � �� , � � � � � � , for all � .

Denote by � � � and � �� the Laplace operators for these two domains. The

proposition (called the Domain Dependence Inequality) will follow if we can showthat � 6 �� 6 �� , for all � . This in turn will follow if we can showthat � 6 �� 6 �� , for all � . To this end we are of course going to apply

Lemma 1.15, but first we have to overcome the small problem that although � � �� and � � �� are compact and positive operators, they are defined on different Hilbert

spaces. To remedy this we regard � ; �� as the subspace of � ; �� consistingof functions which vanish on the complement of � in � , and extend � � �� to an

operator on � ; �� by definining it to be zero on the orthogonal complement of� ; �� . Having done so the proof of the Domain Dependence Inequality reducesto the following lemma.

1.17 Lemma. Suppose that � � � and denote by � � � and � �� the Laplace

operators for these two domains. If � � ; �� then� � � ��

� � � �� .

458 N. Higson

Proof. Let � � � ; �� and denote by � � � ; �� the restriction of � � to � .Write � � � � �

� � � and � � � � � � � � , where � � dom � � � and � � dom � �� .

Sorting out the notation, we see that what we need to prove is that� � �! � � � � � � � � � � � � � � � �

given that � � dom � � � , that � � dom � �� , and that the restriction of � �

� � �to � is equal to � � � � � . These hypotheses certainly imply that

� � � � � � � � �� Now apply the Cauchy-Schwarz inequality for the form

� � � � � on �� tothe left hand side to complete the proof.

Proof of Weyl’s Theorem. Let us first show that if is any bounded open set then

(1.2)Area �� , lim inf��

��

(roughly speaking, this is � � % of Weyl’s Theorem). Let � be a finite disjoint unionof open rectangles � � within . Then �� , � � � � � , by the Domain Depen-dence Inequality. But since � is a disjoint union, we get that�� Moreover for each rectangle � � it follows from Lemma 1.14 that

lim�� Area � � � ��

so thatArea � � �� lim��

� , lim inf��

�After approximating Area �� by Area � � � we get the required inequality (1.2).

To complete the proof, put into a large rectangle � and denote by � thecomplement of (the closure of) in � . According to inequality (1.2),

Area �� ) Area �� , lim inf�� ) lim inf��

��

But in addition � � � � �') � � � � � ��, � �� , so that

lim inf�� ) lim inf��

� , lim sup�� ) lim inf��

�

, lim sup��

� Area ��


Since Area �� ') Area �� Area �� the proof is complete.

1.6 Elliptic Operators

We shall now return to the analysis of the Laplace operator on a closed Riemannianmanifold � . In the remainder of Section 1, which is a bit technical, we shallaccomplish several things:� Show that the domain � ; of the Friedrichs extension of � is precisely the Sobolev

space � ; �� .� Show that the eigenfunctions of � are in fact smooth functions on � .

� Indicate how to develop a similar eigenvalue analysis for operators more generalthan � .

The key to all this is to recognize the following local feature of the operator �which implies strong regularity properties for solutions of the equation � � � � :

1.18 Definition. A linear partial differential operator � of order � is elliptic oforder � if, in every local coordinate system, the local expression for � ,

��

� ��

has the property that �� 7 �� ; � ) *"*"* ) � ;$ �� for every point � in the coordinate chart, some constant

� � �depending on � , and

every� � $ .

1.19 Example. If � is equipped with a Riemannian metric � 9 3�6�� then the associatedLaplace operator � is elliptic of order

. Indeed in local coordinates the formula

for � is

��1$354 687 � 9

3�6 � ;� � 3 � � 6 ) lower order terms �The required inequality is therefore$

3�4 6 7 �:93 6�� 3 � 6 �� ; � ) *"*"* ) � ;$ �

which is an immediate consequence of the fact that the matrix � 93�6�� is positive-

definite.

460 N. Higson

1.7 Basic Estimate

Let � be a smooth closed manifold (equipped with a smooth measure, so we canform � ; �� ).

1.20 Definition. Let � be a linear partial differential operator on � of order � .We shall say that � is basic (this is not standard terminology) if

� �� ) � � � � � ��

for all - � �. Here � is a smooth function on � , and the symbol � means that

the left and right hand side define equivalent norms on the space of all smoothfunctions on � .

1.21 Remark. The comparison�

holds for any order � operator, so the force ofthe definition is that � holds too. The latter we shall refer to as the basic estimatefor � .

We are going to show that all elliptic operators are basic:

1.22 Theorem. If � is an order � elliptic operator on a smooth closed manifold,then � �� ) � � � � � ��

� � � � � � �� We shall prove this in the next subsection, but for motivation let us first show

how the basic estimate implies a strong regularity property for elliptic operators. Tokeep things simple we’ll focus on the Laplace operator � (the general case requiressome small modifications which we shall mention at the end).

1.23 Lemma. Let � �� ; �� and assume that � �� in the sense of distri-butions. If � � � �� , then in fact � � ; �� .

Sketch of the Proof. Assume first that � has support in the interior of a compactset

�in a coordinate neighbourhood. If � is a compactly supported, non-negative

bump function on � $ with total integral � , and if��

is the operator of convolutionwith

� � $ � � � � � � � then it can be shown that

(a) If � � � �� & � � (supported in the coordinate neighbourhood) then� � � � �

in � � �� & � � as� � �

.

(b) � � �� is uniformly bounded in�

as an operator from � �� & � � to � � �� & � � .


The family� �� is called a Friedrichs mollifier. Right now we’ll only use the -��

properties of mollifiers, but later we’ll consider -� �. From the equation

� �� ) � � �� we see that

� � � � � � �� is uniformly bounded in � ; �� . It therefore follows fromthe basic estimate that

� � � � � is uniformly bounded in � ; �� . Since� � � � �

in � ; �� it follows, after a little functional analysis, that in fact� � � � � is actually

convergent in � ; �� , which implies � � ; �� as required.In the general case, let � be supported in a coordinate neighbourhood. Since

� � � � is a differential operator of order � we see that

�� ) � � � � ; �� and therefore � � � ; �� by the special case just considered. Varying � , itfollows that � � ; �� , as required.

1.24 Theorem. Denote by � the Laplace operator on a closed Riemannian mani-fold. The domain of the Friedrichs extension of � is the Sobolev space � ; �� .

Proof. The domain of the Friedrichs extension is precisely the space of those � �� for which � � (taken in the distributional sense) belongs to � ; �� . Soaccording to the lemma, if � dom � then � � ; �� . The reverse inclusion iseasy.

1.25 Theorem. Let � dom � and assume that �� in the sense of distribu-tions. If � � � �� , then � � �� ; �� .

Proof. This can be proved by the same Friedrichs mollifier method we used toprove Lemma 1.23.

1.26 Theorem. Let � be the Laplace operator on a closed Riemannian manifold.There is an orthonormal basis for � ; �� consisting of eigenfunctions for � , whichare in fact smooth functions on � .

Proof. As before, the Rellich Lemma implies that � has compact resolvent, and sothe Spectral Theorem for compact operators applies to provide an eigenbasis. Theeigenfunctions are in dom � � � ; �� , and applying Theorem 1.25 repeatedlyto the equation � � � � � we see that �

� � � �� . Hence � is smooth byLemma 1.9.

462 N. Higson

1.27 Remark. If � is a symmetric elliptic operator of order � , then Theorems1.24, 1.25 and 1.26 above continue to hold, although with the Sobolev space index“” in the statements of 1.24 and 1.25 replaced by “ � ”. The proofs are essentially

the same once we introduce Sobolev spaces � � �� with negative indices - (seeAppendix B). Once this is done, the general version of Lemma 1.23 says that if � � � �� , and if � is a distribution for which �� in the sense of distributions,then in fact � � ��

� �� . The proof is essentially the same, although it makesmore serious use of the language of distributions.

1.8 Proof of the Basic Estimate

Before starting the proof of Theorem 1.22 we note the following fact:

1.28 Lemma. Fix an integer - � �. For every

� � �there is a constant � � �

such that � � � � � � � �� , � � � � � � �� )2� � � � � � �� for all smooth functions � .

Roughly speaking, this says that the � � -norm is much stronger than the � � � � -norm — only a tiny multiple of the former is needed to dominate the latter. Like justabout everything else involving Sobolev spaces, the lemma is proved by reducingto the case of a torus, and doing an explicit Fourier series calculation there.

With the lemma in hand we can proceed.

Proof of Theorem 1.22. It will be helpful to introduce the following piece of termi-nology. We shall say that a differential operator � which is defined on some openset � � � satisfies the basic estimate over � if for every compact subset

�of �

the inequality � �� ) � � � � � �� holds, for some

� � �depending on

�and - , and all � supported in

�.

The first step in the proof is to observe that if � � is a constant coefficient order� elliptic operator, defined in some coordinate neighbourhood � of � , then � �satisfies the basic estimate over � . This is an exercise in Fourier theory.

The next step is this. If � is a general order � elliptic operator, if � � , andif � � is the constant coefficient operator obtained by freezing the coefficients of �at � , then for every

� � �there is a small neighbourhood � of � for which

� �� 1 � � � � � , � � � �


for every � supported in � . This follows from the fact that the coefficients of�21 � � vanish at � , as a result of which, �21 � � can be written as a sum of terms�� , where � is a smooth function vanishing at � and � is an order � operator.

From the first two steps, it follows that for every � � there is a neighbour-hood � of � such that the basic elliptic estimate holds for � over � .

Now cover � by finitely many open sets over each of which the basic ellipticestimate for � holds, and let

� � 6�� be a smooth partition of unity which is subordinateto this cover. Write

� � �� 6 � 6 � �� , 6

� � 6 � �� 6

� �� 6 � � � ) 6� � 6 � � �

, 6� � 6 � � � � ) 6

� � � � 6 � � � � ) 6� � 6 � � �

In the middle inequality we have invoked the basic elliptic estimates over the setsin the cover; everything else is just algebra. Since multiplication by � 6 is continu-ous on each Sobolev space we obtain from the above sequence of inequalities theestimate � � � � � � � � � � � � ) � � � � ) 6

� � � � 6 � � � �Finally, the operators � � � 6 � are of order � 1 � , or less, and as a result

6� � � � 6 � � � � � � � � ��

Combining this with Lemma 1.28 we get

� � � � � � � � � � � � ) � � � � � ) � � � � �� in which we can make

�as small as we like, say

� � � . The theorem now followsjust by rearranging the terms in this inequality.

464 N. Higson

2 Zeta Functions

In this section we shall study further the eigenvalue sequence� � 6 � associated to the

Laplace operator on a closed Riemannian manifold � of dimension � . The mainresult will be Theorem 2.2 below, although not only the result but also the proofwill be important for our later purposes.

Let � � . Define a sort of zeta function for � using the formula

�� 7 � � ��6 �

The definition makes sense in view of the following computation:

2.1 Lemma. There is some� � such that if Re ��

then

� ��7� & � ��6 &�� Proof. According to Weyl’s Theorem we can take

� � �(� dim �� , which isthe optimal value. However, if we are content with some value for

�(not the best)

then we can prove the lemma with less effort. We can take, for example, any eveninteger

�bigger than � . It follows from the basic estimate proved in Section 1

that the operator � � ) � � �

� maps � ; �� into � � �� , and since the inclusionof � � �� into � ; �� is a trace-class operator (see Appendix B) it follows that

� �!) � � � � , viewed as an operator on � ; �� , is trace-class. Its eigenvalue sequence

is therefore summable, and it follows from this that� � �

�6 � � , as required.

Let us disregard Weyl’s Theorem for a moment and refer to the smallest�

withthe property of the lemma as the analytic dimension of � . Our main result willgive an independent proof that

� �� .Basic analysis proves that

�� is analytic in the region Re ��

. We aregoing to prove the following remarkable fact.

2.2 Theorem. Let� � 6�� be the eigenvalue sequence for the Laplace operator on a

closed Riemannian � -manifold � . The zeta function

��

�� 7� � ��6extends to a meromorphic function on the complex plane. The only singularities ofthe zeta function are simple poles, and these are located within the set of integerpoints � 5� 1 �< 5� 1 �"�"� .


2.3 Example. If � � � � then the zeta function�� is precisely twice the famous

Riemann zeta function. This explains our terminology and of course illustrates thephenomenon of meromorphic continuation.

Theorem 2.2 was discovered in the 1940’s, by Minakshisundaram and Plei-jel [23] in connection with attempts to refine Weyl’s Theorem. The relation withWeyl’s Theorem is made clear by the following Tauberian Theorem (see for exam-ple Hardy’s book Divergent Series [16]):

2.4 Theorem. Let�� 6�� be a sequence of positive real numbers and assume that it

is � -summable for all � � � . For��

denote by � � � � the number of � such that� 6 � �. Then

lim!�� 1 �� 687 �

� !6�� +� � lim� � � � * � � � � �+� �Thanks to the Tauberian theorem, putting

� 6 �� 6 we see rather easily that if�

� has a pole at � � � then the eigenvalues of � satisfy the asymptotic relation

� � � � ! Res !�7 $ � � �� * � � �(here � � � � is the counting function from Weyl’s Theorem). It follows that theanalytic dimension of � is equal to � , the topological dimension. Moreover Weyl’sTheorem follows from the meromorphic continuation of

�� , plus a computation

of the residue of the zeta function at � �/� . Or, to put it in a better way, Weyl’sTheorem, plus the Tauberian Theorem, show that the residues of the zeta function�� contain important geometric information about � . This is a theme we shall

be developing throughout the rest of these notes.

2.1 Outline of the Proof

The proof of Theorem 2.2 will involve some Hilbert spectral theory and some al-gebra, notably the fundamental ‘Heisenberg commutation relation’

��

� � �� in the algebra of differential operators. It is closely related to Guillemin’s proof ofWeyl’s Theorem in [15] (for a different proof based on pseudodifferential operatorssee [25]).

466 N. Higson

Here is the basic idea. If � � �"�"� �� $ are local coordinates on � then it followsfrom Heisenberg’s relation that if � is any differential operator of order � or lessthen

(2.1) �:� �� 3 7 � � � ��

3 � �� 3 ) � where � is a differential operator of order � 1 � or less. As a result of this, a littlebit of algebra shows that

(2.2) � � ) � � � �$ 3 7 � � � ��

3 �� 3� )

$ 3 7 � � �� 3 ��

3 � � ) � with the same remainder term � .

Now, we are going to show that the same sort of formula as (2.1) holds if � isreplaced by a more complicated operator, roughly speaking one of the form �� ,to which we shall assign the “order” � 1 Re � � . As for the operator �� , if thereal part of is positive then we can define it to be the unique bounded operatorsuch that on eigenfunctions � �� 6 � � � �6 � 6 (we define the complex powers of thezero eigenvalue to be zero). We shall give a more useful description of this operatorin the next subsection, but for the moment we note the key property

Trace �� 7 � � ��6 when Re � � � � �

Having found an analog of (2.1) for �� , it will follow that �� may besubstituted into (2.2) in place of � to give an equation

� � ) � � �� $ 3 7 � � ��

3 �� 3� )

$ 3 7 � � �� 3 ��

3 �� ) � � The remainder term will be a combination of operators of the same general type as�� but of “order” one less than �� .

Obtaining this formula for �� is the crucial step, and from here on the restof the proof is simple. Taking traces, and bearing in mind that the trace of a com-mutator is zero, we shall get

Trace � �� 1 �) � Trace ��


If we repeat the whole process, with �� replaced by the remainder � � , and thenwith � � replaced by the new remainder, and so on, then we shall get

Trace � �� 687� �� 1 �%1 �) ��

Trace �� where � � has order ��1 1� . But as � gets really large then we see from the Rel-lich Lemma that Trace �� becomes well-defined and holomorphic on an increas-ingly large half-plane in � . So the formula determines meromorphic extension ofTrace � �� to any desired half-plane in � , and hence to � itself.

2.2 Remark on Orders of Differential Operators

If �� and � ; are differential operators then the order of � �#� ; is usually the sum ofthe orders of � � and � ; . However the order of the commutator � � � � ;

� is nevermore than the sum of the orders of � � and � ; minus � . This drop in degree is veryimportant for the arguments that we are going to develop. It implies that taking thecommutator of an operator � with a function lowers the degree of � by one; takingthe commutator of � with a vector field does not change the degree; and taking thecommutator of � with � raises the degree by at most one.

If we work with more general rings of differential operators (for example actingon sections of vector bundles) then the general fact about � � � � ;

� no longer holds,and one must take a little care to check that the consequences listed above hold insufficiently generality for the arguments below to work (they do work).

2.3 The Actual Proof

On a closed manifold there do not exist global coordinates � � �"�"� �� $ . But by usinga partition of unity

� � � � subordinate to a cover of � by coordinate charts, we caneasily find functions � �"�"� � � and vector fields � � �"�"� ��$ such that

� 687 � � �

6 � 6 � �� and

�:�� 687 � � � � 6 � � 6 ) �

where as before � is an operator of order � or less and � has order less than � . (Theoperators 3 are of the form � � * � 3 , where � � is supported in the � ’th coordinate

468 N. Higson

chart and is � on the support of � � , and the operators � 3 are of the form � � *�� .)For the purposes of the commutator argument sketched in the last section the 6and � 6 work just as well as the coordinates � 6 and vector fields �� .

So let us begin by attempting to compute an expression of the form � � �� .For this purpose we shall need a way of looking at the operator � �� which is bettersuited to computation. We shall use the Cauchy formula

� �� 1 � � � � � � �

The integral is a contour integral along a downwards pointing vertical line in �which separates

�from the eigenvalues of � . It is not hard to check that if Re � � ��

and if � � � �� then by applying the integrand to � we get a convergentintegral in each Sobolev space � � �� , so the integral defines an operator from� � �� to � � �� . Cauchy’s formula from complex analysis proves that this isthe same as the operator � �� we defined previously.

Now, onwards with the computation, the first part of which is straightforward:

� � �� 1 � � � � � � � � �

� � � �� 1 � � � � � � � � � � 1 � � � � � � �

� � � �� 1 � � � � � � � � � � � 1 � � � � � �) � � �� 1 � � � � � � � � � � 1 � � � � � � � � � � 1 � � � � � � �

(In the last step we did two things at once: we commuted � past � � 1 � � � � and wethen used the formula � � � � � � � � � � � � � � � � � .) The operators � � � � and � � � �have orders � and

, respectively.

Before going on, we shall introduce some better notation for our contour inte-grals.

2.5 Definition. If � � �"�"� � � are differential operators on the closed manifold � ,then denote by � � � � � �"�"� � � � the integral

� � �� 1 � � � � *"*"* � � � � 1 � � � � � �

(in the integral, copies of � � 1 � � � � alternate with the operators � 6 ). The integralconverges if Re � � � � , in the sense we discussed above, and defines an operatoron � � �� .


Using our new notation, we can write �� and by elaborating veryslightly the computation we just ran through, we see that

� � � � � � � � � � � � � � � � � � �%)�� ) � � � � � � � � � � � � � �So what? Well, after replacing and � by 6 and � 6 , and summing over � , weknow that

� 687 � � � � 6 � � 6 � �:� 1 � and

� 687 � � � � 6 � � 6 � � 1 �

where the ‘remainder’ � has order � . We are going to plug these formulas intoour expression for � � � � � � � � � . To prepare for this, let us introduce the followingterminology:

2.6 Definition. We shall say that � � � � � �"�"� � � � is an integral of type � �� if

order � � � �') *"*"* order � � � �:1 ��,�� 2.7 Lemma. If � � � � � � � � �"�"� � � � is any integral of type � then

� 687 � � � � � 6 � � 6 � �� 1 � � � ) � �

where � � is a finite sum of integrals of type ��1 � .Proof. Let us just consider the case of the integral � � � � � (thus � � �� order � � � );the other cases are no harder. Using the formulas we have already obtained we get

� 687 � � � � � � � � 6 � � 6 �+� � � � � �') � � � � � �') type � 1 � integrals.

So the lemma will be proved if we can deal successfully with � � � � � � . What weneed to show is that

(2.3) � � � � � � ��1 � � � � � at least modulo integrals of order � 1 � . But in fact (2.3) holds exactly. To see whythis is so, note first that from the formula

� � � 1 � � � � � � � � 1 � � � � 1 �

470 N. Higson

along with our definition of the integrals � � it follows that

� � � � � � � � � � � � � �� %1�� So (2.3) is equivalent to the formula

� � � � � � �� 1 � � � � � � �This functional equation is proved using calculus, as follows. Take the integralwhich defines � � � � � � � and differentiate the integrand with respect to � (the inte-grand is of course a function of � ). We get

�

� �� 1 � � � � � � � � 1 �� 1 � � � � 1 � � �� 1 � � � ; �

Using the fact that the integral of this derivative is zero we get

� ��1 � � � � � � 1�� as required.

2.8 Remark. In the general case the functional equation is

� � 1 � � � � � � �"�"� � � � �� 687� � � � � � � � �"�"� � 6 �� 6 � �! �"�"� � � � �

At this stage we have almost proved our meromorphic continuation theorem.Using the algebraic tricks described earlier we can reduce the problem of comput-ing the trace of an integral of type � to the problem of computing the trace of anintegral of type � 1 � . It only remains to relate our notion of “type” to some notionof “order” of operators, so that we can guarantee the traceability of � � , for all in asuitable right half plane.

2.9 Definition. Let � be an integer (positive or negative). We shall say that a linearoperator �� has analytic order � or less if, for every � �such that �

� �and � )2� � �

, the operator � extends to a continuous linearoperator from � �

�! �� to � ! �� .

Thus for example every differential operator of order � or less has analyticorder � or less. If Re � � , 1 � then the operator � � � has analytic order 1 � , orless.

To prove Theorem 2.2 using our commutator strategy it remains to prove thefollowing two results:


2.10 Lemma. If � � � � � �"�"� � � � is an integral of type � then

analytic order � � � � � � �"�"� � � � � ,�� 1 Re � � �2.11 Remark. The integrand which we use to define � � � � � �"�"� � � � is

� � � � � � � 1 � � � � *"*"* � � � � 1 � � � � �This has order � 1 � ) . So when Re � � is negative (recall that the integral isdefined as long as Re � � ) � � �

) the order estimate in the lemma (which is sharp)is considerably better than one would expect by looking at the integrand alone.

To understand the content of the following lemma, recall that the integral defin-ing � � � � � �"�"� � � � is convergent when Re � � � � , and that we have not up tothis point defined the integral for other values of . However, thanks to the previ-ous lemma, the quantity Trace � � � � � � �"�"� � � � � is defined in the domain Re � � �max

� � $ � �

;� (this is where the integral makes sense and converges to an operator

of order less than 1 � ).

2.12 Lemma. If � � � � � �"�"� � � � is an integral of type � then the function

�� Trace � � � � � � �"�"� � � � � extends to a holomorphic function on the half-plane Re � � � $ � �

; .

Lemmas 2.10 and 2.12 are both proved by the same explicit computation. Toget the basic idea, let’s pretend that the operators � � commute with the operator � .In this case the integral � � � � � �"�"� � � � can be written as

� � � � � �"�"� � � � � 1 � � � � � � � � � � �

The “constant” � � �"�"� � � can be pulled out from under the integral sign, and whatis left can be evaluated by Cauchy’s integral formula. We get

� � � � � �"�"� � � � �� 1 - � � � *"*"* � � � � � � � �

With this formula in hand, both lemmas are obvious.

Proof of Lemmas 2.10 and 2.12. The idea of the proof is to try to move all theterms � � 1 � � � � which appear in the basic quantity

" � � � 1 � � � � *"*"* " � � � 1 � � � �

472 N. Higson

toward the right using the identity

� � 1 � � � � � � � � � 1 � � � � ) � � � 1 � � � � � �� 1 � � � � ) � � 1 � � � � � � � � � � 1 � � � � �

The formula leads to the formal expansion

� � 1 � � � � � � �� 1 � � � � � �

where we have used the notation

� � � � �� and � � � � � � � � � � � � � � for - � � .

The series does not converge, but instead it is an asymptotic formula in the follow-ing sense: if � and � � depend on a parameter � , then we shall write � �

� � � � if,for every ��

, every sufficiently large finite partial sum agrees with � up to anoperator of analytic order � or less, whose norm as an operator from � ! �

� �� to � ! �� is � � & � & � � . In our case if we truncate our series at - � �

, then theremainder term is

� � 1 � � � � � � � � � � � � 1 � � � � � �and the asymptotic expansion condition is easily verified. The reason for includingthe � � & � & � � condition is that we shall then be able to integrate with respect to � ,and obtain an asymptotic expansion for the integrated operator.

More generally one has, for any non-negative integer � , an asymptotic expan-sion

� � 1 � � �� 1 ��

� 1��- � # � � � � � 1 � � � � � �(this can be proved by induction on � ).

Before beginning the actual computation let us also define the quantities

� � -��! �"�"� - 6 � � � -�� ) *"*"* ) - 6 ) � �-�� *"*"* - 6 �� -�� ) �� *"*"* � - � ) *"*"* ) - 6 ) � � which depend on non-negative integers - � �"�"� - 6 . These have the property that� � - � � � � , for all - � , and

� � -�� "�"� - 6 �� -�� "�"� - 6 � � � �

� -�� ) *"*"* ) - 6 ) �%1 �- 6 �


(to be explicit, the right hand fraction is the product of the - 6 successive integersfrom � - � ) *"*"* ) - 6 � ��) � � to � -�� ) *"*"* ) - 6 ) � 1 �� , divided by - 6 ).

Now we can begin. Using this notation we obtain an asymptotic expansion

� � 1 � � � � ��

� � -�� 1 � � � � � � � � � and then

� � 1 � � � � �� 1 � � � � � ; � ��

� � - � � � � � � �� 1 � � � � � � � ; � " ;� �� 4� �� -�� - ; � �

� �� ; � � � � � � 1 � � � �

�� ; �

where & - & � -�� ) - ; , and finally

� � 1 � � � � �� *"*"* � � 1 � � � � � � � �� - � �� *"*"* � � � �� 1 � � � �

��

where we have written - � � - � �"�"� - � � and & - & � -�� ) *"*"* ) - � . Premultiplyingby � � , postmultiplying by � � 1 � � � � , and integrating with respect to � we get

� � �� 1 � � � � *"*"* � � � � 1 � � � � � �

� �� - � � � � � � � �� *"*"* � � ��

� 1 & - & ) � � � � � � � � � � � �The terms of this expansion have analytic order

� 1 - 1 � Re � �:) � � � ��1 - 1 Re � �

or less. This proves Lemma 2.10. If Re � � � �; �8� 1 � � then all the terms in the

asymptotic expansion are trace-class. This proves Lemma 2.12.

Having proved the lemmas, the proof of Theorem 2.2 follows by using themethod outlined in Subsection 2.1. Let us add one or two small remarks aboutthe vanishing of traces of commutators. It is a fundamental property of the tracethat if " and # and bounded, and if one of them is trace-class, then Trace � " # � �Trace � # " � . The situation here is a little more complicated because we are consid-ering the traces of commutators of possibly unbounded operators. To see that thetraces still vanish, we use Sobolev spaces, as follows. First, we may assume that

474 N. Higson

Re � � � �(if the trace of the commutator vanishes here it will vanish everywhere

the commutator is defined, by unique analytic continuation). Next we note that thetrace of an operator � of analytic order 1 � � 1 �

is the same, whether we regard �as an operator on � ; �� or on any Sobolev space � � �� with - � � .3 Indeed ifwe denote by � � � � �� ; �� the inclusion, and by � � the operator � actingon � � �� , then we can write

Trace �� Trace � � � � � � � �Since � � � � �� ; �� is a bounded operator and � � � � �� ; �� is trace-class (when � �

� ) - ) we get

Trace � � � � � � �� Trace � � � � � � � � � � Trace �� Finally, if we wish to show that Trace � " # � � Trace � # " � when say " has boundedorder � and # has order � � 1 �

we can think of " # and # " as compositions ofbounded operators and trace-class

� ; ��

// � � ��

// � ; ��

and

� � ��

// � ; ��

// � � �� and apply the basic trace property together with the previous remark to � � # " .

An Improvement of the Main Theorem

In this concluding subsection we shall improve a little Theorem 2.2 by proving thata number of the singularities of Trace � � �� "�"� � � � � , including in particularthe singularity at 2� �

, are removable. As we shall see in Section 5, this isquite significant for index theory. Moreover the appearance of the Gamma functionin the following lemma will prepare the way for our later computations in cycliccohomology.

2.13 Lemma. If � � � � � �"�"� � � � is an integral of type - then the function

�� Trace � � � � � � �"�"� � � � � is holomorphic in the domain Re � � � $ � �; .

3With a bit more effort one can show that the same thing holds for all �� and all .


The content of the lemma is that Trace � � � � � � �"�"� � � � � has zeros at the non-positive integer points in its domain Re � � � $ � �; , which cancel out the simplepoles of the

-function. The factor � 1 �� is present for tidiness; it also plays a

useful role in subsequent developments within cyclic cohomology (see Section 6).

Proof. The argument used to prove Lemma 2.12 produces the formula

� 1 �� Trace � � � � � � �"�"� � � � �� 1 ��

� � � 1 & - & ) � � � � - � Trace� � � � � � � �� *"*"* � � ��

��

The symbol � , which we are now applying to functions of , means that, given anyright half-plane in � , any sufficiently large finite partial sum of the right hand sideagrees with the left hand side (on the common domain of the functions involved)modulo a function of which is holomorphic in that half-plane. It follows from thefunctional equation for

� � that

� 1 �� 1 & - & ) � � � � 1 �� ) � )�& - & � �� & - & ) � � �

So we get

� 1 �� Trace � � � � � � �"�"� � � � �� 1 ��

�� ) � )2& - & � �

� & - & ) � � � � - �� Trace

� � � �� *"*"* � � � �� This completes the proof.

Repeating the argument from the previous subsection we obtain the followingresult:

2.14 Theorem. Let � � � � � �"�"� � � � be an integral of type - . The function

� 1 �� Trace � � � � � � �"�"� � � � �extends to a meromorphic function on � with only simple poles. The poles arelocated within the sequence � ) - 5� ) - 1 �< �"�"� .

476 N. Higson

3 Abstract Differential Operators

In this section we shall first introduce a more abstract notion of differential op-erator, and then develop a corresponding theory of pseudodifferential operators.Apart from the standard example coming from standard differential operators on asmooth, closed manifold, we shall also consider a more elaborate example relatedto foliation theory, and a collection of examples derived from Alain Connes’ notionof spectral triple.

3.1 Algebras of Differential Operators

Let � be a complex Hilbert space. We shall assume as given an unbounded, posi-tive, self-adjoint operator � on � . As the notation might suggest, the main exampleto keep in mind is the Laplace operator on a closed Riemannian manifold, but thereare many other examples too. We shall soon introduce a notion of “order”, gener-alizing the notion of order of a standard differential operator, and we should keepin mind that � need not have order

. In fact let us now fix an integer � � �

, whichwill play the role in what follows of order �� .

For - � �denote by � � the domain of the operator � � � . In the standartd

example, where � is the Laplace operator and � � , it follows from the basic

elliptic estimate that the Hilbert space � � may be identified with the Sobolev space� � �� .

Let � �.� �� 7 � � � . We shall assume as given an algebra�

of linear operatorson the vector space � � . In the standard example,

�will be the algebra of all linear

differential operators on � . Let us also assume that the algebra is filtered: thus itis given as an increasing union of linear subspaces

� � � � � � *"*"* � �in such a way that

��* � � � �

� �� . We shall write order � " ��, � if " � � .

3.1 Definition. We shall say that the pair comprised of � and�

is differential4 ifthe following conditions hold:

(i) If " � , then also � � " � � , and

order � � � " � ��, order � " �')�� 1 � �4Strictly speaking we should include the integer � somewhere here.


(ii) If " � , and if order � " � , � , then there is a constant� � �

such that

� �� ) � � � �� " � � � � � �

(the norm is that of the Hilbert space � ).

3.2 Remark. If we introduce the natural norm on the space � � � dom �� , namely

� � � ; � � � �� ; ) � � � ;

then the estimate in item (ii) can be rewritten as

� � � � ) � � � �� " � � � � � �(for perhaps a different

�). In the standard example this is easily recognizable as

the basic estimate of elliptic regularity theory.

3.3 Lemma. If " � �� , and if " has order � or less, then for every ��

theoperator " extends to a bounded linear operator from � ! � � to � ! .Proof. If � is an integer multiple of the order � of � then the lemma follows im-mediately from the elliptic estimate above. The general case (which we shall notactually need) follows by interpolation.

This begs us to make the following version of Definition 2.9 in our new abstractcontext:

3.4 Definition. A linear transformation �� has analytic order � �� iffor all �

� �such that � ) � ��

it extends to a bounded linear operator �� ! � � �� ! .

3.2 An Example

Let � be a smooth manifold. Assume that an integrable smooth vector sub-bundle� � �� is given, along with metrics on the bundles�

and �� (the metricswill play only a very minor role in what we are going to do here). The bundledertermines a foliation of � by say � -dimensional submanifolds.

Let�

be the algebra of linear partial differential operators on � with compactsupports. Define a filtration on

�, which makes use of the foliation on � , as

follows:

(i) If � is a � � -function on � then order �� .

478 N. Higson

(ii) If " is a � � -vector field on � then order � " � , .(iii) If " is a � � -vector field on � which is everywhere tangent to

�then order � " � ,� .

From now onwards in this subsection we shall use the above non-standard notionof order while discussing operators in

�.

When discussing local coordinates on � we shall use coordinates which iden-tify a neighbourhood � in � with an open set in � � � � � in such a way that theplaques of the foliation (the connected components of the interections of the leaveswith the chart) are of the form � � � � pt � . Let us call these foliation coordinates. If" � , then in local foliation coordinates we can write " as a sum

" � � � � ��

� ��

If order � " ��, - , then we can separate the sum into a part of order - , plus a part oflower order,

" � � � � 7

��

�� )

� � ��

��

where� � � is defined by the formula

� � � � � � ) *"*"* )�� ) � � � � ) *"*"* ) � � �3.5 Definition. An operator " � is elliptic of order � , relative to

�, if in every

coordinate system, as above, and at every point � in the domain of the coordinatesystems the the order � part of " has the property that�� 7 � � � ��

� �� & � � & ; ) *"*"* & � � & ; )2& � � � � & � ) *"*"* )2& � � & � �

for some� � � �

and all�.

If� � � � then this coincides with the usual definition of ellipticity. If we

define a Sobolev norm in a foliation chart by the formula

� � � ;� � ��4 �� !

��

�� ;

then every order - operator is continuous from � ! � � �� to � � �� . Moreoverthe arguments used to prove the elliptic estimate in Section 1 easily adapt to showthat if " is elliptic of order � relative to

�, then

(3.1)� " � � ; ! ) � � � ; � �� !


for some� � � �

and every smooth, compactly supported � .Passing from coordinate charts to global situation on � using partitions of

unity, we obtain global Sobolev spaces � � �� and the corresponding globalversion of the elliptic estimate (3.1). Observe also that

� ; � �� from which it follows that

�� +� � ��

We obtain the following result:

3.6 Theorem. Let � be a smooth manifold and let�

be a smooth, integrablesubbundle of � � . If � is a positive and elliptic operator on � (relative to

�), and

if � and its powers are essentially self-adjoint, then � � � � is a differential pair inthe sense of Definition 3.1.

We can define an explicit elliptic operator

�� ; � ) � # composed of a “leafwise” operator � � and a “transverse” operator � # on � , asfollows. Using the given metric on

�we can define a leafwise Laplace operator

� � which acts just by differentiation along the leaves of the foliation. Using localfoliation coordinates we can identify a foliation chart � in � with an open setin � � � � � , and after having done so, we can use the given metric on � �� todefine Riemannian metrics on each transversal

�pt � � � � , which together determine

a “transverse” Laplace operator on � . The operator � # 4 � so constructed dependson our choice of foliation coordinates. However by covering � by charts � � andchoosing a partition of unity

� � � � we can form a non-canonical operator

� # � � �� # 4 ��

��

We are requiring operators in our algebra�

to be compactly supported, but if weput this requirement to one side for a moment and think of � as an element of

�

then we can say that � has order � , and that up to operators of lower order, both �and � # are independent of all the choices made in their construction.

For the particular differential pair � � � � we have just constructed it is a sim-ple matter to adapt the arguments of Section 2 to prove that all the zeta functions

480 N. Higson

Trace � �� admit meromorphic extensions to � , with only simple poles. Theproof begins from the basic formula

- �� 3 7 � � � ��3 � �� 3 )

$3 7� � �

� � �� 3 � �� 3 ) � for an order - operator, where � is a differential operator of order - 1 � or less (ascomputed in the given filtration of

�). This implies that

� - ) � ) � � � � � 3 7 � � � ��3 �� 3

� ) � 3 7 � � �� 3 ��3 � �

) $3 7� � �

� � �� 3 �� 3� ) $

3 7� � �

� �� 3 ��3 � � ) �

with the same remainder term � . From here the proof proceeds exactly as in Sec-

tion 2. The result is that, if � has order - , then the zeta function Trace � �� hasa meromorphic extension to � , with at most simple poles located at the sequenceof points - ) � ) �� - ) � ) � 1 �< �"�"� �In particular the basic zeta function Trace �� has poles at � ) �� ) ��1�< �"�"� . An interesting feature of this result is that the ‘analytic dimension’ of �� (measured as in Weyl’s Theorem by the asymptotic behavious of the eigenvaluesequences of elliptic operators) is not � , the dimension of the manifold, but � )�� ) � .

An important feature of the differential pair � � � � is the invariance of � , mod-ulo operators of lower order, under diffeomorphisms of � which preserve

�and

which moreover preserve the metics on�

and �� . As Connes and Moscoviciobserve in [10], starting with a manifold

�and any group � of diffeomorphisms

of�

, it is possible to build a new manifold � which fibers over � along withmetrics on the vertical tangent bundle

�and the quotient bundle �� , in such at

way that the action of � lifts to � , preserving the given metrics. Starting from thisobservation Connes and Moscovici are able to develop elliptic operator theory andindex theory on very complex spaces, for example the transverse spaces of foliatedmanifolds.

3.3 Pseudodifferential Operators

Let us return now to our general notion of differential pair. Starting from this con-cept we can reproduce many of the computations we did in Section 2, for example


those used to prove Lemmas 2.10 and 2.12 in our new context (we already sug-gested as much at the end of the last subsection). However we shall leave this tothe reader to check, and instead we shall develop the following closely notion ofabstract pseudodifferential operator.

3.7 Definition. Let �� be a differential pair. Fix a positive operator�

of ana-lytic order 1 � (this means that

�maps � into � � ) such that the operator

� � � � ) �

is invertible. A basic pseudodifferential operator of order - �� is a linear operator�� with the property that for every � � the operator � may bedecomposed as

� � " � � �� ) � where " � � � � , � � , and � � � � � � � , and where

order � " �') � ,�- and order �� , � �A pseudodifferential operator of order - �� is a finite linear combination of basicpseudodifferential operators of order - .

3.8 Remark. The introduction of the operator�

is more or less a matter of con-venience; for example we could have changed � � to � �%) � � without changing theclass of pseudodifferential operators determined by the definition. In particular thechoice of

�has no effect on the definition. (We should add that using spectral

theory it is easy to find a suitable operator�

.)

3.9 Example. If � is a pseudodifferential operator of order - , then � � � � is a pseu-dodifferential operator of order - ) � 1 � .3.10 Example. All of the integrals � � � � � �"�"� � � � for integral are pseudodiffer-ential operators. This follows from the asymptotic expansion formula used in theproof of Lemma 2.10.

We are going to show that the linear space of all pseudodifferential operatorsis an algebra. For this purpose we shall need to develop some of the asymptoticexpansions used in Section 2 in our new, abstract context.

3.11 Definition. If � and � 6 are operators on � � , then let us write

� �� 687 � � 6

482 N. Higson

if, for every � , there exists � � such that if � � � � , then the difference � 1 � �687 � � 6is an operator of order � or less.

3.12 Lemma. If � is a pseudodifferential operator, and if � � , then

� � � � � � �� 687 �

� �� 6 � � � � 6� �

3.13 Remark. We define � � �� , for Re � � � �, by a Cauchy integral, as we did in

Section 2. Since � � is invertible we can choose the contour of integration to be the(downwards pointing) imaginary axis.

Proof of the Lemma. We compute as follows:

� � � � � � � � � �� 1 � � � � � � � � �

� � � �� 1 � � � � 0 � �� 1 � � � � � ��

where we have written 0 � �� . The integral converges as long as Re � � ��(it converges absolutely to an operator on the Frechet space � � ), and for the

moment let us confine our attention to such . Continuing, we can write

� � � � � � � � � �� 1 � � � � � 0 � �� 1 � � � � � � �

� � � �� 0 � �� 1 � � � � ; � �

) � � �� 1 � � � � � 0 � �� 1 � � � � � � �

� � � �� 0 � �� 1 � � � � ; � �

) � � �� 1 � � � � � 0 ; � �� 1 � � � � ; � ��

and more generally

� � � � � � � � � �� 0 � �� 1 � � � � ; � � ) � �

�� 0 ; � �� 1 � � � �

� � �

) *"*"* ) � � �� 0 � � �� 1 � � � � � � � � �

) � � �� 1 � � � � � 0 �� 1 � � � � � ��


Using the Cauchy integral formula we can now compute

� � � � � � �� 1 �

� 0 � �� )� 1 � 0 ; � �� ; ��

) *"*"* )� 1 - � 0 � � ��

) � � �� 1 � � � � � 0 �� 1 � � � � � ��

If order �� , then the remainder integral in the final display converges whenRe � � � - ) � to an operator of order ��1 - 1 � Re � � . This proves the lemma.

3.14 Proposition. The set of all pseudodifferential operators is a filtered algebra.

Proof. The set of pseudodifferential operators is a vector space. The formula

" � �

�� * # � �� 687�� ; � � " 0 6 � � # � � � � �

� �6�

shows that it is closed under multiplication and moreover that the product of twopseudodifferential operators of orders - and � is a pseudodifferential operator oforder - ) � .

The algebra of pseudodifferential operators is a good context in which to studythe residues of the zeta functions Trace � �� , thanks to the following beautifulfact:

3.15 Lemma. Assume that for every differential operator � � , and all �with sufficiently large real part, the operator �� is trace-class. Assume that, inaddition, for every � � the zeta function Trace � �� extends to a meromor-phic function on � with only simple poles. Then the residue functional

� �� '� Res �7� Trace ��

is a trace on the algebra of pseudodifferential operators.

3.16 Remark. If � itself has discrete spectrum and compact resolvent � � ) � � � � ,and if we define � � � as we did in Section 2, integrating down a vertical line whichseparates

�from the positive spectrum of � , then the residues of Trace � �� and

Trace �� are equal.

484 N. Higson

Proof of the Lemma. We want to show that

Res �7� Trace �� Res �

7� Trace �� Using the trace property of the operator trace, this amounts to showing that

Res �7 � Trace

�� 1 � � ��

Using Lemma 3.12 we get

� � � �� 1 � � � �� 1� 687 �

� 1 �

� � � � 6 � � � � 6�As a result,

Res �7� Trace

�� 1 � � �� 21 � 687 � Res �

7� � � 1 �

� Trace �� 6 � � � � 6� ��

This is a finite sum since all but finitely many of the residues of Trace �� 6 � � � � 6� �are zero. But in fact since each trace function has at worst a simple pole, all theresidues in the sum are zero: the possible pole of Trace �� 6 � � � � 6� � at � �

iscanceled out by the factor of in the binomial coefficient

� � �6 � .3.17 Remark. This result of Wodzicki [28] was first observed in the following al-gebraic context (compare for example [26] for a clear account). Let be a complexalgebra and let � be a derivation on . The main example is where is the algebraof smooth functions on unit circle and � is ordinary differentiation:

� � � � ��

�� The space � � � of formal polynomials

� �$ 7 � � $ � $ in � with coefficients in isan associative algebra, with multiplication law derived from

� � � � � � � � � �In the main example this is the algebra of differential operators on the circle. Con-sider now the algebra � � � of formal series

�$ 7 � � �

$ � $


in � with coefficients in . Infinitely many of the negative coefficients may benonzero, but we require that each series contain only finitely many positive powersof � . This is an associative algebra with multiplication derived from the formula

� $ * � �� 687 �

� �� 6 � � � � $ � 6 �

Let �� be a trace functional which vanishes on the range of � . Thus � is a

linear functional for which

�� ) � � � ��

In the main example, where is the algebra of smooth functions on the circle � isthe ordinary integral:

� � � � ��

The following is then an algebraic counterpart of Lemma 3.15:

3.18 Lemma. The functional � � � � � � � defined by

�� 3 �3� � � � � � � � �

is a trace on the algebra � � � .3.4 Spectral Triples

Further examples of differential pairs � � � � are furnished by Connes’ notion ofspectral triple. In this subsection we shall briefly review the basic definitions.

3.19 Definition. A spectral triple is a triple � � � � , composed of a complexHilbert space � , an algebra of bounded operators on � , and a self-adjoint oper-ator � on � with the following two properties:

(i) If � then the operator � * � � ) � ; � � � is compact.

(ii) If � then � * dom � � � � dom � � � and the commutator � � � � extends toa bounded operator on �

Various examples are listed in [10]; in the standard example is the algebraof smooth functions on a complete Riemannian manifold � , � is a Dirac-typeoperator on � , and � is the Hilbert space � ; �� of square-integrable sections ofthe vector bundle on which � acts.

486 N. Higson

3.20 Definition. Let � � � � be a spectral triple. Denote by�

the unboundedderivation of � �� given by commutator with & � & . Thus the domain of

�is the set

of all bounded operators � which map the domain of & � & into itself, and for whichthe commutator extends to a bounded operator on � .

3.21 Lemma. Let � � be a core of & � & (a subspace of the domain on which theoperator is essentially self-adjoint). If � maps � � into itself, and if � & � & � � isbounded on � � , then � lies in the domain of

�.

3.22 Definition. A spectral triple is regular if and � � � � belong to �$ 7 � � $ .

The notion of regular spectral triple � � � � plays a useful role in the detailedanalysis of Alain Connes’ spectral triples and their Chern characters. See for exam-ple [14]. The purpose of this subsection is to show that regularity is equivalent tothe basic elliptic estimate which appears in item (ii) of Definition 3.1 (the relevantpair � � � � will be described in a moment). This equivalence is essentially provedin [10, Appendix B], although in disguised form.

3.23 Definition. Let � � � � be a spectral triple with the property that every � maps � � into itself. Denote by � the operator � ; . The algebra of differentialoperators associated to � � � � is the smallest algebra

�of operators on � �

which contains and � � � � and which is closed under the operation � �� .3.24 Remarks. If the spectral triple � � � � is regular, then the condition *� � � � � is automatically satisfied. The above description of

�is in some

sense the minimal reasonable definition of an algebra of differential operators. Notehowever that the operator � is not necessarily included in

�.

The algebra�

of differential operators is filtered, as follows. We require thatelements of and � � � � have order zero, and that the operation of commutatorwith ��2� ; raises order by at most one. Thus the spaces

�� of operators of order- or less are defined inductively as follows:

(a)� � � algebra generated by ) � � � � .

(b)� �� ) � � � � � � � .

(c)�� 687 � � 6 * � � � 6 ) � � � � � � � ) � � � � � � � � � .

We want to prove the following result.

3.25 Theorem. Let � � � � be a spectral triple with the property that every

� maps � � into itself. It is regular if and only if � � � � is a differential pairin the sense of Definition 3.1.


3.26 Remark. As should be clear, we assign to � the order � � . Condition (i) ofDefinition 3.1 is then automatically satisfied.

We shall begin by proving that a regular spectral triple � � � � satisfies thebasic estimate.

3.27 Definition. Let � � � � be a regular spectral triple. Denote by � � � � thealgebra of operators on � � generated by all the spaces

� $ � � and� $�� , for

all � ��.

Note that, according to the definition of regularity, every operator in � � � �extends to a bounded operator on � . The notation “� � � � ” is chosen to suggest“pseudodifferential operator of order

�” (it is indeed the case that � � � � is an

algebra of order�

pseudodifferential operators associated to the differential pair� � � � ).3.28 Lemma. Assume that � � � � is a regular spectral triple. Every operatorin�

of order - may be written as a finite sum of operators � & � & � , where � belongsto the algebra � � � � and where � ,�- .

Proof. Define � , a space of operators on � � , to be the linear span of the operatorsof the form � & � & � , where - � �

and � � � � � . The space � is an algebra since� � � � � � � � � � � � and since

� �"& � & � � * � ; & � & � � ��

687�� -�� 6 � � ; � & � & �

��

6 �Filter the algebra � by defining � � to be the span of all operators � & � & � with � , - .The formula above shows that this does define a filtration of the algebra � . Nowthe algebra

�of differential operators is contained within � , and the lemma we are

trying to prove amounts to the assertion that�� . Clearly

� � � � � . Usingthe formula

� � � & � & � � � � � � & � & ; � & � & � � � � � � � � � & � & � ) � ; � � � & � & � � � along with our formula for

�� , the inclusion

�� is easily proved by induc-

tion.

We can now prove that every regular spectral triple satisfies the basic estimate.According to the lemma, it suffices to prove that if - � � and if " � � & � & �

, where� � , then there exists� � �

such that� � � � � ) � � � �� " � �

488 N. Higson

for every � � � . But we have

� " � � � � � & � & � � � , � � � * � & � & � � � � � � � * � � � � � And since by spectral theory for every � ,�- we have that

� � � � � ; , � � � � � ; ) � � � ; ,� � � � � � ) � � � � ;

it follows that � � � � � ) � � � � �� ) �� " � �

as required.We turn now to the proof of the second half of Theorem 3.25. Assume from

now on that � � � � is a spectral triple for which * � � � � � and for which� � � � is a differential pair. Starting from the differential pair we can form thealgebra of pseudodifferential operators, as in Subsection 3.3.

3.29 Lemma. If � is a pseudodifferential operator then so is� �� , and moreover

order � � �� , order �� .Proof. We compute that

� �� '�+& � & � 1�� & � & � �� 1��

��

�� 687 �

� �; � � 0 6 � ��

�� 6 �

This computation reduces the lemma to the assertion that if � is a pseudodifferentialoperator of order - then 0 � �� is a pseudodifferential operator of order - ) � orless. Since 0 � �� this in turn follows from the observation made inExample 3.9.

Proof that � � � � is regular. By the basic estimate, every pseudodifferential op-erator of order zero extends to a bounded operator on � . Since every operator in or � � � � is pseudodifferential of order zero, and since

� �� is pseudodifferentialof order zero whenever � is, we see that if � or � �� then for every �the operator

� $ � � � extends to a bounded operator on � . Hence the spectral triple� � � � is regular, as required.


3.5 Dimension Spectrum

3.30 Definition. A spectral triple � � � � is finitely summable if there is some-� �such that the operator � * � � ) � ; � � � is trace-class, for every � .

Suppose that the spectral triple � � � � is regular, and denote by�

the asso-ciated algebra of differential operators. If � � � � is finitely summable then forevery " � the zeta function Trace � " � �� is defined in a right half-plane in � ,and is holomorphic there (as before, � � is an invertible operator obtained from � byadding a positive, order 1 � operator). The following concept has been introducedby Connes and Moscovici [10, Definition II.1].

3.31 Definition. Let � � � � be a regular and finitely summable spectral triple.It has discrete dimension spectrum if5 there is a discrete subset

�of � with the

following property: for every operator � in the algebra � � � � of Definition 3.27,

the zeta function Trace �� extends to a meromorphic function on � with allpoles contained in

�.

If � � � � has discrete dimension spectrum then for every differential, or

indeed pseudodifferential, operator " , the zeta function Trace � " � �� extends to ameromorphic function on � . Moreover if " has order - then the poles of this zetafunction are located in

� ) � . Conversely, if � � � � is a regular spectral triple,

and if, for every differential operator " of order - , the zeta function Trace � " � � �� extends to a meromorphic function on � whose poles are located within

� ) � , then� � � � has discrete dimension spectrum

�.

A final item of terminology:

3.32 Definition. A regular and finitely summable spectral triple has simple dimen-sion spectrum if it has discrete dimension spectrum and if all the zeta-type functionsabove have only simple poles.

It is an interesting and as yet unsolved problem to find algebraic conditions ona regular spectral triple which will imply that it has discrete or simple dimensionspectrum.

5Connes and Moscovici add a technical condition concerning decay of zeta functions along verti-cal lines in

�.

490 N. Higson

4 Computation of Residues

We saw in the Section 2 that if � is the Laplace operator on a closed Riemannianmanifold � and if � is any differential operator on � then the function

(4.1) Trace � �� is meromorphic on � . Moreover if � has order � then the poles of this zeta functionare all simple and are located at the integer points � ) � � ) � 1 �< �"�"� .

The purpose of this section is to explain how the residues of Trace � �� aregiven by complicated but in principal explicit and computable formulas involvingthe coefficients of � and � . This ‘local computability’ of residues is a very im-portant conceptual point: in the next section we shall consider a family of globallydefined index invariants of manifolds, and it will be a significant and nontrivial factthat these global invariants are given by explicit (albeit complicated) local residueformulae.

We shall not take the shortest route toward our goal of producing local formulaefor residues. Instead we shall follow a method, based on commutators, which isloosely related to our proof of meromorphic continuation in Section 2. Nor shallwe give a very detailed or sophisticated account of this topic. Instead, for the fullstory the reader is referred to [28] or [15].

4.1 Computation of the Leading Residue

We are going to find a formula for the residue at �� ) � of the functionTrace � �� , where � is an order � differential operator. This is the residueat the leading or rightmost pole in � . Note that

Res �7 $ � � Trace � �� Res ! 7� Trace � ��

where � is the residue trace on the algebra of pseudodifferential operators (seeLemma 3.15). So the leading residue is the residue trace of the order 1 � pseudod-ifferential operator �� . We are going to use the trace property of � to producea formula for the residue trace of any order 1 � pseudodifferential operator.

In order to produce such a formula we first need to extract from a pseudodif-ferential operator its symbol, which is a function on the cotangent sphere bundle� � � .

4.1 Definition. Let � be a differential operator of order � . Its principal symbol isthe function ��

�� defined in local coordinates by the formula�� 7 ��

� �


where

��

� ��

In other words, to define � �� we just exchange each partial deriva-

tive �� in the leading order terms of � for the corresponding coordinate function� 3 on the cotangent bundle. The reason for dropping the lower order terms of �is that the principal symbol is then independent of the choice of local coordinateson � , and so well defined on all of � � � . This would not be the case if the lowerorder terms of � were retained.

The overall factor � is conventional. It ensures that, for example, the symbolof � , a positive operator, is a positive function. In fact the symbol of � is theconstant function � . For a general operator � of order � , the symbol extends to afunction on � � � which is polynomial and homogeneous of order � in each fiber ofthe cotangent bundle (in the case of the Laplace operator � this extension is just thenorm-squared function

� �� ; obtained from the Riemannian metric). Going inthe other direction, if � � � � � � � is polynomial and homogeneous of order � ineach fiber, then it is the symbol of some order � differential operator.

4.2 Definition. Let � be an order � pseudodifferential operator. Its principal sym-bol is the function � # �� obtained by representing � in the form

� �2�� ) � where order �� +� , and then setting

� # � ��

The symbol is well-defined. This follows in the first place from the fact that thesymbol of � is the constant function � on � � � , so that if we write � �2�� * � � � �then �� , and in the second place from the fact that the analytic order of adifferential operator is exactly equal to its differential order.

We are going to prove the following result.

4.3 Theorem. There is a constant � such that if � is any order 1 � pseudodifferen-tial operator then

� �� :� � ��

� # � vol �Here is roughly how we are going to proceed. We shall show that if the integral

vanishes, then the symbol � # can be written as a linear combination of ‘derivatives’.

492 N. Higson

As we shall see, this will imply that � can be written as a linear combination ofcommutators of pseudodifferential operators, modulo an operator � of lower order.From the trace property of � it will follow that � �� , and since � has orderless than 1 � it follows that � �� All this will show that if the integral vanishesthen so does � �� . Since the integral and the trace are both linear functionas on thespace of order 1 � operators, it will follow that � is a constant multiple of theintegral, as required.

To start the argument, we consider the complex of differential forms on � � �which are polynomial in the fiber direction (this means that the forms are localcombinations of forms � �� , where � is polynomial in the

�-variables;

here � � �"�"� �� $ , together with� � �"�"� � $ , are the standard coordinate functions on

� � � ). This complex computes the de Rham cohomology6 of � � � . The volumeform on � � � is given by the formula

vol � � � �$ 687 � � 1 ��

6� � � 6 � � � *"*"* � � $ � � � *"*"* � � � 6 *"*"* � � $

and so belongs to our complex. If the integral in Theorem 4.3 is zero then � * vol � � �is exact, say

(4.2) � * vol � � � � � � �We are now going to transfer this equation to the space � � � , obtained from � � �by deleting the zero section. Of course, � � � is a submanifold of � � � . We extend� to a function on � � � by requiring it to be homogeneous of order 1 � in eachfiber. We extend vol � � � to the form

� � ��

$ 6 7 � � 1 ��

6� � � 6 � �%� *"*"* � � $ � � � *"*"* � � � 6 *"*"* � � $ �

Here � � � � � � � is the function � � � � � � � � . By collapsing each positive ray in� � � to a point we get a projection to � � � , and using it we pull back � to a form�

on � � � . From (4.2) we get

� � � � * � �6This part of the argument would be simpler if we used the classical notion of pseudodifferential

operator from analysis, in which case the relevant class of functions on ��

would be the class ofall smooth functions, and the relevant complex would be the standard de Rham complex.


Multiplying both sides by the closed form�� , and observing that

�� * � � vol � � �

we get�� * vol � � �

where� � � * � � . Writing this equation in local coordinates we arrive at the

following:

4.4 Lemma. If � � � �� vol � �

, then � , viewed as a function on �� , is a sum

of functions each of which is supported in a coordinate chart and is of the form

� �� 6 or � �� 6 �The functions � and � are quotients of functions which are polynomial in each fiberof � � � by powers of � .

Proof of Theorem 4.3. Let � be an order 1 � operator. It suffices to prove that ifthe integral of the symbol of � over � � � is zero then the residue trace of � is zero.If the integral over � � � of the symbol of � is zero then the symbol is a sum ofderivatives of the type � �� or � �� , as in Lemma 4.4. If we construct operators and � with symbols � or � then we find that the commutators � �� and � � �� 6 �have symbols � �� or � �� , respectively. Conclusion: the operator � is a sum ofcommutators, modulo an operator of order less than 1 � . Since the residue tracevanishes on commutators, and also on operators of order less than 1 � , it followsthat the residue trace of � is zero, as required.

4.5 Remark. It is not difficult to see that the constant � depends only on dim �� (note that � is determined by the residue trace of an operator supported in a coordi-nate neighbourhood; given two different connected manifolds, apply Theorem 4.3to a third manifold which contains coordinate neighbourhoods isometric to neigh-bourhoods in the first two manifolds). By checking an explicit example, like theflat torus, one can see that � $ � � � � � $ .

4.2 The Lower Residues

Let � be an order � differential operator. The problem of computing the residuesof Trace � �� (or similarly of

� �; � Trace � �� ) at the “lower” poles � ) ��1�< 5� ) � 1 �"�"� can be reduced to the problem of computing the highest residueby the scheme used in our proof of meromorphic continuation.

Before starting the computation, it is useful to note that our basic meromorphiccontinuation theorem can be strengthened in the following way. An elaboration

494 N. Higson

of the Sobolev theory that we developed in Section 1 and Appendix B shows thatevery � , every operator �� of sufficiently large negative ordermay be represented by a � � integral kernel function defined on � � � :

� � �� '� ��- ��

It follows that if is restricted to a suitable left half-plane in � , then any integral� � of the type considered in Definition 2.5 may be represented by a kernel function- � �%� � � � � which is � -times continuously differentiable. Consider now thebasic identity from the proof of Theorem 2.2: if � � is an integral of type - then

� 3 7 � � � � � 3 � 3 � ) �

3 7 � � �3 � 3 � � � � � - 1 � � � ) � �

where � � is a finite sum of integrals of lower type. As we know, the identity isequivalent to the identity

(4.3)

� 3 7 � � � � � 3 � � 3 ) � � � � � - ) � 1 � � � ) � � �

Now, let us represent the integral � � by an integral kernel - � �� , and compute theleft hand side of (4.3). The vector field � 3 is a skew-symmetric operator, modulooperators of lower order: this means that there is a smooth function � 3 � � � �so that

�� * � 3 � �

vol �21 �� 3 � * � �

vol ) �� 3 � * � �

vol �We can therefore write the left-hand side of (4.3) as an integral operator

� �� - � �� 3 � �'1 3 �� 3 � � � � ) � � - � �� 21 �

� 3 � - � �� 3 � �'1 3 �� ) � � 3 � � - � �� 3 � �'1 3 �� <� � � � ) � � - � ��

(the vector field � 3 acts on the -variable). Finally,

� 3 � - � �� 3 � � 1 3 �� 3 � - � �� 3 � � 1 3 �� ) - � �� 3 � 3 � � � �


But � 3 � 3 � � � is the scalar function � � 3 � 3 � . Setting � � in the above formulasand using the fact that

� � � 3 � 3 � � � we now see that the left hand side of (4.3)is represented by an integral kernel which vanishes identically along the diagonal� � in � �� . The proof of Theorem 2.2 now provides the following well-known meromorphic continuation of trace-densities for complex powers of � :

4.6 Theorem. Let � be a positive, elliptic operator on a smooth, closed manifold� , and let " be a differential operator on � . For Re � � � �let

�

� �� be the restriction to the diagonal in � � � of the integral kernel - � �� for the

operator " � � �� . For every � the map ��

� extends to a meromorphicfunction from � into the � -times continuously differentiable functions on � .

We see that the residues we are trying to compute are the integrals over � ofresidue densities Res �

7 � Trace � �� Res �7 � �

� �� . The leading residuedensity is given by the formula

(4.4) Res �7 $ � � Trace � ��

� � � $�

� ��

��

which integrates the symbol of �� over the cotangent sphere at � .To compute the lower residue densities near � let us choose 3 and � 3 to be of

the form � 3 and �� near � . Setting � � �� and using the formula

� � ) � 1 � � � � $ 3 7 � � � � �� 3 �� 3

� )$ 3 7 � � �� 3 ��

3 � � � ) � � we see as before that the trace densities of � � )�� and � � are equal (since notonly are the traces of the commutators zero, but their trace densities are identicallyzero). It follows that the residue densities of � ��) � 1 � � � are equal to the residuedensities of � � � � � 1 � � � 1 � $ 3 7 � � � �� 3 � �� . In particular, looking at the residuejust one below the leading residue we get

Res �7 � � � � �

�Trace � � � � �� Res �

7 � � � � ��

Trace �� But on the right hand side we are computing the leading residue of � � , so that wecan invoke the explicit formula (4.4). As a result, since � � is explicitly computablein terms of � � , we obtain an explicit (but complicated) formula for the residue den-sity of � � at � ) � 1 � .

Repeating this argument we get explicit formulas (which get more and morecomplicated) for all the residue densities of Trace � � � � .

496 N. Higson

5 The Index Problem

In this section we shall introduce the problem in Fredholm index theory whosesolution will occupy the remained of the notes. This will require us to introducecyclic cohomology. Since there are several good introductions to the latter subject(for example [22] or [4]) we shall do so quite rapidly.

5.1 Index of Elliptic Operators

From now on we are going to work in the � � -graded situation which is standard inindex theory. We shall assume that � , a linear partial differential operator on closedmanifold � , is the square of a self-adjoint, first order, elliptic partial differentialoperator � . We shall assume that � acts not on scalar functions but on the sectionsof some smooth vector bundle � over � . We shall assume moreover that � iswritten as a direct sum � � � � �� (in other words that � is � � -graded), andthat, with respect to this direct sum decomposition, the operator � has the form

� ��

so that

��2� ; ��

� � � � ��

Denote by�

the grading operator

� �� 1 � � �

As is customary in the � � -graded world we shall call operators which commutewith

�even and those which anticommute with

�odd. Even operators are diagonal

in the � matrix notation and odd operators are off-diagonal.

5.1 Definition. An unbounded Hilbert space operator �� is Fredholm

if it is Fredholm as a linear transformation from dom � into � � . In other words, �is Fredholm if and only if its kernel is a finite dimensional subspace of dom � andits range has finite codimension in � � . In this case the index of � is the integer

Index �� '� dim ker �� 1 dim coker �� 5.2 Lemma. The unbounded operator � � � � ; �� ; �� is Fredholm.


Proof. We want to show that, when viewed as a bounded operator from its domaininto � � , � � is a Fredholm operator in the usual sense, meaning that its kernel andcokernel are finite-dimensional. By the basic elliptic esitmate, the domain of � � isthe Sobolev space �.� �� of �� -sections of � � . Denote by

�� ; �� ; ��

compression of the operator � � ) � � � . Thus in matrix form we have

� � ) � � � �� ; �� ; �� ; �� ; ��

By the basic elliptic estimate again, the range of�

is contained within �(� �� .If we regard

�as an operator from � ; �� to �� then it follows from the

Rellich Lemma that�

is an inverse of � � , modulo compact operators. As is wellknown, an operator which is invertible modulo the compact operators is Fredholm(this is Atkinson’s Theorem), so the lemma is proved.

5.3 Remark. By elliptic regularity theory, the kernel of � � consists of smoothfunctions. Moreover the cokernel identifies with the kernel of � � , which againconsists of smooth functions.

We can therefore pose the very famous problem of computing the Fredholmindex of � � . The full solution to the problem is provided by the Atiyah-Singerindex theorem [2], and is known to involve in a very subtle way information notonly about the operator � but also about the global topology of the underlyingmanifold � . But Atiyah and Bott [1] pointed out a very simple formula for theindex involving residues of zeta functions, as follows. Fix an even, positive, order1 � operator

�such that the sum

� � � � ) �

is invertible.

5.4 Proposition. Index � � � � � Res �7� � � � Trace � � � �� .

Proof. It is not difficult to see that the residue is independent of the choice of�

,and therefore we may take

�to be the orthogonal projection on to the kernel of � .

We shall work with this choice below.Let

� � 6�� be an orthonormal eigenbasis for � acting the orthogonal complementof ker �� in � ; �� . Define

� 6 � �� 6 �� 6 � ; ��

498 N. Higson

It is easy to check that � � 6 � � 6 � 6 and that the collection of all � 6 constitutes anorthonormal basis for � acting on the orthogonal complement of ker �� in � ; �� .Computing the trace in these orthonormal basis we see that

Trace � � � �� dim ker �� & � � � � � � �'1 dim ker �� & � � � �

� � � � Index � � � � �The formula

Index � � � � � Res �7� � � � Trace � � � � ��

follows immediately from this.

The significance of this result is that, as we saw in Subsection 2.3 and Section 4,the residue of

� � Trace � � � �� can in principle be determined by a completelymechanical computation, involving ultimately integrals over the cosphere bundleof � of various polynomial combinations of the symbol of � and its partial deriva-tives. This is quite remarkable since a priori the index problem is very global innature, and is not at all obviously reducible to a definite sequence of computationsin coordinate patches.

From this point onwards a viable approach to the index theorem is to de-velop means to organize the complicated computations involved in determiningthe residue at � �

of � � Trace � � � � �� , so as to put the result of the computa-

tions into recognizable form. See for example [13]. But rather than carry that out,we shall spend the remaining parts of these notes developing a considerable elabo-ration of Proposition 5.4, in which the numerical index of an elliptic operator � isreplaced by a much more detailed invariant in cyclic cohomology.

5.2 Square Root of the Laplacian

Let � � � � be a general differential pair, in the sense of Definition 3.1. In orderto develop index theory in this context we shall now assume that � is the squareof a self-adjoint operator � . We shall assume that the underlying Hilbert space �is � � -graded; that the operator � is odd; and that the algebra

�is stable under

multiplication by the grading operator�.

We shall also assume that an algebra � � �� is specified, consisting ofoperators of differential order zero (the operators in are therefore bounded oper-ators on � ) which are even with respect to the grading. We shall assign the order�

; to � (recall that � is the order of � ), and we shall assume that if � , thenorder

� � � � � � , order � � �'1 � .In the standard example of a smooth manifold, will be the ring of smooth,

compactly supported functions on � .


5.5 Example. In the case of a differential pair which is generated from a regularspectral triple � � � � , we shall assume that the spectral triple is even, whichmeans that � is � � -graded, is comprised of even operators, and � is odd.We enlarge the algebra

�of Definition 3.23 by guaranteeing it to be closed under

multiplication by the grading operator�. Then we can let � itself be our square

root of � , and take to be the algebra of order zero operators.

5.3 Cyclic Cohomology Theory

In this subsection we shall establish some notation and terminology related to cycliccohomology theory. We shall follow Connes’ approach to cyclic cohomology,which is described for example in his book [4, Chapter 3], to which we refer thereader for more details.

Let be an associative algebra over � and for the moment let us assume that has a multiplicative unit. If�

is a complex vector space and � is a non-negativeinteger, then let us denote by � � � � � space of � � ) �� -multi-linear maps from into

�. Usually one is interested in the case where

� � � , but for our purposes itis useful to consider other cases too.

We are going to define the periodic cyclic cohomology of with coefficientsin

�, and to do so we introduce the operators

� � � � � � � � � � � � � � � and � �%� � � � � � � � � � � � � which are defined by the formulas

(5.1) � � � � � �"�"� � � � � � � � 687 � � 1 ��6 � � �

� �"�"� �6�6� � �"�"� � � � � �

) � 1 �� "�"� � � �

and

(5.2) � � � �� "�"� � � � �

� 687� � 1 �� 6 � � �< �

6 �6� � �"�"� �

6� � �

) � 687� � 1 �� 6� � � � � �

6 �6� � �"�"� �

6� � ��

5.6 Lemma. � ; � �, � ; � �

and � � ) � � � �.

500 N. Higson

As a result of the lemma, we can assemble from the spaces � � � � � the followingdouble complex, which is continued indefinitely to the left and to the top.

......

......

�"�"� � // � �

� � �

� OO

� // � ; � � �

� OO

� // � � � � �

� OO

� // � � � � �

� OO

�"�"� � // � ; ��

� OO

� // � � � � �

� OO

� // � � � � �

� OO

�"�"� � // � � � � �

� OO

� // � � � � �

� OO

�"�"� � // � � � � �

� OO

5.7 Definition. The periodic cyclic cohomology of , with coefficients in�

is thecohomology of the totalization of this complex. Thanks to the symmetry inherentin the complex, all even periodic cyclic cohomology groups are the same, as are allthe odd groups. So we shall use the notations � � � even � � � and � � � odd � � � .

A cocycle for � � � even � � � is a sequence

�� #� ; #�� "�"� �

where � ; � � ; � �� , � ; � �

�for all but finitely many - , and

� � ; � ) � � ; �� ; � �

for all - ��. A cocycle for � � � odd � � � is a sequence

�� #� � #�� "�"� � where � ; �� ; ��

� � � � , � ; ��

for all but finitely many - , and

� � ; �� ) � � ; ��

for all - ��(and in addition � � � � �

).The periodic cyclic cohomology groups of can be computed from a variety

of complexes, so we shall refer to cocycles of the above sort as � �� -cocycles,with coefficients in

�.


If we totalize the � �� -bicomplex by taking a direct product of cochain groupsalong the diagonals instead of a direct sum, then we obtain a complex with zero co-homology. We shall refer to cocycles for this complex (consisting in the even caseof sequences �� #� ; #�

� �"�"� � all of whose terms may be nonzero) as improper� � � � -cocycles. On their own, improper periodic � �� -cocycles have no coho-mological significance, but nevertheless the concept will be a convenient one forus.

If the algebra has no multiplicative unit then by a � � � � -cocycle for weshall mean a � � � � -cocycle

� � ; �� or

� � ; �� for the algebra ˜ obtained from

by adjoining a unit, which gives the value zero when the value � � is placed inany but the first argument of any of the multilinear maps � 6 (in the even case onealso requires that � � � ��

). This vanishing condition defines a subcomplex ofthe � �� -bicomplex.

5.8 Example. Let � be a smooth, closed manifold and denote by � � �� thealgebra of smooth, complex-valued functions on � . For � � �

denote by �the space of � -dimensional de Rham currents (dual to the space � of smooth � -forms). Each current � � determines a cochain � � �� for the algebra� � �� by the formula

� � �� "�"� � � � � � � �� *"*"* � � � �

One has that � � � � �and � � � � � * � � � �

where� ��

� � � � is the operator adjoint to the de Rham differential. Thisleads one to consider the following bicomplex:

......

......

�"�"� ��

�// �

� OO

�

��

// ;

� OO

; ��

// �� OO

��

// �

� OO

�"�"� �

��

// ;

�OO

; ��

// ��

OO

��

// ��

OO

�"�"� ; ��

// ��

OO

��

// ��

OO

�"�"� ��

// ��

OO

502 N. Higson

A fundamental result of Connes [3, Theorem 46] asserts that this complex computesperiodic cyclic cohomology for +� � � �� :5.9 Theorem. The inclusion � �� of the above double complex into the � � � � -bicomplex induces isomorphisms

� � � evencont � � � �� !� � � �� ; �� *"*"*

and� � � odd

cont � � � �� !� � � �� *"*"*Here � � � �

cont � � � �� denotes the periodic cyclic cohomology of � , computedfrom the bicomplex of continuous multi-linear functionals on � � �� .

It follows that an even/odd � �� -cocycle for � � �� is something very like afamily of closed currents on � of even/odd degrees. This close connection with deRham theory makes the � �� -description of cyclic cohomology particularly wellsuited to index theory problems.

5.10 Definition. A multi-linear functional � � � � � � � is said to be cyclic if

� � � �� "�"� � � � � � 1 ��

� �"�"� � � � � � for all �

� �"�"� � � in .

If � � is cyclic then it is clear from the formula (5.2) that � � � � �. As a result,

if in addition � � � � �, then we obtain a � �� -cocycle

� � �"�"� � #� � � �"�"� �by placing � � in position � and

�everywhere else. These are the cyclic cocycles of

Connes [3], using which Connes first formulated the definition of cyclic cohomol-ogy.

5.11 Lemma. Every � �� -cocycle is cohomologous to a cyclic cocycle of somedegree � .

5.4 Chern Character and Pairings with K-Theory

One of the most important cyclic cocycles is defined as follows. Let be an algebraof bounded operators on a Hilbert space � and let

�be a bounded operator on �

such that� ; � � . Assume in addition that the Hilbert space � is � � -graded, and

that consists of even operators, while�

is odd.


5.12 Theorem. Let � be an even integer and assume that for all �� "�"� � $ in

the product � � �� *"*"* �

� � $ � lies in the trace ideal. The formula

(5.3) ch�$ � �

� �"�"� � $ � � � $ ; ) �� * � Trace

� � � � � �� "�"� �

� � $ � �defines a cyclic � -cocycle whose class in periodic cyclic cohomology is indepen-dent of � (as can be seen by inserting

�at the front of the formula above for

� $ � � ).This is Connes’ (even) cyclic Chern character of

�. The constant in front of

the trace is chosen in such a way that the periodic cyclic cohomology class of the� � � � -cocycle determined by ch

�$ is independent of � . To see that this is so, onecan define

� $ � � � �� "�"� � $ � � � �

� $ ; ) �

�8� ) � Trace� � �

� � � � � � � �� ; � �"�"� �

� � $ � � � � �and then compute that � � $ � � ��1 ch

�$ � ; while � � $ � � � ch�$ .7

Each even � �� -cocycle determines a homomorphism from the algebraic�

-theory group

� � � � to � , depending only on the periodic cyclic cohomology classof the cocycle. If � is an idempotent in then we can form the element � � �

� � � � . Under the pairing between cyclic theory and�

-theory the class � � � ismapped by an even � � � � -cocycle � � �� #� ; �"�"� � to the scalar

(5.4) � � � � � � � � � � � � ) ��7 � � 1 ��

� - �- � ; � � ��1� � � �"�"� � � �

Compare [12]. In the case of the even cyclic Chern character defined in the lastsection, the pairing is

(5.5) ch� � � � � � � Index � � � � � � � � � � � � �

where � � and � � are the degree zero and degree one parts of the � � -gradedHilbert space � .

This connection with index theory makes it a very interesting problem to com-pute the cyclic Chern character in various instances, and it is this problem to whichwe want to turn our attention. For example, in the case of an ordinary elliptic oper-ator on a closed manifold, where the cyclic cohomology of +� � � �� identifies

7This formula actually proves Theorem 5.12 since � �� and the image of the differential � iscomprised entirely of cyclic multi-linear functionals.

504 N. Higson

with the de Rham homology of � , the identification of the class of the Chern char-acter with a specific homology class on � is equivalent to the Atiyah-Singer IndexTheorem.

Our goal will be not so much to compute the Chern character in this (or anyother) specific instance. Instead we aim to show that in general the Chern characteris cohomologous to a cocycle constructed entirely out of residues of zeta functions.As we saw in Section 4, in at least the classical case this leads to complicated butexplicit formulas from which Fredholm indexes may in principle be computed. Theproblem of actually organizing and simplifying these formulas in various cases isboth interesting and important, but we shall not consider it in these notes.

5.5 Zeta Functions

Let us continue to assume as given an differential pair � � � � of generalized differ-ential operators, along with a square-root decomposition ��2� ; .

We are now going to define certain zeta-type functions associated with the al-gebra. To simplify matters we shall now assume that the operator � is invertible.This assumption will remain in force until Section 8, where we shall consider thegeneral case.

5.13 Definition. The differential pair � � � � has finite analytic dimension if thereis some

� � �with the property that if " � has order � or less, then for every

� with real part greater than�� the operator " � � � extends by continuity to

a trace-class operator on � (here � is the order of � , as described in Section 3.1).

Assume that � � � � has finite analytic dimension�

. If " � �� and iforder � " � , � then the complex function Trace � " � � � � is holomorphic in the righthalf-plane Re � � �

�� .

5.14 Definition. An differential pair � � � � which has finite analytic dimension hasthe meromorphic continuation property if for every " � �� the analytic functionTrace � " � �� , defined initially on a half-plane in � , extends to a meromorphicfunction on the full complex plane.

Actually, for what follows it would be sufficient to assume that Trace � " � � ! �has an analytic continuation to � with only isolated singularities, which could per-haps be essential singularities.


5.15 Definition. Let � � � � be a differential pair which has finite analytic dimen-sion. Define, for Re � � � �

and " � �"�"� " � � ,8 the quantity

(5.6)� " � " � �"�"� " � � � �

� 1 �� Trace

� �� " � � � 1 � � � � " � � � 1 � � � � *"*"* " � � � 1 � � � � � � �

(the factors in the integral alternate between the " 6 and copies of � � 1 � � � � ).The contour integral is evaluated down a vertical line in � which separates

�and

Spectrum �� .5.16 Remark. If order � " � � ) *"*"* ) order � " � ��, � and if the integrand in equation(5.6) is viewed as a bounded operator from � ! � � to � ! , then the integral convergesabsolutely in the operator norm whenever Re � � ) � � �

. In particular, if Re � � � �then the integral converges to a well defined operator on � � .

Of course, apart from the insertion of the grading operator�, this is precisely

the sort of integral we encountered in our discussion of meromorphic continuationin Section 2. In our former notation,

� " � " � �"�"� " � � � � � 1 �� " � " � �"�"� " � � �Using the arguments we developed in Section 2 we obtain the following results:

5.17 Proposition. Let � � � � be a differential pair and let " � �"�"� " � � . As-sume that

order � " � �') *"*"* ) order � " � ��, � �If � � � � has finite analytic dimension

�, and if Re � � ) � � �� ) � � , then the

integral in Equation (5.6) extends by continuity to a trace-class operator on � , andthe quantity

� " � �"�"� " � � � defined by Equation (5.6) is a holomorphic functionof in this half-plane. If in addition the algebra � � � � has the meromorphiccontinuation property then the quantity

� " � �"�"� " � � � extends to a meromorphicfunction on � .

5.18 Definition. Let - � � - � �"�"� - � � be a multi-index with non-negative integerentries. Define a constant � � - � by the formula

� � - � � � - � ) *"*"* ) - � ) � �-�� *"*"* - � � -�� ) �� *"*"* � -�� ) *"*"* ) - � ) � � �8Occasionally we shall take one or more of the �

to lie within a larger algebra, for example the

algebra generated by � , � and � .

506 N. Higson

5.19 Proposition. Let � � � � be a differential pair with the meromorphic continu-ation property and let " � �"�"� " � � . There is an asymptotic expansion

� " � �"�"� " � � � � �� 1 ��

�� ) ��)2& - & � �

� & - & ) � � � � - �� Trace

� � " � " � � �� *"*"* " � � � � � � ��

where the symbol � means that, given any right half-plane in � , any sufficientlylarge finite partial sum of the right hand side agrees with the left hand side moduloa function of which is holomorphic in that half-plane.

5.6 Formulation of the Local Index Theorem

The following result is the local index formula of Connes and Moscovici:

5.20 Theorem. Let � � � � be a differential pair with the meromorphic continua-tion property and let � be a square root of � . The formula

� � � �� "�"� � � � �

��

� 1 �� - �� & - & ) � �

� Res !�7 �� ) � )2& - & � Tr

� � � � � � � � � � � � � *"*"* � � � � � � � � � � � � � � � � � � ! � �defines an periodic � �� -cocycle

� � ; �� for which is cohomologous to the cyclic

Chern character of the operator� �2� & � & � � .

5.21 Remark. If & - & ) � ��

then the � � - � -contribution to the above sum ofresidues is actually zero. Hence for every � the sum is in fact finite (and the sum is�

when � ��

).

5.22 Remark. If all the poles of the zeta functions Trace � " �� are simple thenthe above cocycle can be rewritten as

� � � �� "�"� � � ��

�� 4 � Res !�7 � Tr

� � � � � � � � � � � � � *"*"* � � � � � � �� ! � where

� � � � � 1 �� - � & - & ) � ; �� -�� ) �� -�� ) - ; ) � *"*"* � -�� ) *"*"* ) - � ) � � �


(Note: the constant � � � � � � � is not well defined in our formula since�

is a poleof the

-function. To cope with this problem we must treat the � � �

, - � �term

separately and replace � � � Res !�7 � � Tr � � �� !�� with Res ! 7� � �� Tr � � �

�� !�� .)

508 N. Higson

6 The Residue Cocycle

6.1 Improper Cocycle

In this section we shall assume as given a differential pair � � � � with the mero-morphic continuation property, a square root � of � , and an algebra � �

, as inthe previous section.

We are going to define a periodic cyclic cocycle �� ; �"�"� � for thealgebra . The cocycle will be improper—all the � � will be nonzero. Moreoverthe cocycle will assume values in the field of meromorphic functions on � . But inthe next section we shall convert it into a proper cocycle with values in � itself.

We are going to assemble � from the quantities� " � �"�"� " � � � defined in Sub-

section 5.5.9 We begin by establishing some ‘functional equations’ for the quan-tities

� *"*"* � � . In order to keep the formulas reasonably compact, if " � thenwe shall write � 1 �� to denote either ) � or 1 � , according as " is an even or oddoperator on the � � -graded Hilbert space � .

6.1 Lemma. The meromorphic functions� " � �"�"� " � � � satisfy the following func-

tional equations:

� " � �"�"� " � � � " � � � � � � � 687 �� " � �"�"� " 6 � � �< " 6 �"�"� " � � �(6.2)

� " � �"�"� " � � � " � � � � � 1 �� " � " � �"�"� " � � � � �(6.3)

Proof. The first identity follows from the fact that

�

� �� " � � � 1 � � � � *"*"* " � � � 1 � � � � �� 1 �� " � � � 1 � � � � *"*"* " � � � 1 � � � �

1 � 687� � � � " � � � 1 � � � � *"*"* " 6 � � 1 � � � ; " 6 � � *"*"* " � � � 1 � � � �9In doing so we shall follow quite closely the construction of the so-called JLO cocycle in entire

cyclic cohomology (see [21] and [12]), which is assembled from the quantities

(6.1) � � �� JLO� Trace

� ��

(the integral is over the standard � -simplex). The computations which follow in this section are moreor less direct copies of computations already carried out for the JLO cocycle in [21] and [12].


and the fact that the integral of the derivative is zero. As for the second identity,if � � �

then the integrand in Equation (5.6) is a trace-class operator, and Equa-tion (6.3) is an immediate consequence of the trace-property. In general we canrepeatedly apply Equation (6.2) to reduce to the case where � � �

.

6.2 Lemma.

(6.4)� " � �"�"� � � ; " 6 � �"�"� " � � � �� " � �"�"� " 6 � � " 6 �"�"� " � � � 1 � " � �"�"� " 6 " 6 � � �"�"� " � � �

Proof. This follows from the identity

" 6 � � � � 1 � � � � � � ; " 6 � � � 1 � � � � " 6 � �� " 6 � � � � 1 � � � � " 6 " 6 � � 1 " 6 � � " 6 � � 1 � � � � " 6 � � �6.3 Lemma.

(6.5)� 687 � � 1 ��

� �� " � �"�"� � � " 6 � �"�"� " � � � � �

Proof. The identity is equivalent to the formula

Trace

� � � � � � � � " � � � 1 � � � � *"*"* " � � � 1 � � � � � �� which holds since the supertrace of any (graded) commutator is zero.

With these preliminaries out of the way we can obtain very quickly our im-proper � �� -cocycle.

6.4 Definition. If � is a non-negative and even integer then define a � � ) �� -multi-linear functional on with values in the meromorphic functions on � by the for-mula

� � � �� "�"� � � � �

�� "�"� � � � � � � ! � � �

6.5 Theorem. The even � � � � -cochain � � � � � � ; �� *"*"* � is an improper � �� -

cocycle with coefficients in the space of meromorphic functions on � .

510 N. Higson

Proof. First of all, it follows from the definition of � and Lemma 6.1 that

� � � � ; � �� "�"� � � � � � �

� � � 6 7 � � 1 ��6 � �< � � �

6 � �"�"� � � �6� � � � ! � � � ��

� � ��

6 7 ��

� � �"�"� � � �6� � � �< � � �

6 � �"�"� � � � � � � � � ! � � � ��

� � � � � � � �"�"� � � � � � � � � ! � � � �Next, it follows from the definition of � and the Leibniz rule � � �

6�6� � � � �

6 � � �6� � � )

� � �6 � �6� � that�� "�"� � � � � � � � �

�� ; � �"�"� � � � � � � � � ! � � �1 � �

� � � � � � ; � �"�"� � � � � � � � � ! � � � �1 � �� ; � � �

� � �"�"� � � � � � � � � ! � � �1 � �� ; � � �

� � �"�"� � � � � � � � � ! � � � �) *"*"*) � �

�� "�"� � � � � � � � � � � ! � � �1 � � � � � �

� � � � � � �"�"� � � � � � � � � ! � � � � �Applying Lemma 6.2 we get

�� "�"� � � � � � � � ��

687 � � 1 ��6� � � �

� � � � � � �"�"� � � ; �6 � �"�"� � � � � � � � � ! � � �

Setting " � � � � and " 6 � � � �6 � for � � � , and applying Lemma 6.3 we get

� � � � ; � �� "�"� � � � � �') �� "�"� � � � � �

� � ��

687� � 1 �� " � �"�"� � � " 6 � �"�"� " � � � � ! � � � � � �

6.2 Residue Cocycle

By taking residues at � � �we map the space of meromorphic functions on �

to the scalar field � , and we obtain from any � �� -cocycle with coefficients in


the space of meromorphic functions a � �� -cocycle with coefficients in � . Thisoperation transforms the improper cocycle � that we constructed in the last sectioninto a proper cocycle Res !�7 � � . Indeed, it follows from Proposition 5.17 that if �is greater than the analytic dimension

�of � � � � then the function

� � � �� "�"� � � � ! �

�� "�"� � � � � � � ! � � �

is holomorphic at � � �.

The following proposition identifies the proper � �� -cocycle Res !�7 � � withthe residue cocycle studied by Connes and Moscovici. The proof follows immedi-ately from our computations in Section 2, as summarized in Subsection 5.5.

6.6 Theorem. For all � ��and all �

� �"�"� � � ,

Res ! 7� � � � �� "�"� � � � �

��

� 1 �� - �� & - & ) � �

� Res !�7 �� ) � )2& - & � Tr

� � � � � � � � � � � � � *"*"* � � � � � � � � � � � � � � � � � � ! � � where

� � - � � � - � ) *"*"* ) - � ) � �-�� *"*"* - � � -�� ) �� *"*"* � -�� ) *"*"* ) - � ) � � �

6.3 Complex Powers in a Differential Algebra

In this subsection we shall try to sketch out a more conceptual view of the impropercocycle which was constructed in Section 6.1. This involves Quillen’s cochainpicture of cyclic cohomology [24], and in fact it was Quillen’s account of theJLO cocycle from this perspective which first led to the formula for the quantity� " � �"�"� " � � � given in Definition 5.15. Since our purpose is only to view the co-cycle � in a more conceptual way we shall not carefully keep track of analyticdetails.

As we did when we looked at cyclic cohomology in Subsection 5.3, let us fixan algebra . But let us now also fix a second algebra � . For � � �

denote byHom $ � � � the vector space of � -linear maps from to � . By a

�-linear map

from to � we shall mean a linear map from � to � , or in other words just anelement of � . Let Hom

� � � � � be the direct product

Hom� � � � � � ��

$ 7� Hom$ � � � �

512 N. Higson

Thus an element � of Hom� � � � � is a sequence of multi-linear maps from to

� . We shall denote by � � � � �"�"� � $ � the value of the � -th component of � on the� -tuple � � � �"�"� � $ � .The vector space Hom� � � � � is � � -graded in the following way: an element� is even (resp. odd) if � � � � �"�"� � $ � � �

for all odd � (resp. for all even � ). Weshall denote by deg � �� the grading-degree of � . (The letter ‘ � ’ standsfor ‘multi-linear’; a second grading-degree will be introduced below.)

6.7 Lemma. If � �� Hom� � � � � , then define

�� "�"� � $ � � � �� 7 $ � � �

� �"�"� � � � � � � � � � �"�"� � $ �and

�

� � � � � �"�"� � $ � � � �$ 3 7 � � 1 ��

3� � � � � � �"�"� �

3�3� � �"�"� � $ � � � �

The vector space Hom� � � � � , so equipped with a multiplication and differential,

is a � � -graded differential algebra.

Let us now suppose that the algebra � is � � -graded. If � Hom� � � � � then

let us write deg � �� if � � � � �"�"� � $ � belongs to the degree-zero part of � for

every � and every � -tuple � � � �"�"� � $ � . Similarly, if � Hom� � � � � then let us

write deg � �� if � � � � �"�"� � $ � belongs to the degree-one part of � for every� and every � -tuple � � � �"�"� � $ � . This is a new � � -grading on the vector spaceHom

� � � � � . The formula

deg �� deg � �� ') deg � �� defines a third � � -grading—the one we are really interested in. Using this last� � -grading, we have the following result:

6.8 Lemma. If � �� Hom� � � � � , then define

�� /� � 1 �� deg � �� deg � � � � ��

and� �� 1 �� deg � ��

These new operations once again provide Hom� � � � � with the structure of a

� � -graded differential algebra (for the total � � -grading deg �� deg � �� %)deg � �� ).


We shall now specialize to the situation in which and � � � ; are as inprevious sections, and � is the algebra of all operators on the � � -graded vectorspace � � � � .

Denote by � the inclusion of into � . This is of course a � -linear map from to � , and we can therefore think of � as an element of Hom

� � � � � (all of whose� -linear components are zero, except for � � � ). In addition, let us think of �as a

�-linear map from to � , and therefore as an element of Hom

� � � � � too.Combining � and � let us define the ‘superconnection form’

� �2� 1 � Hom� � � � �

This has odd � � -grading degree (that is, deg �� ). Let�

be its ‘curvature’:

� � � ��) � ; which has even � � -grading degree. Using the formulas in Lemma 6.8 the element

�may be calculated, as follows:

6.9 Lemma. One has� � � 1 � Hom

� � � � � where � � � � is the � -linear map defined by the formula

� � � � � � � �� In all of the above we are following Quillen, who then proceeds to make the fol-

lowing definition, which is motivated by the well-known Banach algebra formula

� � � � � �$ 7��

� � � � �

� � � � � � � � *"*"* � � � � � � �� 6.10 Definition. Denote by � �

� Hom

� � � � � the element

� �� $ 7�

��

�� "�"� � � � � � � ��

The � -th term in the sum is an � -linear map from to � , and the series shouldbe regarded as defining an element of Hom

� � � � � whose � -linear component isthis term. As Quillen observes, in [24, Section 8] the exponential � �

�defined in

this way has the following two crucial properties:

6.11 Lemma (Bianchi Identity).� � � �

��') � � �

� � � � �.

514 N. Higson

6.12 Lemma (Differential Equation). Suppose that�

is a derivation of Hom� � � � �

into a bimodule. Then � � � �� 21 � � � � � �

� modulo (limits of) commutators.

Both lemmas follow from the ‘Duhamel formula’

� � � ��

� �� which is familiar from semigroup theory and which may be verified for the notionof exponential now being considered. (Once more, we remind the reader that weare disregarding analytic details.)

Suppose we now introduce the ‘supertrace’ Trace� � " � � Trace � � " � (which is

of course defined only on a subalgebra of � ). Quillen reinterprets the Bianchi Iden-tity and the Differential Equation above as coboundary computations in a complexwhich computes periodic cyclic cohomology (using improper cocycles, in our ter-minology here). As a result he is able to recover the following basic fact about theJLO cocycle — namely that it really is a cocycle:

6.13 Theorem (Quillen). The formula

� ; $ � �� "�"� � ; $ � ��

� � Trace� � � � � � �

��

� � � � � � ��

� � � ; � �"�"� � � � $ � � ��

defines a � �� -cocycle.

With this in mind, let us consider other functions of the curvature operator�

,beginning with resolvents.

6.14 Lemma. If � � Spectrum �� then the element � � 1 � � Hom� � � � � is

invertible.

Proof. Since � � 1 � � � � � 1 � �') � we can write

� � 1 � � � � � � � 1 � � � � 1 � � 1 � � � � � � � 1 � � � �) � � 1 � � � � � � � 1 � � � � � � � 1 � � � � 1 *"*"*

This is a series whose � th term is an � -linear map from to � , and so the sumhas an obvious meaning in Hom

� � � � � . One can then check that the sum defines� � 1 � � � � , as required.


6.15 Definition. For any � with positive real part define� �� Hom

� � � � �by the formula

� �� 1 � � � � � ��

in which the integral is a contour integral along a downward vertical line in �separating

�from Spectrum �� .

The assumption that Re � � � �guarantees convergence of the integral (in each

component within Hom� � � � � the integral converges in the pointwise norm topol-

ogy of � -linear maps from to the algebra of bounded operators on � ; the limitis also an operator from � � to � � , as required). The complex powers

� � � sodefined satisfy the following key identities:

6.16 Lemma (Bianchi Identity).� � � �� ') � � � � � � � �

.

6.17 Lemma (Differential Equation). If�

is a derivation of Hom� � � � � into a

bimodule, then � � � �� 21 � � � � � �� modulo (limits of) commutators.

These follow from the derivation formula

� � � � � � � � � �� 1 � � � � � � � � � � 1 � � � � � � �

In order to simplify the Differential Equation it is convenient to introduce theGamma function, using which we can write

� � � � � � � � �21 � � � � � ) �� (modulo limits of commutators, as before). Except for the appearance of ) �in place of in the right hand side of the equation, this is exactly the same asthe differential equation for � �

�. Meanwhile even after introducing the Gamma

function we still have available the Bianchi identity:

�� ) � � � � ��

The degree � component of � � � �� is the multi-linear function

� � � �"�"� � $ � �� 1 �� $ � � �

�� 1 � � � � � � � � � �"�"� � � � $ � � � 1 � � � �

� �� Quillen’s approach to JLO therefore suggests (and in fact upon closer inspectionproves) the following result:

516 N. Higson

6.18 Theorem. If we define

� !� � �� "�"� � � � �� 1 �� 1 � ; � � Trace

� �� ! � �

�� 1 � � � � � � � � � �"�"�

� � � � � � � 1 � � � ��

then �� !� ) � � !� � ; � �.

This is of course precisely the conclusion that we reached in Section 6.1.


7 Comparison with the Chern Character

Our goal in this section is to identify the cohomology class of the residue cocycleRes !�7 � � with the cohomology class of the Chern character cocycle ch

�$ associatedto the operator

� � � & � & � � (see Section 5.3). Here � is any even integer greaterthan or equal to the analytic dimension

�. It follows from the definition of analytic

dimension and some simple manipulations that

� � �� *"*"* � � � $ �

� � �� for such � , so that the Chern character cocycle is well-defined.

We shall reach the goal in two steps. First we shall identify the cohomologyclass of Res ! 7� � with the class of a certain specific cyclic cocycle, which involvesno residues. Secondly we shall show that this cyclic cocycle is cohomologous tothe Chern character ch

�$ .The following result summarizes step one.

7.1 Theorem. Fix an even integer � strictly greater than� 1 � . The multi-linear

functional

� �� "�"� � $ � ��

� $ 687� � 1 ��

6� � � � � �

� � �"�"� � � �6 � � � � �

6� � � �"�"� � � � $ � � � � � �

is a cyclic � -cocycle which, when considered as a � �� -cocycle, is cohomologousto the residue cocycle Res !�7 � � .

7.2 Remark. It follows from Proposition 5.17 that the quantities� � � �

� � �"�"� � � �6 � � � � �

6� � � �"�"� � � � $ � � �

which appear in the theorem are holomorphic in the half-plane Re � � � 1 $;)�� 1 �8� ) �� . Therefore it makes sense to evaluate them at ��21 $ ; , as we have

done. Appearances might suggest otherwise, because the term � � which appears

in the definition of� �"�"� � � has poles at the non-positive integers (and in particular at

� 1 $;

if � is even). However these poles are canceled by zeroes of the contourintegral in the given half-plane.

Theorem 7.1 and its proof have a simple conceptual explanation, which we shallgive in a little while (after Lemma 7.7). However a certain amount of elementary,if laborious, computation is also involved in the proof, and we shall get to work onthis first. For this purpose it is useful to introduce the following notation.

518 N. Higson

7.3 Definition. If " � �"�"� " � are operators in the algebra generated by�

, thendefine

� � " � �"�"� " � � � � � ��7� � 1 ��

� � �� " � �"�"� " � � " �� "�"� " � � � which is a meromorphic function of � � .

The new notation allows us to write a compact formula for the cyclic cocycleappearing in Theorem 7.1:

� �� "�"� � $ � ��

;� � � � �

� � �"�"� � � � $ � � � � � � �We shall now list some properties of the quantities

� � *"*"* � � � which are analo-gous to the properties of the quantities

� *"*"* � � that we verified in Section 6.1. Thefollowing lemma may be proved using the formulas in Lemmas 6.1 and 6.2.

7.4 Lemma. The quantity� � " � �"�"� " � � � � satisfies the following identities:

� � " � �"�"� " � � � � � � � " � " � �"�"� " � � � � � �(7.1)

� 6 7 �� " � �"�"� " 6 �< " 6 � � �"�"� " � � � � � �� " � �"�"� " � � � �(7.2)

In addition,

(7.3)� � " � �"�"� " 6 � � " 6 �"�"� " � � � � 1 � � " � �"�"� " 6 " 6 � � �"�"� " � � � �� " � �"�"� � � ; " 6 � �"�"� " � � � � 1 � 1 �� " � �"�"� � � " 6 � �"�"� " � � � �

(In both instances within this last formula the commutators are graded commuta-tors.)

We shall also need a version of Lemma 6.3, as follows.

7.5 Lemma.

(7.4)� 687� � 1 ��

� � �� " � �"�"� � � " 6 � �"�"� " � � � ��

�7 �� " � �"�"� " � � � � ; " � �"�"� " � � � �


Proof. This follows from Lemma 6.3. Note that � � � � � � ; , which helps ex-plain the factor of

in the formula.

The formula in Lemma 7.5 can be simplified by means of the following com-putation:

7.6 Lemma.

� 687�� " � �"�"� " 6 � ; " 6 � � �"�"� " � � � � � ) � � � " � �"�"� " � � �

Proof. If we substitute into the integral which defines� " � �"�"� � ; �"�"� " � � � the

formula � ; � � 1 � � 1 � �we obtain the (supertrace of the) terms

� 1 �� " � � � 1 � � � � *"*"* � � � 1 � � � � *"*"* " � � � 1 � � � � � �

1 � 1 �� " � � � 1 � � � � *"*"* " � � � 1 � � � � � �

Using the functional equation � � � � 1 �� 1 �� we therefore obtain the

quantity

� 1 �� " � �"�"� " 6 �< " 6 � � �"�"� " � � � � � ) � " � �"�"� " � � �(the change in the sign preceding the second bracket comes from the fact that thebracket contains one fewer term, and the fact that � 1 �� 21 � 1 �� ). Adding upthe terms for each � , and using Lemma 6.1 we therefore obtain

� 6 7 �� " � �"�"� " 6 � ; " 6 � � �"�"� " � � � � � 1 �� " � �"�"� " � � � ) � � ) �� " � �"�"� " � � �

� � ) � � � " � �"�"� " � � �as required.

Putting together the last two lemmas we obtain the formula

(7.5)� 687� � 1 ��

� � �� " � �"�"� � � " 6 � �"�"� " � � � � � � ) � � � " � �"�"� " � � � �With this in hand we can proceed to the following computation:

520 N. Higson

7.7 Lemma. Define multi-linear functionals � � on , with values in the space ofmeromorphic functions on � , by the formulas

� � � �� "�"� � � � �

� �� "�"� � � � � � � � ! � � �

��

Then��

� �"�"� � � � ��

� � �"�"� � � � � � � � ! � � � �and in addition�� "�"� � � �:) �� "�"� � � � � � � � � � � �"�"� � � �for all � � and all �

� �"�"� � � .

Proof. The formula for �� "�"� � � � is a simple consequence of Lemma 7.4.

The computation of �� "�"� � � � is a little more cumbersome, although stillelementary. The reader who wants to see it carried out (rather than do it himself) isreferred to [19].

7.8 Remark. The statement of Lemma 7.7 can be explained as follows. If wereplace � by

� � and � by�; � in the definitions of

� *"*"* � � and � � , so as to obtain anew improper � �� -cocycle � � � � � � � � �

; �"�"� � , then it is easy to check from the

definitions that� �

� � �� "�"� � � � �

� � ;! � � � �

� �"�"� � � � �Now, we expect that as

�varies the cohomology class of the cocycle � �

should notchange. And indeed, by borrowing known formulas from the theory of the JLOcocycle (see for example [12], or [14, Section 10.2], or Section 7.1 below) we canconstruct a � �� -cochain � such that

�� ) �� ) �

�� This is the same � as that which appears in the lemma.

The proof of Theorem 7.1 is now very straightforward:

Proof of Theorem 7.1. According to Lemma 7.7 the � �� -cochain

�Res ! 7� � �

�� Res ! 7� � �

�� "�"� Res !�7 � � �

�� $ � � � � � �"�"� �

cobounds the difference of Res !�7 � � and the cyclic � -cocycle Res ! 7� � �;! �� $ � � � .

Since

Res ! 7� � �� $ � � � � �

� �"�"� � $ � � � � � � � � � � �"�"� � � � $ � � � � � �the theorem is proved.


We turn now to the second step. We are going to alter � by means of thefollowing homotopy:

� � �2� & � & � �

� � , � , ��

(the same strategy is employed by Connes and Moscovici in [9]). We shall similarlyreplace � with � � � � ;� , and we shall use � � in place of � in the definitions of� *"*"* � � and of

� � *"*"* � � � .To simplify the notation we shall drop the subscript

�in the following computa-

tion and denote by ˙� �21�� * log & � & the derivative of the operator � � with respectto�.

7.9 Lemma. Define a multi-linear functional on , with values in the analyticfunctions on the half-plane Re � �') � � � � �

;, by the formula

� �$ � �� "�"� � $ � �

� ��

˙� � � � � � �"�"� � � � $ � � � � �Then � � �$ is a cyclic �8� 1 �� -cochain and

� � � �$ � �� "�"� � $ �

��

�� "�"� � � � $ � � � � ) � ) � � $ 687� � ˙� � � � 6 � �"�"� � � 6 � � � � � �

7.10 Remark. Observe that the operator log & � & has analytic order�

or less, forevery

� � �. As a result, the proof of Proposition 5.17 shows that the quantity is a

holomorphic function of in the half-plane Re � �:) � � � � �;

. But we shall not beconcerned with any possible meromorphic continuation to � .

Proof. See [19].

We can now complete the second step, and with it the proof of the Connes-Moscovici Residue Index Theorem:

7.11 Theorem (Connes and Moscovici). The residue cocycle Res !�7 � � is coho-mologous, as a � �� -cocycle, to the Chern character cocycle of Connes.

Proof. Thanks to Theorem 7.1 it suffices to show that the cyclic cocycle

(7.6)� � � � � � � � �"�"� � � � $ � � � � � �

522 N. Higson

is cohomologous to the Chern character. To do this we use the homotopy � � above.Thanks to Lemma 7.9 the coboundary of the cyclic cochain

� ��

�$ � �� "�"� � $ � � �

��

is the difference of the cocycles (7.6) associated to � � � � and � � � �. For � �

we have � ; � � � � � � and so

� � � � ��! � � � �"�"� � ��! � $ � � � �� $

6 7 � � 1 ��6� � � 1 �� $ � � � � � �

Trace

� ��

� � *"*"* � � �6 � � *"*"* � � � $ � � � 1�� $ � ; �

� � �Since

�anticommutes with each operator � � �

6 � this simplifies to

� $ 687 �

� 1 �� $ � � � � � Trace

� ��

� � *"*"* � � � $ � � � 1 � � � � $ � ; ��

The terms in the sum are now all the same, and after applying Cauchy’s formulawe get

� ) � � 1 �� $ � � � � * Trace� � � � � � � � *"*"* � � � $ � � *

� 1 � ) ��

Using the functional equation for the

-function this reduces to � ) � ) �� * � Trace� � � � � � � � *"*"* � � � $ � �

and evaluating at ��21 $;

we obtain the Chern character of Connes.

7.1 Homotopy Invariance and Index Formula

By combining the Theorem 7.11 with the formula (5.5) for the pairing betweencyclic theory and

�-theory we obtain the index formula

Index��!� � � � � � � � � � � � �

Res !�7 � � � � � � � �


for a projection � in (there is a similar equation for projections in matrix algebrasover ). In this section we shall very briefly indicate a shorter route to this formula.

The starting point is the following transgression formula for the basic impropercocycle � that we have been studying: informally, if � � �+� ;� is a smooth familyof operators satisfying our basic hypotheses (for a fixed algebra ) then

��

� � � ) ��

� � � )�

��

� � � where(7.7)

� � � �� "�"� � � � �

� 687� � 1 ��6� � � �

� �"�"� � � �6 � ˙� � � �

6� � � �"�"� � � � � � � ! � � �

��

It is a little tedious to precisely formulate and prove this result in any generality(one problem is to understand the analytic continuation property for algebras whichcontain the operators �

�

�� ). But fortunately we are only interested in a very easy

special case, where

� � �2� ) � " and where " is a differential order-zero operator in the algebra

�. The formula (7.7)

can be proved without any real difficulty in this case by following the methods usedin the proofs of Lemmas 7.7 and 7.9.

With the transgression formula (7.7) in hand the proof of the index formula canbe finished rather quickly, using a trick due to Connes. Given a projection � , define

�� !� ��) � � � � � �2� ) " where " is of course an order zero operator in

�, and let � � � � ) � " , as above.

Thanks to the transgression formula, it suffices to show that the residue cocycleRes !�7 � � � of � � , paired with the

�-theory class of the projection � , gives the

Fredholm index of �!� � (considered as an operator from � � � to � � � ). Now, byEquation (5.4),

Res ! 7� � � � � � � � � Res ! 7� � �� ) �� 1 ��

� - �- Res ! 7� ��; � � ��1� � �"�"� � � �

524 N. Higson

But the terms in the series are all zero since they all involve the commutator of �with � � , which is zero. Hence

Res ! 7� �� Res ! 7� �� Res ! 7� � �� Trace � � � � � ��

!� �

� Index��!� � � � � � � � � � �

as required (the last step is the index computation made by Atiyah and Bott that wementioned in the Subsection 5.1).


8 The General Case

Up to now we have been assuming that the self-adjoint operator � is invertible (inthe sense of Hilbert space operator theory, meaning that � is a bijection from itsdomain to the Hilbert space � ). We shall now remove this hypothesis.

To do so we shall begin with an operator � which is not necessarily invertible(with � ; � � ). We shall assume that all our assumptions from Subsection 5.2concerning the differential pair � � � � , the the square root � , and the algebra hold. Fix a bounded self-adjoint operator

�with the following properties:

(i)�

commutes with � .

(ii)�

has analytic order 1 � (in other words,� * � � � � ).

(iii) The operator � ) � ; is invertible.

Having done so, let us construct the operator

� � ��

� 1 � �acting on the Hilbert space � � � opp, where � opp is the � � -graded Hilbert space� but with the grading reversed. It is invertible.

8.1 Example. If � is a Fredholm operator then we can choose for�

the projectiononto the kernel of � .

Let � � � � � � � ; and denote by� � the smallest algebra of operators on

� � � opp which contains the � matrices over

�and which is closed under

multiplication by operators of analytic order 1 � .The conditions set forth in Subsection 5.2 for the pair � � � � � � , the square root� � and the algebra , which we embed into

� � as matrices� � �� .

8.2 Lemma. Assume that the operators� � and

�

; both have the properties (i)-(iii) listed above. Then

� � � � � � � . Moreover the algebra has finite analyticdimension

�and has the analytic continuation property with respect to � � � if and

only if it has the same with respect to � ��. If these properties do hold then the

quantities� " � �"�"� " � � � associated to � � � and � �

�differ by a function which is

analytic in the half-plane Re � � ��1 � .

526 N. Higson

Proof. It is clear that� � � � � �

�. To investigate the analytic continuation property

it suffices to consider the case where� � is a fixed function of � , in which case

� �and

�

; commute. Let us write

" � � � � � � �� " � � 1 � � � � � �

for Re � � � �. Observe now that

� � 1 � � � � � � 1 � � 1 � �� ! � � � 1 � � � � � ; 1�� 1 � � � � � � ) *"*"*

where � � � � � 1 � ��

(this is an asymptotic expansion in the sense describedprior to the proof of Proposition 5.17). Integrating and taking traces we see that

(8.1) Trace� " � � �� 1 Trace

� " � �� 1 �� 1 - � Trace

� " � � � � � �� which shows that the difference Trace

� " � �� 1 Trace� " � �� has an analytic con-

tinuation to an entire function. Therefore � � � has the analytic continuation prop-erty if and only if � �

�does (and moreover the analytic dimensions are equal).

The remaining part of the lemma follows from the asymptotic formula

� " � �"�"� " � � � � �� 1 ��

�� ) ��)2& - & � �

� & - & ) � � � � - �� Trace

� � " � " � � �� *"*"* " � � ��

that we proved earlier.

8.3 Definition. The residue cocycle associated to the possibly non-invertible oper-ator � is the residue cocycle Res ! 7� � associated to the invertible operator � � , asabove.

Lemma 8.2 shows that if � � �then the residue cocycle given by Definition 8.3

is independent of the choice of the operator�

. In fact this is true when �.� �too. Indeed Equation (8.1) shows that not only is the difference Trace � � �

�� !� � �'1

Trace � � �� !�� analytic at � � �

, but it vanishes there too. Therefore

Res ! 7� � � �� :1 Res !�7 � � � ��

��

� Res !�7 � �� Trace � � �� !� � �:1 Trace � � �

�� !��


8.4 Example. If � happens to be invertible already then we obtain the same residuecocycle as before.

8.5 Example. In the case where � is Fredholm, the residue cocycle is given by thesame formula that we saw in Theorem 5.20:

Res ! 7� � � � �� "�"� � � �� 4� Res !�7 � Tr

� � � � � � � � � � � � � *"*"* � � � � � � � � � � � � � � � � � � ! � �The complex powers � � � are defined to be zero on the kernel of � (which is alsothe kernel of � ). When � � �

the residue cocycle is

Res ! 7� � �� Trace � � �� !�� ) Trace � � �

��

where the complex power � �!

is defined as above and � is the orthogonal projec-tion onto the kernel of � .

Now Connes’ Chern character cocycle is defined for a not necessarily invertibleoperator � by forming first � � , then

� � �2� � & � � & � � , then ch��$ . See Appendix 2,

and also Section 5, of [3, Part I]. The following result therefore follows immediatelyfrom our calculations in the invertible case.

8.6 Theorem. For any operator � , invertible or not, the class in periodic cycliccohomology of the residue cocycle Res !�7 � � is equal to the class of the Cherncharacter cocycle of Connes.

8.7 Remark. There is another way that the index theorem can be generalized —by considering the ‘odd-dimensional’ case instead of the even-dimensional one thatwe have been examining. This involves the construction of an odd cyclic cocyclestarting from data the same as we have been using, except that all assumptionsabout the � � -grading of the Hilbert space � are dropped. There is a completelyanalogous local index formula in this case (indeed it was the odd case that Connesand Moscovici originally considered). For remarks on how to adapt our approachto the odd case see [19].

528 N. Higson

A Appendix: Compact and Trace-Class Operators

In this appendix we shall take a rapid walk through the elementary theory of com-pact Hilbert space operators.

A.1 Definition. A bounded linear operator on a Hilbert space � is compact (orcompletely continuous, in old-fashioned terms) if it maps the closed unit ball of �to a (pre)compact set in usual norm topology.

We write ‘(pre)compact’ because it turns out that if the image of the closedunit ball has compact closure, then it is in fact already closed, and therefore com-pact. Here are various ways of using the compactness of a bounded Hilbert spaceoperator � :

(i) If� � 6 � is a bounded sequence of vectors, then the sequence

� �� 6 � contains anorm-convergent subsequence.

(ii) If� � 6�� is a bounded sequence of vectors, and if it converges weakly to � (this

means that� � 6 �� converges to

� �� , for every � ), then� �� 6 � converges in

the norm topology to �� .(iii) The quadratic functional � �� is continuous from the closed unit ball

with its weak topology into � . Since the closed unit ball is compact in theweak topology, the functional has extreme values.

The first two items are actually equivalent formulations of compactness. The lastitem, has a very important consequence:

A.2 Lemma. If � is a compact and self-adjoint operator (which means that� �� , for all � and � ), then � has a non-zero eigenvector.

Proof. Let � be a unit vector which is an extreme point of the functional in item(iii). If � is a unit vector orthogonal to � , then by differentiating the function

� �� cos �� ) sin �� cos �� ) sin �� at � � �

(which is an extreme point) we find that �� is orthogonal to � . Hence ��must be a scalar multiple of � , which is to say an eigenvector.

We can now restrict the operator � of the lemma to the orthogonal complementof � , and then apply the lemma again to get a second eigenvector. Continuing inthis way we get Hilbert’s Spectral Theorem for compact, self-adjoint operators:


A.3 Spectral Theorem. If � is a compact and self-adjoint operator on a Hilbertspace � , then there is an orthonormal basis for � consisting of eigenvectors for � .The corresponding eigenvalues are all real, and converge to zero.

Conversely, if a bounded operator � has such an eigenbasis, then it is readilychecked that � must be compact. Examples of compact operators tend to comeeither from this source, or from one of the following two observations:

(i) If � is a norm limit of finite-rank operators, then � is compact (moreoverevery compact operator is a norm limit of finite-rank operators).

(ii) If � is an operator on � ; � " � , and if � can be represented as an integral operator

� � �� '� � � - �� where the kernel - �� is square-integrable on " � " , then � is compact(these are the Hilbert-Schmidt operators; not every compact operator on � ; � " �is of this type).

It follows from the Spectral Theorem that the theory of compact self-adjointoperators has much in common with the theory of real sequences which converge to�. It is therefore quite natural to consider subclasses of compact operators for which

the eigenvalue sequence is summable, � -summable, and so on, and to develop, forexample, Holder inequalities, and so on. This program has in fact been carried outvery far.

We can apply many of the same ideas to non-self-adjoint compact operators bymeans of the following device.

A.4 Definition. Let � be a bounded operator on a Hilbert space � . The singularvalues

� � �� ; �� "�"� of � are the non-negative scalars defined by the formula

� 6 �� '� infdim ��

7 6� � sup��

� �� Thus

� � �� is the norm of � , and� 6 �� measures the norm of � acting on all

codimension �%1 � subspaces of � . Observe that� � ��

; �� "�"� and that

� is compact � lim6�� 6 �� :� � �(If � is not compact, then the singular value sequence is typically not very interest-ing — often it is constant.)

530 N. Higson

A.5 Lemma. Let � be compact and positive Hilbert space operator (this means� �� for all � , which implies that � is self-adjoint). Let

� � 6�� be the eigenvaluesequence for � , arranged in decreasing order, and with multiplicities counted. Then� 6 �� '�� 6 �� , for all � .Proof. This follows readily from the Spectral Theorem, which gives a concreterepresentation for � as a diagonal matrix:

� � ��

� ; � �

. . .

��

Apart from being quite meaningful for arbitrary compact operators, the advan-tage of the singular values over the eigenvalues is that by virtue of their definitionit is rather easy to prove inequalities involving them. For example:

A.6 Lemma. Let � � and � ; be compact operators on a Hilbert space and let � bea bounded operator. Then� 6 ��:) � ; � ,

� 6 �� ) � 6 �� ; ��,�;6 ��:) � ; � �

and � 6 �� 6 �� , � � � � 6 �� With these inequalities to hand we can make the following definition:

A.7 Definition. Let � be a Hilbert space and denote by � �� the algebra ofbounded operators on � . The trace ideal in � �� is

� � �� & � 6 �� Every trace-class operator is compact. Thanks to the inequalities in Lemma A.6

the trace ideal really is a two-sided ideal in the algebra � �� . It is not closed in thenorm-topology, in fact its closure is the ideal of all compact operators.

From the definition of the singular values� 6 �� it follows that if

� �� "�"� �� is any orthonormal set in � , then

� 6 7 � &� � 6 � � 6 �#& ,

� 687 �

� 6 �� As a result of this new inequality we can make the following definition.


A.8 Definition. If � � � �� , then the trace of � is the scalar

Trace �� '� � 6 7 �� 6 �� 6 �

where the sum is over an orthonormal basis of � .

The series converges absolutely, so our definition makes some sense. Simplealgebra (reinforced by the guarantee of absolute convergence of all the series in-volved in the argument) shows that Trace �� does not depend on the choice oforthonormal basis, and that

� � �� Trace �� :� Trace �� Thus the operator-trace has the fundamental property of the trace on matrices, tothe fullest extent it can have it.

If � and � are Hilbert-Schmidt operators, then it may be shown that � � is atrace-class operator (incidentally, an operator � on � ; � " � belongs to the Hilbert-Schmidt class if and only if

� � 6 �� ; � � ). The trace of many integral operatorsmay be computed using the following result:

A.9 Lemma. Let � be a closed manifold which equipped with a smooth measure.If - is a smooth function on � � � , then the operator � defined by the formula

� � �� '� ��- ��

is a trace-class operator. Moreover

Trace �� '� ��- ��

A.10 Remark. One can replace ‘smooth’ by ‘differentiable sufficiently many times’,but the order of differentiability depends on the dimension of the manifold (assum-ing that the kernel - is merely continuous is not enough).

B Appendix: Fourier Theory

If � �� $ � � is a smooth function on the � -torus � $ !� � $ � � $ then its FourierTransform is the function

�� $ � � defined by

�� ; � 3#6 � � � � �

532 N. Higson

The Fourier transform � �� extends to an isometric isomorphism of Hilbertspaces

� ; � � $ � // � ; �� $ � �This is the Plancherel Theorem. If � � � $ � � is smooth then the Fourier trans-forms of its partial derivatives �

� � may be computed from the formula

�

��

��

Thanks to this and the Plancherel Theorem, the norm in the Sobolev space � � � �$ �

may be computed from the formula� � � ;�� 6�� ) � ; � � � & �� & ; �

B.1 Lemma. If - � �then the inclusion of � � � �

$ � into � ; � � $ � is a compactoperator.

Proof. If � � $ then denote by � 6 the function � ; � 3#6 � � on � . Using our formulafor the norm in � � � �

$ � we see that the Hilbert spaces � ; � � $ � and � � ��$ � have

an orthonormal bases� � 6�� and

� � 6 � � � ) � ; � � � � � 6 � respectively. Using these bases, the inclusion of � � � �

$ � into � ; � � $ � takes theform

� 6 �� ) � ; � � � � � 6 �If -��

then the scalar coefficient sequence converges to zero, and so the inclusionoperator is compact.

B.2 Remark. If - � � then the coefficient sequence is summable, and thereforethe inclusion is a trace-class operator.

If � is a smooth function on � $ then according to the Plancherel Theorem,

� � 6�� 6 where as above � 6 �� ; � 3 6 � � . To begin with, the series converges in � ; � � $ � , sothat the coefficient family

� � � � � � � is square-summable, but from the formula

�� 6��

�� 6 � 6��

�� 6


we see that the coefficient family� � � � � � � remains square-summable after multipli-

cation by any polynomial in � . So by the Cauchy-Schwarz inequality the series� 6�� 6 is in fact absolutely summable, and therefore converegent in theuniform norm. A refinement of this computation proves the following lemma:

B.3 Lemma. If � and - are non-negative integers, and if -�� ) $;

then � � � �$ � �� $ � .

Proof. Let � be a smooth function on � $ . We have that� � �� max� � � �

�sup� � � � & � � � �� & �

Since �� 6��

�� 6 �� we get

& �� & , 6�� & �� & � 6�� & � & � * & �� & �

If -�� ) $ ; then the Cauchy Schwarz inequality implies that

6�� & � & � * & �� & � � � � �� and therefore

� � �� , as required.

The Fourier Transform of a smooth, compactly supported function � � � $ � �is the function

�� $ � � given by the formula

� � � � � � � � � � �� ; � 3 � � � � � �Once again, the Fourier transform extends to an isometric isomorphism of Hilbertspaces, but this time from � ; �� $ � to itself. If is an open set in � $ then theSobolev norm

� �� of Definition 1.5 can be given, up to equivalence, by the

formula � � � ;�� ) � ; � � � & � � � � � & ; � � �

With this formula available we can obviously now define Sobolev spaces � � �� for any real - � just by completing the smooth, compactly supported functionsin the above norm. Using partitions of unity and local coordinates we can nowdefine Sobolev spaces � � �� for any - � and any closed manifold, just as wedid in the case where - was a non-negative integer. These are the spaces we brieflyreferred to in Remark 1.27.

534 N. Higson

References

[1] M. F. Atiyah. Global aspects of the theory of elliptic differential operators.In Proc. Internat. Congr. Math. (Moscow, 1966), pages 57–64. Izdat. “Mir”,Moscow, 1968.

[2] M. F. Atiyah and I. M. Singer. The index of elliptic operators I. Annals ofMathematics, 87:484–530, 1968.

[3] A. Connes. Noncommutative differential geometry. Inst. Hautes Etudes Sci.Publ. Math., (62):257–360, 1985.

[4] A. Connes. Noncommutative Geometry. Academic Press, 1994.

[5] A. Connes. Hypoelliptic operators, Hopf algebras and cyclic cohomology.In Algebraic

�-theory and its applications (Trieste, 1997), pages 164–205.

World Sci. Publishing, River Edge, NJ, 1999.

[6] A. Connes. Cyclic cohomology, quantum group symmetries and the localindex formula for � � � � � . Preprint, 2002. math.QA/0209142.

[7] A. Connes and D. Kreimer. Hopf algebras, renormalization and noncommu-tative geometry. Comm. Math. Phys., 199(1):203–242, 1998.

[8] A. Connes and G. Landi. Noncommutative manifolds, the instanton algebraand isospectral deformations. Comm. Math. Phys., 221(1):141–159, 2001.

[9] A. Connes and H. Moscovici. Transgression du caractere de Chern et co-homologie cyclique. C. R. Acad. Sci. Paris Ser. I Math., 303(18):913–918,1986.

[10] A. Connes and H. Moscovici. The local index formula in noncommutativegeometry. Geom. Funct. Anal., 5(2):174–243, 1995.

[11] A. Connes and H. Moscovici. Modular hecke algebras and their hopf symme-try. Preprint, 2003. math.QA/0301089.

[12] E. Getzler and A. Szenes. On the Chern character of a theta-summable Fred-holm module. J. Funct. Anal., 84(2):343–357, 1989.

[13] P. B. Gilkey. Invariance Theory, the Heat Equation, and the Atiyah-SingerIndex Theorem, volume 11 of Mathematics Lecture Series. Publish or Perish,Inc., 1984.


[14] J. M. Gracia-Bondıa, J. C. Varilly, and H. Figueroa. Elements of noncommu-tative geometry. Birkhauser Advanced Texts: Basler Lehrbucher. [BirkhauserAdvanced Texts: Basel Textbooks]. Birkhauser Boston Inc., Boston, MA,2001.

[15] V. Guillemin. A new proof of Weyl’s formula on the asymptotic distributionof eigenvalues. Adv. in Math., 55(2):131–160, 1985.

[16] G. H. Hardy. Divergent series. Editions Jacques Gabay, Sceaux, 1992. Witha preface by J. E. Littlewood and a note by L. S. Bosanquet, Reprint of therevised (1963) edition.

[17] N. Higson. Meromorphic continuation of zeta functions associated to ellipticoperators. Preprint, 2003, In preparation.

[18] N. Higson. A note on regular spectral triples. Preprint, 2003.

[19] N. Higson. On the Connes-Moscovici residue cocycle. Preprint, 2003.

[20] N. Higson. Slides to accompany lectures given at the Trieste summer school,2002. http://www.math.psu.edu/higson/trieste.

[21] A. Jaffe, A. Lesniewski, and K. Osterwalder. Quantum�

-theory. I. The Cherncharacter. Comm. Math. Phys., 118(1):1–14, 1988.

[22] J.-L. Loday. Cyclic homology, volume 301 of Grundlehren der Mathema-tischen Wissenschaften [Fundamental Principles of Mathematical Sciences].Springer-Verlag, Berlin, 1992. Appendix E by Marıa O. Ronco.

[23] S. Minakshisundaram and A. Pleijel. Some properties of the eigenfunctions ofthe Laplace-operator on Riemannian manifolds. Canadian J. Math., 1:242–256, 1949.

[24] D. Quillen. Algebra cochains and cyclic cohomology. Inst. Hautes EtudesSci. Publ. Math., 68:139–17, 1988.

[25] R. T. Seeley. Complex powers of an elliptic operator. In Singular Integrals(Proc. Sympos. Pure Math., Chicago, Ill., 1966), pages 288–307. Amer. Math.Soc., Providence, R.I., 1967.

[26] G. Segal and G. Wilson. Loop groups and equations of KdV type. Inst. HautesEtudes Sci. Publ. Math., (61):5–65, 1985.

536 N. Higson

[27] H. Weyl. David Hilbert and his mathematical work. Bull. Amer. Math. Soc.,50:612–654, 1944.

[28] M. Wodzicki. Noncommutative residue. I. Fundamentals. In�

-theory, arith-metic and geometry (Moscow, 1984–1986), volume 1289 of Lecture Notes inMath., pages 320–399. Springer, Berlin, 1987.

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