In this chapter we concentrate upon the measure theoretic aspects of foliated spaces, including especially the notion of transverse measures.

We begin with a general study of groupoids, first in the measurable and later in the topological context. Our examples come from the holonomy groupoid of a foliated space (2.20) and a discrete version corresponding to a complete transversal. We introduce transverse measures ν with a given modulus and discuss when these are invariant.

Next we look in the tangential direction, defining a tangential measure λ to be a collection of measures λ = { λx} (one for each leaf in the case of a foliated space) which satisfies certain invariance and smoothness conditions. For instance, a tangential, tangentially elliptic operator D yields a tangential measure lD as follows. Restrict D to a leaf l. It follows from Chapter I that KerD` and Ker D*lare locally finite-dimensional and hence the local index.

Next we specialize to topological groupoids and continuous Radon tangential measures. In the case of a foliated space we recount the Ruelle–Sullivan construction of a current associated to a transverse measure and we show that the current is a cycle if and only if the transverse measure is invariant.

This book grew out of lectures and the lecture notes generated therefrom by the first named author at UC Berkeley in 1980 and by the second named author at UCLA, also in 1980. We were motivated to develop these notes more fully by the urgings of our colleagues and friends and by the desire to make the general subject and the work of Alain Connes in particular more readily accessible to the mathematical public. The book develops a variety of aspects of analysis and geometry on foliated spaces which should be useful in many contexts. These strands are then brought together to provide a context and to expose Connes’ index theorem for foliated spaces [Connes 1979], a theorem which asserts the equality of the analytic and the topological index (two real numbers) which are associated to a tangentially elliptic operator. The exposition, we believe, serves an additional purpose of preparing the way towards the more general index theorem of Connes and Skandalis [1981; 1984]. This index theorem describes the abstract index class in K0(C*r(G(M))), the index group of the C*-algebra of the foliated space, and is necessarily substantially more abstract, while the tools used here are relatively elementary and straightforward, and are based on the heat equation method.

We must thank several people who have aided us in the preparation of this book. The origins of this book are embedded in lectures and seminars at Berkeley and UCLA (respectively) and we wish to acknowledge the patience and assistance of our colleagues there, particularly Bill Arveson, Ed Effros, Marc Rieffel and Masamichi Takesaki. More recently, we have benefitted from conversations and help from Ron Douglas, Peter Gilkey, Jane Hawkins, Steve Hurder, Jerry Kaminker, John Roe, Jon Rosenberg, Bert Schreiber, George Skandalis, Michael Taylor, and Bob Zimmer

We owe a profound debt to Alain Connes, whose work on the index theorem aroused our own interest in the subject. This work would not exist had we not been so stimulated by his results to try to understand them better.

Global analysis has as its primary focus the interplay between the local analysis and the global geometry and topology of a manifold. This is seen classically in the Gauss–Bonnet theorem and its generalizations, which culminate in the Atiyah–Singer Index Theorem [Atiyah and Singer 1968a]. This places constraints on the solutions of elliptic systems of partial differential equations in terms of the Fredholm index of the associated elliptic operator and characteristic differential forms which are related to global topological properties of the manifold.

Our object in this chapter is to develop the notion of what we call locally traceable operators-or, more or less equivalently, the notion of locally finite-dimensional subspaces relative to an abelian von Neumann algebra A. The underlying idea here is that certain operators, although not of trace class in the usual sense, are of trace class when suitably localized relative to A. The trace, or perhaps better, the local trace of such an operator is not any longer a number, but is rather a measure on a measurable space X associated to the situation with A =L∞(X). This measure is in general infinite but σ-finite, and it will be finite precisely when the operator in question is of trace class in the usual sense, and then its total mass will be the usual trace of the operator. Heuristically, the local trace, as a measure, will tell us how the total trace—infinite in amount — is distributed over the space X. Once we have the notion of a locally traceable operator, and hence the notion of locally finite-dimensional subspaces, one can define then the local index of certain operators. This will be the difference of local dimensions of the kernel and cokernel, and will therefore be, as the difference of two σ-finite measures, a σ-finite signed measure on X. One has to be slightly careful about expressions such as ∞-∞ that arise, but this is a minor matter and can be avoided easily by restricting consideration to sets of finite measure. These ideas are developed to some extent in [Atiyah 1976] for a very similar purpose to what we have in mind here, and we are pleased to acknowledge our gratitude to him.

To be more formal and more exact about this notion, we consider a separable Hilbert space H with an abelian von Neumann algebra A inside of ℬ(H), the algebra of all bounded operators on H. (We could dispense in part with this separability hypothesis, but it would make life unnecessarily difficult; all the examples and applications we have in mind are separable.) For example, suppose that X is a standard Borel space (see [Arveson 1976] and [Zimmer 1984, Appendix A] for definitions and properties of such spaces). It is a fact that X is isomorphic to either the unit interval [0,1] with the usual σ-field of Borel sets or is a countable set with every subset a Borel set; see [Arveson 1976] for details. Now let µ be a σ-finite measure on X and let Hn be a fixed n-dimensional Hilbert space where n=1, 2 ....,∞.

Mikhael Gromov and Blaine Lawson, in their classic paper [1980], use Dirac operators with coefficients in appropriate bundles and associated topological invariants to investigate whether or not a given compact nonsimply connected manifold can support a metric of positive scalar curvature. In this appendix we consider the analogous problem for foliated spaces. We use appropriate tangen-tial Dirac operators to investigate the existence of a tangential Riemannian met-ric with positive scalar curvature along the leaves of a compact foliated space. Gromov and Lawson use the Â-genus and the Atiyah–Singer index theorem; we shall use the tangential Â-genus and the Connes index theorem.

This chapter is devoted to the study of tangential pseudodifferential operators and their index theory. The chapter has four topics, treated in turn:

• •the general theory of pseudodifferential operators on foliated spaces (VII-A on page 169);differential operators and finite propagation (VII-B on page 190);Dirac operators and the McKean–Singer formula (VII-C on page 197);

• •superoperators and the asymptotic expansion of the heat kernel (VII-D on page 202).

We discuss each of them briefly before revisiting them in their respective sections.

Pseudodifferential operators. We begin the chapter by introducing the machinery of tangential differential operators, smoothing operators, and pseudo-differential operators, first in a local setting and then globally. We demonstrate that a tangentially elliptic pseudodifferential operator has an inverse modulo compactly smoothing operators.

In this chapter we mimic as closely as possible [Milnor and Stasheff 1974], itself an expos´e of the Chern–Weil construction of characteristic classes in terms of curvature forms. See also [Dupont 1978; Husemoller 1975; Lawson and Michelsohn 1989; de Rham 1955].

The Chern–Weil procedure begins with a vector bundle with a certain structural group G. In our situation we consider complex (tangentially smooth) bundles with structural group GL(n,ℂ) real vector bundles with structural group GL(n, ℝ), and oriented real even-dimensional vector bundles with structural group SO.(2n). Choose a tangential connection ∇, see below, that respects the structure. The associated curvature form K determines a closed tangential 2- form whose tangential cohomology class is independent of choice of the connection. Then any polynomial or formal power series P which is G-invariant determines a characteristic form. In the case X = M is a manifold with FX = TM then this yields the usual characteristic classes in de Rham cohomology H*M.

We shall assume throughout that all bundles over foliated spaces are tangentially smooth and that leaf-preserving maps between foliated spaces are also tangentially smooth; this is not a real restriction, in view of our smoothing results (Proposition 2.16). We use Milnor and Stasheff’s sign conventions for characteristic classes.

The purpose of this Appendix is to discuss the conclusion of the foliation index theorem in the context of foliations whose leaves are two-dimensional. Such foliations provide a class of reasonably concrete examples; while they are cer-tainly not completely representative of the wide range of foliations to which the theorem applies, they are sufficiently complicated to warrant special attention, and possess the smallest leaf dimension for which the leaves have interesting topology. There is another, more fundamental reason for studying these folia-tions: given any leafwise C∞-Riemannian metric on a two-dimensional foliation ℱ there is a corresponding complex-analytic structure on leaves making ℱ into a leafwise complex analytic foliation. Thus, two-dimensional foliations automatically possess a Teichm¨uller space, and for each point in this space of complex structures, there is an associated Dirac operator along the leaves. The foliation index theorem then assumes the role of a Riemann–Roch Theorem for these complex structures.

We begin in Section A1 with a discussion of the average Euler characteristic of Phillips–Sullivan, which is the prototype for the topological index character of the foliation index theorems for surfaces. In Section A2, the index theorem is reformulated for the N@-operator along the leaves of a leafwise-complex foli-ation. The Teichm¨uller spaces for two-dimensional foliations are discussed in Section A3, and a few remarks about their properties are given. In Section A4, some homotopy questions concerning the K-theory of the symbols of leafwise elliptic operators are discussed, with regard to the determination of all possibleTopological indices for a fixed foliation. Finally Section A5 describes some of the “standard” foliations by surfaces, especially of three-manifolds, and the calculation of the foliation indices for them.

We turn now to the discussion of operator algebras that can be associated with groupoids and in particular to the groupoid of a foliated space. For this discussion we start with a locally compact second countable topological groupoid G and we assume given a continuous tangential measure λ (see Chapter IV for the definition). Thus for each x in the unit space X of G we have a measure λx on Gx = r-1(x) with certain invariance and continuity properties as described in Chapter IV. For the moment we do not need to assume that the groupoid has discrete holonomy groups as in Chapter IV, but all the examples and all the applications will satisfy this condition. If in addition the support of the measure λx is equal to r-1(x), as is usual in our examples, then λ is called a Haar system.

In this chapter we construct the C*-algebra of the groupoid and we determine this algebra in several important special cases. (As general references for C*- algebras, we recommend [Davidson 1996; Fillmore 1996; Arveson 1976; Pedersen 1979].) We describe the Hilsum–Skandalis stability theorem. Assuming a transverse measure, we construct the associated von Neumann algebra and develop its basic properties and important subalgebras. This leads us to the construction of the weight associated to the transverse measure; it is a trace if and only if the transverse measure is invariant. Finally, we introduce the K theory index group K0(C*r(G)) and construct a partial Chern character.