In this article, we study the index theory of longitudinal elliptic operators on open foliated manifolds. Our study is inspired by the longitudinal index theory on a manifold M with a regular foliation F due to Connes and Connes-Skandalis (see for example [5,6,8]). Examples of longitudinal elliptic operators on open foliated manifolds appear naturally, both in their own right and also by associating non-compact manifolds to closed manifolds as in the next two examples.
As a first example, in order to define the transverse fundamental class on (M, F ) in [6], Connes introduced a noncompact manifold M, the space of metrics on the normal bundle of F , together with a regular foliation F of M. A longitudinal elliptic operator on (M, F ) has a canonical lifting to a longitudinal elliptic operator on (M, F ). The lifted operator on (M, F ) plays a key role in Connes' theory of transverse fundamental classes.
A second source of examples of longitudinal elliptic operators on open manifolds is our efforts to incorporate the fundamental group into index theory on a foliated manifold (M, F ). Here, we consider the covering space M of M together with the lifted regular foliation F . A longitudinal elliptic operator on (M, F ) naturally lifts to a longitudinal elliptic operator on ( M, F ).
To allow the possibility that the rank of F can vary on M, we develop our index theory for elliptic differential operators on a general Lie groupoid G with unit space G 0 . Our theory applies to a manifold M with a regular foliation F when the groupoid G is the holonomy groupoid associated to a foliation F on M = G 0 .
If G 0 is a closed manifold, the index of a longitudinally elliptic differential operator on G defines an element in the K-theory of the (reduced) groupoid C * -algebra C * r (G) of the groupoid G. However, this is generally not the case when G 0 is not closed. In general, we show how to use a Euclidean metric on the Lie algebroid A to define a coarse structure E on G in the sense of [10] under a natural completeness assumption (see Proposition 2.7 for details). Following ideas that start in [21], we then introduce a Roe C * -algebra C * E (G) for G associated to the coarse structure E; its K -theory provides a natural home for indices of longitudinal elliptic operators in the non-compact case. If G 0 happens to be closed, Chapter 3]; thus our theory generalizes both of these cases.
Assume now that the Lie algebroid A associated to G is equipped with a Euclidean metric, and G with the coarse structure it induces. We also associate an algebra C e (A * ) of continuous functions on A * whose K -theory provides a natural home for symbols of geometrically defined operators. A coarse version of the groupoid C * -algebra of Connes' tangent groupoid T G of G (c.f. [7]) leads to the following coarse analytic index map
This approach to 'coarse index theory' via tangent groupoids seems to be new even in the classical case when G is the pair groupoid associated to a Riemannian manifold.
To understand the analytic index map Ind a , we consider what we call a smooth symbol
) . Roughly speaking a smooth symbol consists of a pair of smooth vector bundles V 1 , V 2 over G 0 and a smooth endomorphism σ between their pullbacks to A * that is invertible at infinity; so far this is analogous to the symbol data appearing in the classical Atiyah-Singer theorem, but in our non-compact setting we need an additional assumption on boundedness of the derivatives of σ (see Definition 4.1 for details). Using groupoid pseudodifferential operators as introduced in [15,16], we consider a 'quantization' {Q λ (σ )} λ∈ [0,1] of the symbol, and construct a multiplier Q(σ ) of the tangent groupoid Roe C * -algebra. We prove the following theorem under a natural bounded geometry assumption; roughly, this says that all the curvature tensors on the target fibers of G have uniformly bounded derivatives, and injectivity radius bounded below (see Assumption 4.3 for details). Given a smooth symbol (V 1 , V 2 , σ ) representing an element in K 0 (C * e (A * )), the opera-
We apply our theorem to study positive scalar curvature metrics as a generalization of the Lichnerowicz vanishing theorem [13]. In 1963, Lichnerowicz proved that the index of the Dirac operator on a closed spin manifold with a positive scalar curvature metric must vanish. Around the same time, the Atiyah-Singer theorem [2] was proved; it in particular implies that the index of the Dirac operator is equal to the A-genus of the manifold. The two results together imply that non-triviality of the topologically invariant A-genus of the manifold is an obstruction for the manifold to carry a Riemannian metric with positive scalar curvature. These breakthroughs started much progress in the study of geometry and topology of manifolds in the past 50 years. See [3,14,28,29] for recent developments. Our study is inspired by Connes' far reaching generalization [6] of the Lichnerowicz theorem to closed foliated manifolds in the 80s. Here, we attempt to generalize Connes' result to open foliated manifolds. In this direction, we prove the following result; this requires the same boundedness assumptions on the derivatives of the curvature tensors and injectivity radius as before (see Assumption 4.3 for details).
Theorem 1.2 Let G be a Lie groupoid, and assume that the associated Lie algebroid A has even rank, and is equipped with a spin structure such that the associated Riemannian metric g A satisfies Assumption 4.3. Then the triple (S + , S -, σ ( / D + )) associated to the Dirac operator is a smooth symbol in the sense of Definition 4.1. Moreover, if g A has uniformly positive scalar curvature, the analytic index of the K-theory class [S + , S -, σ ( /
Ind a [S + , S -, σ ( / D + )] = 0.
Applying this theorem to the regular foliations (M, F ) and ( M, F ), in Corollary 5.2 we obtain obstructions to the existence of leafwise positive scalar curvature metric on (M, F ) as a generalization of the Lichnerowicz vanishing theorem, [6,28]. We would like to remark that in the above theorem we have considered the longitudinal index of a foliation, which is different from the study in [14,29], where the transverse index of a (singular) foliation from a group action is considered. The article is organized as follows. In Sect. 2, we discuss coarse structures on a locally compact Hausdorff groupoid, and show how they arise from suitable Riemannian structures on Lie groupoids (in particular, on holonomy groupoids). In Sect. 3, we introduce the groupoid Roe algebra associated to a locally compact groupoid with an appropriate coarse structure and Haar system; if the groupoid is in addition a Lie groupoid, we then use Connes' tangent groupoid to define an analytic index for longitudinal elliptic operators. In Sect. 4, we introduce the notion of a smooth symbol and use the groupoid pseudodifferential operator theory to study the analytic index of such a symbol; this is the technical heart of the paper, and overviews of the arguments involved can be found in that section. Finally, in Sect. 5, we apply our groupoid index theory to study (regular) foliations with leafwise positive scalar metrics.
We fix some basic notation for groupoids. Let G ⇒ G 0 be a locally compact groupoid with unit space G 0 . We will always assume that such a groupoid G is paracompact and Hausdorff.
Let s, t : G → G 0 be the source and target maps of G, which we always assume to be open. Throughout this paper, we think of an arrow as from the left to the right: thus a pair (g, h) of arrows is composable if t (g) = s(h), and the composition is gh. The s-fibers and t-fibers of x ∈ G 0 are denoted respectively by G x := s -1 (x) and G x := t -1 (x).
If G is in addition a Lie groupoid then s and t are assumed to be submersions. Thus s and t are open maps, and the s-and t-fibers are closed submanifolds of G. We will usually assume in this case that all s-fibers (and therefore all t-fibers) are connected. The Lie algebroid of G ⇒ G 0 is A := ker t * ⊂ T G| G 0 . Noting that the right action of G on itself permutes the t-fibers, sections of A can be identified with right G-invariant vector fields on G that are tangent to the t-fibers.
We are mainly interested in Lie groupoids (and in fact, in holonomy groupoids) in this paper; nonetheless, some definitions and results are stated in more generality than this where it seems potentially useful for future applications, and where the extra generality causes no extra difficulties.
In [10, Sect. 2], Higson, Pedersen and Roe introduced a notion of coarse structure on a locally compact, Hausdorff groupoid G in the following way. 1. the inverse of any entourage is contained in an entourage; 2. the (groupoid) product of two entourages is contained in an entourage; 3. the union of two entourages is contained in an entourage; 4. entourages are proper: that is, for any entourage E and any compact subset C of the objects G 0 , E ∩ s -1 (C) and E ∩ t -1 (C) are relatively compact; 5. the union of all the entourages is G.
If there is an entourage that contains the unit space G 0 , the coarse structure is called unital. Two coarse structures E and F are equivalent if every element of E is contained in some element of F and vice versa.
Example 2.2 Say X is a locally compact topological space, and G = X × X is the associated pair groupoid. Then an equivalence class of coarse structures on G in the sense above is essentially the same thing as a coarse structure on X that is compatible with the topology as described in [23,Definition 2.22]. More specifically, the difference between the definition of [23,Definition 2.22] and the specialization of Definition 2.1 above to pair groupoids is analogous to the difference between a maximal atlas and an atlas for a manifold; it does not make any substantial difference to the resulting theory.
Remark 2.3 Note that the collection of all open relatively compact subsets of G always constitutes a coarse structure. Moreover, conditions (3) and (5) from Definition 2.1 imply that every relatively compact subset of G is contained in some entourage, and thus (up to equivalence) the collection of relatively compact subsets is contained in any coarse structure.
On the other hand, if the unit space G 0 is compact, then condition (4) implies that all entourages in any coarse structure are relatively compact. Hence if G 0 is compact, any coarse structure on G is equivalent to the coarse structure consisting of all open relatively compact subsets of G. Thus (the equivalence class of) a coarse structure gives no more information than the topology of G when G 0 is compact.
In the remainder of this subsection we assume that G is a Lie groupoid with connected t-fibers, and will describe how to define a coarse structure on G starting with an appropriate Euclidean metric on the Lie algebroid A. The following example is an important, and particularly straightforward, special case of the general construction.
Let G be a Lie groupoid, and A the associated Lie algebroid, equipped with a Euclidean metric { , x } x∈G 0 as above. Then we may regard A itself as a Lie groupoid with unit space G 0 , and all arrows having the same source and target. Write points in A as (x, v) where x ∈ X and v is in the fiber A x over x, and define
Then the collection {E r } r >0 clearly defines a coarse structure on the Lie groupoid A.
Fix a smooth Euclidean metric { , x } x∈G 0 on the bundle A with base space G 0 . Let s * A be the pullback of A to G via the source map. For each fixed x ∈ G 0 with associated t-fiber G x , the restriction (s * A)| G x identifies canonically with the tangent bundle T G x ; in this way, the Euclidean metric on A gives rise to a Riemannian metric on each of the t-fibers G x . Moreover, these metrics are invariant under the right action of G on itself: precisely, right multiplication by each g ∈ G defines a diffeomorphism G s(g) → G t (g) , and invariance means that these diffeomorphisms are all isometries.
For each x ∈ G 0 , let d x denote the length metric on the t-fiber G x induced by the Riemannian metric defined above; note that as each G x is assumed connected, each d x is everywhere finite. Moreover, the family of metric spaces (G x , d x ) is invariant for the right action of G: precisely, for any g ∈ G and g 1 ,
(
Define now a function ρ : G → [0, ∞) by ρ(g) := d t (g) (g, t (g)).
We will show that under one additional condition (discussed below), the sets {ρ -1 ([0, r )) | r > 0} define a coarse structure on G. The bulk of the proof of this is in the next two lemmas.
The function ρ is subadditive and symmetric, meaning
for all g, h ∈ G for which these expressions make sense.
Proof The triangle inequality and the right invariance of the metric imply
and the right-invariance again implies
Lemma 2.6 Let r > 0. For any g ∈ G with ρ(g) < r there is an open neighborhood U of g in G such that ρ(h) < r for all h ∈ U .
Proof Let > 0 be such that 2 < r -ρ(g). Fix a smooth path from t (g) to g in G t (g) with length l at most r -(this is possible by definition of the length metric d t (g) ). As t : G → G 0 is a submersion and this path is compact, we may cover it by finitely many open sets (for the topology on G) that identify with product sets of the form V × W , where V is an open neighborhood of t (g) in G 0 , and W is an open subset of G t (g) . Using smoothness of the Euclidean metric on A and another compactness argument based on moving along the original path through these product neighborhoods, it is not difficult to see that there is a neighborhood U of g (for the topology on G) of the same form such that for every h ∈ U there is a path from t (h) to h in G t (h) of length at most l + . This completes the proof.
Proposition 2.7 Assume that the Riemannian manifolds G x constructed above from a Euclidean metric on the Lie algebroid A are all complete. Then the collection
defines a coarse structure on G.
The assumption that each G x is complete is not automatic (Example 2.9 below), but holds in the examples of most interest to us (Proposition 2.10 below).
Proof Symmetry and subadditivity as in Lemma 2.5 imply that E -1 r = E r and the product of E r and E s is contained in E r +s . Each E r is open by Lemma 2.6. To see properness, note that completeness of each G x implies that there is a globally defined, continuous exponential map exp : A → G. For any compact subset C of G 0 , we have that
and the set on the right is clearly compact. The fact that s -1 (C) ∩ E r is relatively compact follows by symmetry.
Example 2.8 Going back to Example 2.4, note that if we consider A as a Lie groupoid, then the Lie algebroid is just A again. For each x ∈ G 0 , the metrics d x defined above just identify with the metric on the fiber A x defined by the norm • x , and thus the coarse structure defined on the groupoid A by Proposition 2.7 is the same as the one in Example 2.4.
Example 2.9 Let M be a connected Riemannian manifold, and G = M × M the associated pair groupoid. Then the Lie algebroid A identifies naturally with the tangent bundle T M, and thus inherits a Euclidean metric. The coarse structure on G defined by the process above then identifies with the coarse structure on M induced by the original metric as in [23,Example 2.5]. Moreover, each t-fiber G x is isometric to M.
In particular, completeness of the t-fibers G x is not automatic in general (as it would be if G were a Lie group).
For regular foliations, the construction above specializes as follows. This is the key example of this paper. Proposition 2.10 Let F be a regular foliation on a complete Riemannian manifold M. Let G be the associated holonomy groupoid, which we assume is Hausdorff.
Identify the Lie algebroid A with F , and equip it with the Euclidean metric defined by restricting the Riemannian metric from M. Then each of the Riemannian manifolds G x defined above is complete, so we get a coarse structure on G as in Proposition 2.7.
Proof We first claim that each leaf L is complete when equipped with the length metric d L induced from the restriction of the Riemannian metric on M. This is presumably well-known, but as we could not find a proof in the literature we provide one for the reader's convenience.
Let then (x n ) be a Cauchy sequence for d L . The metric d L dominates the restriction of the length metric d M on M to L, whence (x n ) is Cauchy for d M as well, and thus convergent to some x ∈ M by completeness. Let U be an open foliation neighborhood (i.e. a neighborhood of x on which the foliation is a product) of x in M such that for some > 0, theneighborhood (defined using d M ) of U is still a foliation neighborhood of x. Then the metric space (L ∩U , d L ) splits into (possibly infinitely many) connected components, each of which is closed in U , and all of which are all at least apart from each other for the d L metric. Hence all but finitely many of the x n are in the same connected component of L ∩ U , whence x is also in this connected component, and so in particular in L. Hence (L , d L ) is complete as claimed.
To finish the argument, recall that for each x ∈ M the t-fiber G x is a connected covering space of the leaf L through x, with the Riemannian metric pulled back from L. As the covering map G x → L is a local isometry for the induced length metrics, G x is also complete.
Example 2.11 Apart from regular foliations, actions of Lie groups are another interesting source of examples. We give a basic example here to illustrate some issues that arise, but no doubt much more could be said. Let SO(2) act on R 2 by rotations in the usual way, and let G := R 2 SO(2) be the associated crossed product groupoid; we write points of the crossed product as pairs (x, z), where x ∈ R 2 and z ∈ SO (2). One natural definition of a coarse structure on G is just to take E c := {G}, which reflects the fact that the acting group SO( 2) is compact (' c ' is for 'compact'), and so itself has no interesting coarse geometry. On the other hand, identifying SO (2) with the unit circle in C, we can define a different coarse structure E m to consist of the sets
for all r > 0; the coarse structure E m reflects the fact that the orbits of the SO(2) action get larger as one moves away from the origin in R 2 (' m ' is for 'metric'). The coarse structures E m and E c are not equivalent, and have quite different properties.
Neither E c nor E m arises directly from a Euclidean metric on the associated Lie algebroid A as in Proposition 2.7, but both are equivalent to coarse structures arising in that way for appropriate choices of Euclidean metric on A. Indeed, identify A with R 2 × so(2), where so(2) is the Lie algebra of SO(2), a copy of R. Fixing a metric on so(2), a Euclidean metric on A is essentially the same thing as a choice of smooth function s : R 2 → (0, ∞), which governs how much the fixed metric on so(2) is 'scaled'. The coarse structure E c is equivalent to that arising from the constant scaling factor s(x) = 1, while the coarse structure E m is equivalent to that arising from the scaling factor s(
Remark 2.12 In order to define a coarse structure on G, it is sufficient to specify a smooth symmetric family of proper metrics on {G x } x∈G 0 . This observation allows one to define coarse structures on general Lie groupoids that are not even source connected; however, to keep in contact with our main motivation-regular foliations-we will not use this.
We assume throughout this subsection that G is a locally compact, second countable, Hausdorff groupoid equipped with a coarse structure. The reader should bear in mind the case that G is a pair groupoid as in Example 2.9, or that G is a holonomy groupoid with coarse structure defined using a metric on the underlying manifold M as in Proposition 2.10. Definition 3.1 A Haar system on G is a family {μ x } x∈G 0 satisfying the following conditions:
1. each μ x is a regular Borel measure on the corresponding t-fiber G x with full support; 2. the family is continuous, meaning that for each f ∈ C c (G), the function
Example 3.2 If G is a Lie groupoid, Haar systems always exist. Indeed, take any (smooth) Euclidean metric on the Lie algebroid A, lift it to a metric on each t-fiber as in the previous section, and define μ x to be the measure on G x canonically defined by the Riemannian metric. We will use this choice for μ x whenever it is convenient.
In order to define a C * -algebra C * E (G), we will need to impose extra conditions on the Haar system as follows.
Definition 3.3 Say G is a locally compact, Hausdorff groupoid with a coarse structure E. A Haar system {μ x } for G has bounded geometry if for all entourages E 4 Let G be a Lie groupoid with a fixed Euclidean metric on the Lie algebroid A. Use this to equip each G x with a Riemannian metric as in the previous section, which we assume complete, and let G have the associated coarse structure as in Proposition 2.7. Let μ x be the associated measure on G x as in Example 3.2. Assume that there is a global lower bound (possibly negative) on the Ricci curvatures of the Riemannian manifolds G x , independently of x. Then the Bishop-Gromov theorem (see for example [4,Theorem 107]) implies that the Haar system {μ x } has bounded geometry.
In particular, say M is a complete Riemannian manifold, and say G is the holonomy groupoid associated to some regular foliation on M, the leaves of which has a global lower bound on the Ricci curvature. Equip each G x with the associated Riemannian metric, and G with the corresponding coarse structure, as in Proposition 2.10. Then the family of measures {μ x } associated to the Riemannian metrics has bounded geometry. Example 3.5 Say X is a locally finite discrete metric space, and G := X × X the associated pair groupoid equipped with the coarse structure defined by
Taking μ x to be the counting measure on G x for all x ∈ G 0 defines a Haar system. This Haar system has bounded geometry if and only if for any r > 0, there is a uniform bound on the cardinalities of all r -balls in X . This is the usual definition of bounded geometry in the setting of discrete metric spaces, and one motivation for the terminology we have adopted.
Let G be a locally compact, Hausdorff groupoid equipped with a coarse structure and a bounded geometry Haar system. The Roe * -algebra of G, denoted C E (G), has as underlying vector space the collection of continuous, bounded, complex-valued functions on G that are supported in an entourage. The adjoint and multiplication on C E (G) are defined by
It follows readily from the properties in Definitions 2.1 and 3.1 that C E (G) is a well-defined * -algebra. Indeed, note first that the adjoint is clearly a well-defined map, while condition (4) from Definition 2.1 implies that the integrand appearing in the multiplication formula is integrable, so this formula also make sense. On the other hand, the formulas defining the multiplication and adjoint above are the standard ones used to define groupoid convolution * -algebras (cf.
As elements of C E (G) are bounded and continuous, and as inverses of entourages are contained in entourages, the bounded geometry assumption implies that f I is finite. Moreover, as each μ x has full support, the I -norm is an honest norm rather than a semi-norm.
We are now ready to define the family of representations that we will use to make C E (G) into a C * -algebra: these are based on the family of regular representations of a groupoid algebra C c (G) as in [20,Sect. 2.3.4].
and in particular extends to a bounded operator on L 2 (G x , μ x ). Moreover,
Proof Let f be an element of C E (G) and ξ be an element of C c (G x ). Then
Moving the absolute value inside the inner integral and applying Cauchy-Schwarz to the product
Using right invariance of the Haar system, the first factor is bounded above by f 1/2 I , and substituting into line (3) gives
Switching the order of integration and using right invariance again bounds this above by
giving the desired bound. The fact that π x defines a * -homomorphism follows from the same sort of computations that show that the operations on C E (G) or C c (G) define a * -algebra structure: see for example [19,Proposition II.1.1].
The following definition now makes sense. Definition 3.9 Let G be a locally compact, second countable, Hausdorff groupoid equipped with a coarse structure and a bounded geometry Haar system. The Roe C * -algebra of G,
We conclude this subsection with some remarks and examples relating Definition 3.9 to other constructions in the literature. Remark 3.10 Say G is as in the above definition, and that G 0 is compact. Then Remark 2.3 implies (whatever the coarse structure is) that C E (G) = C c (G), and that the Roe C * -algebra
Thus our Roe C * -algebras only give something new when the unit space G 0 is not compact.
Note in general that C E (G) always contains C c (G), and that the
Example 3.11 Say G = X × X is the pair groupoid associated to a discrete bounded geometry metric space X , equipped with the coarse structure and Haar system from Example 3.5.
Then C E (G) identifies with the * -algebra of finite propagation, bounded kernels on X , often denoted C u [X ]. Moreover, all the representations π x are unitarily equivalent to the standard representation of this algebra on l 2 (X ), and thus C * E (G) can be canonically identified with the uniform Roe algebra C * u (X ) of [23,Sect. 4.4]. More generally, say G = M × M is the pair groupoid associated to a complete Riemannian manifold M. The associated Lie algebroid identifies with T M, whence it inherits a Euclidean metric from the Riemannian structure on M, and this defines a coarse structure on G as in Example 2.9. If we assume that M has Ricci curvature bounded below, then the associated Haar system {μ x } has bounded geometry. It is not too difficult to see that Remark 3. 13 The above completion of C E (G) is an analogue of the reduced completion of the classical convolution algebra C c (G). It might be interesting to consider other completions, for example the natural maximal completion as was done in [9] for classical Roe algebras. We do not currently have any applications of this, so do not pursue it here. Remark 3.14 Skandalis et al. [25] (see also Tu's generalization [27]) constructed a coarse groupoid G(X ) associated to a discrete bounded geometry metric space as in Example 3.11 above. It is not too difficult to check that C E (G) identifies canonically with C c (G(X )) in this case. It seems reasonable to expect that there should be an analogous construction of a 'coarse groupoid' associated to G with a coarse structure as in our current setting, but we did not pursue this.
In this subsection, we recall the construction of the tangent groupoid [7, Sect. II.5], and build a coarse version of the associated C * -algebra. Throughout this section, G is a Lie groupoid and A the associated Lie algebroid. A is assumed equipped with a Euclidean metric, and G and A with the associated coarse structure and bounded geometry Haar system as in Proposition 2.7, Examples 2.8 and 3.2.
Let S be a closed submanifold of a manifold X and let ν(S) := T X| S /T S be its normal bundle. One can define a manifold with boundary B S →X to be the disjoint union
as a set, and use a choice of exponential map exp : ν(S) → X to produce a natural smooth structure on B that restricts to those on ν(S) and (0, 1] × X , c.f. [11,Sect. 3.1] (the choices make no difference to the resulting smooth structure).
We apply this general construction to a Lie groupoid G and the closed submanifold G 0 . Observe that the normal bundle ν(G 0 ) is isomorphic to the Lie algebroid A as a vector bundle over G 0 . We obtain the tangent groupoid of G, which as a set is given by
The manifold T G can be viewed as a smooth family of groupoids over the interval [0, 1]. Over the point 0 ∈ [0, 1], the fiber is A viewed as a bundle of vector groups, and over λ ∈ (0, 1], the fiber is the groupoid G. These groupoid structures are compatible with the smooth structure on T G; thus T G is a Lie groupoid over the unit space G 0 × [0, 1].
Let A T G denote the Lie algebroid of T G, a bundle over G 0 × [0, 1]; it is isomorphic to the Lie algebroid A × [0, 1], but in a slightly unnatural way, so we will not use this. For each λ ∈ [0, 1], write A λ for the restriction of A T G to G 0 × {λ}, so each A λ identifies naturally with a copy of A. For λ ∈ (0, 1], equip A λ with the original Euclidean metric rescaled by a factor of 1/λ 2 , and equip A 0 with the original Euclidean metric; thus the metric is 'blown up' as λ tends to zero. These structures fit together to define a smooth Euclidean metric on A T G . We use this metric structure to equip T G with the coarse structure and Haar system from Proposition 2.7 and Example 3.2. And following Example 3.4, we assume that there is a global lower bound (possibly negative) on the Ricci curvatures of the Riemannian manifolds G x , independently of x.
The following lemma is clear from the definitions: the basic points are that t-fibers of T G (equipped with metric and measure structure) identify canonically with t-fibers of either G or A, and that the 'blow-up' of the metric is exactly compensated for by the corresponding blow-ups of volume. As in Definition 3.9 above, we thus get an associated Roe C * -algebra C * E (T G). Note that there are natural * -homomorphisms
(where we have used the same notation for the coarse structures on G, T G and A) defined by restricting to the fibers over 0 and 1 respectively in the tangent groupoid and using the compatibility of the coarse structures involved. It is clear from the definition of the norms on the associated Roe C * -algebras that these homomorphisms extend to the completions, giving surjective * -homomorphisms.
Our next task is to compute the kernel of ev 0 . To this end, we first need a technical lemma about the norm on C * E (A). Lemma 3. 16 Let B be any C * -algebra completion of C E (A) for a norm that is bounded above by the I -norm as in Definition 3.7. Then the identity map on C E (A) extends to a surjective * -homomorphism
Proof Let I E (A) denote the completion of C E (A) for the I -norm. We will prove the lemma by showing that C * E (A) is the enveloping C * -algebra of the Banach * -algebra I E (A). Let U be a relatively compact open subset of G 0 over which A is trivialized, and A U the restriction of A to U , so A U is diffeomorphic to U × R d where d is the fiber dimension of A. Then it is straightforward to check that the closure of
inside I E (A) identifies isometrically via the diffeomorphism mentioned above with C 0 (U, L 1 (R d )). We will start by showing that the enveloping
. For this, it suffices to show that any (non-zero) irreducible * -representation
We first claim that there exists x ∈ U such that for every open set V containing x there exists f ∈ C 0 (U, L 1 (R d )) supported in V such that π( f ) = 0. Indeed, as π is non-zero and continuous for the I -norm there must exist
is supported in an element of U. However, as the support of g is compact, we may use a partition of unity to write g as a finite sum of functions supported in elements of U, which contradicts that g is non-zero.
Let then x ∈ U satisfy the condition in the claim above. We claim next that the kernel of π factors through the homomorphism x :
defined by evaluating at the fiber over x. If not, then there is some
As π is continuous for the I -norm, we may assume that there is a neighborhood V x such that f is zero when restricted to V . Let g ∈ C 0 (U, L 1 (R d )) be supported in V and such that π(g) = 0. As the supports of f and g are disjoint, the ideals they generate in C 0 (U, L 1 (R d )) are orthogonal. It is therefore impossible that both π( f ) and π(g) are non-zero by irreducibility of π; this contradiction establishes the clam.
To complete the proof that the enveloping algebra of
, note that we now know that π factors as a composition ρ • x , where ρ is an irreducible continuous * -representation of L 1 (R d ). The result follows as the enveloping C * -algebra of
, by definition of the latter algebra.
Continuing, let V be an open subset of G 0 given as a disjoint union V = i∈I U i where each U i is an open relatively compact subset of G 0 over which A is trivialized. Let A V denote the restriction of A to V , so A V is diffeomorphic to V × R n . Then the I -norm completion of
identifies with a Banach * -subalgebra, say B V , of the Banach * -algebra
of bounded sequences of elements of the different C 0 (U i , L 1 (R d )) with the norm given by the supremum over i ∈ I of the norms on the individual C 0 (U i , L 1 (R d )). Let π be any *representation of B V , and let π i denote the restriction of π to the i th factor. Then orthogonality of the factors implies that for any
by what we have already proven. Hence the enveloping C * -algebra of B V has norm dominated by the norm from
which is in turn clearly equal to the norm B V inherits as a * -subalgebra of C * E (A). To complete the proof, let f ∈ I E (A) be arbitrary, and π be a * -representation of I E (A). Say G 0 has dimension m. Then using paracompactness, we may write G 0 as a union V 0 ∪ • • • ∪ V m , where each V i has the same properties as V above. Using a partition of unity, we 123 may write f = m i=0 f i , where each f i is supported in V i , and
where the second inequality follows from what we showed above about the enveloping C *norm on each B V i . This shows every * -representation of I E (A) is continuous for the C * E (A) norm, completing the proof. Proposition 3.17 There is a canonical short exact sequence
(where ' ⊗' denotes the spatial tensor product of C * -algebras).
Proof Write I E (T G) for the completion of C E (T G) in the I -norm, and similarly write I E (G) for the completion of C E (G) in the I -norm, and I E (A) for the completion of C E (A) in the I -norm. Then it is straightforward to see that there is a short exact sequence
of Banach algebras, where the ideal is defined to consist of continuous functions from [0, 1] to I E (G) that vanish at 0. It is clear that the C * -algebra norm that the ideal C 0 ((0, 1],
and thus that the closure of the ideal C 0 ((0, 1],
where J is the kernel of the quotient map ev 0 :
, the left hand vertical map is an injection, and the right hand vertical map a surjection. Note that the right hand vertical map extends the identity on the dense * -subalgebra C E (A) of the algebras involved, by the existence of the diagram in line (5). As both
are C * -algebra completions for norms on C E (A) that are dominated the I -norm, the surjection must be an isomorphism by Lemma 3.16. Hence the left-hand vertical map is also an isomorphism, and we are done.
Using the K -theory exact sequence and Lemma 3.17, we conclude this subsection with the following corollary.
) induced on K -theory by the evaluation-at-zero map is an isomorphism.
We keep the notation from the previous section: G is a Lie groupoid with T G the associated tangent groupoid and A the associated Lie algebroid; moreover, A is equipped with a Euclidean metric making each t-fiber G x a complete Riemannian manifold, and all three of these objects are equipped with the associated metric, coarse, and Haar structures. We assume moreover that the Haar structure is of bounded geometry. Throughout this section, we also write A * for the dual vector bundle of A, and equip it with the Euclidean metric induced from that on A (the Euclidean metric can be used to identify A and A * , of course, but it is perhaps more natural to keep the distinction).
Our first task in this section is to identify the Roe C * -algebra C * E (A) with an algebra of functions on A * . Let C e (A * ) be the C * -algebra of continuous bounded functions from A * to C such that:
1. for each x ∈ G 0 , the restriction f x of f to the fiber A *
x over x vanishes at infinity; 2. the family { f x } x∈G 0 of restrictions is 'equicontinuous' in the sense that for any > 0 there exists
Proof For each x ∈ G 0 recall that A x denotes the fiber of A over x, and let
, f → F ( f ) denote the Fourier isomorphism. For each r > 0, let C E,r (A) denote the subspace of C E (A) consisting of functions supported in
Let C b (A * ) denote the C * -algebra of bounded continuous functions on A * . Then the formula
Moreover, the fact that an element on the left is supported in some A (r ) implies by standard Fourier theory that the family {F( f x )} is equicontinuous and vanishes at infinity, whence the image is actually in C e (A * ). Standard facts about the Fourier transform imply moreover that this map is isometric for the norm induced on the left hand side by C * E (A), and thus it extends to an injective * -homomorphism φ : C * E (A) → C e (A * ). It remains to show that the image of φ is dense.
For this, let f be an element of C e (A * ), which we write as a family { f x } x∈G 0 of restrictions to fibers. Let d be the fiber dimension of A, and let g be a smooth, positive function on R d with integral one, that only depends on the norms of elements, and that has compactly supported inverse Fourier transform. For δ > 0, define g δ (ξ ) = δ -d g(δξ ). Note that as each g δ depends only on the norm on R d , we may define a function g δ : A * → C by the formula (x, ξ) → g δ ( ξ x ) in the obvious sense, and that g δ thus defined is an element of C e (A * ). Now, note that for each δ > 0 we may define an element f * g δ of C e (A * ) by taking the fiberwise convolution. Equicontinuity of the family { f x } implies that lim δ→0 f * g δf C e (A * ) = 0.
Let now h δ : A → C be given fiberwise as the inverse Fourier transform of f * g δ , and note that each h δ is an element of C E (A) by compact support of the inverse Fourier transform of g (and hence of each g δ ). We then have
which shows that the image of φ is dense as claimed. Lemma 3.19 shows that C * E (A) is isomorphic to C e (A * ) and thus ev 0 * can be lifted to an isomorphism
Combining the above map with the evaluation map ev 1 * , we are finally able to define our analytic index map.
Let G be a groupoid equipped with the usual additional metric, coarse, and measure structures. The analytic index map is the homomorphism defined by
Throughout this section, G is a Lie groupoid and A the associated Lie algebroid. A is assumed equipped with a Euclidean metric, and G and A with an associated coarse structure and bounded geometry Haar system as in Proposition 2.7, Examples 2.8 and 3.2. Moreover, T G is the tangent groupoid of G, equipped with the corresponding coarse and measure structures as in Sect. 3.2. When G 0 is closed, it is well known (c.f. [24]) that elements of K 0 (C 0 (A * )) can be represented by smooth symbols, i.e. a pair of smooth vector bundles V 1 and V 2 over G 0 and a smooth endomorphism σ between their pullbacks to A * that is invertible at infinity. Generalizing this idea to K 0 (C e (A * )), in Definition 4.1, we introduce a special collection of elements of K 0 (C e (A * )) with particularly good representatives. Such a representative is called a smooth symbol. As G 0 is non-compact, we need an additional assumption on the boundedness of the derivatives of σ . In the rest of this section, we will study the behaviour of the analytic index map of Definition 3.20 on elements of K 0 (C e (A * )) with smooth symbols. We compute the analytic index of a smooth symbol using the idea of quantization of symbols. In particular, we construct a multiplier Q(σ ) of the tangent groupoid Roe C * -algebra using groupoid pseudodifferential operators. Under a natural bounded geometry assumption we prove in Theorem 4.8 and Corollary 4.9 that the analytic index of a smooth symbol (V 1 , V 2 , σ ) as defined in Definition 3.20 agrees with the index defined by Q(σ ). Throughout, we work for simplicity with the case of K 0 only, although it should certainly be possible to extend the results to K 1 .
In order to carry out our quantization program, we will need to work with a restricted class of cycles for K 0 (C e (A * )); we now describe some necessary preliminaries. Compare the discussion in [18,Sect. 4.2] for a treatment of similar geometric issues.
Recall that the Lie algebroid A is equipped with a smooth Euclidean metric { , x } x∈G 0 , and that for each x ∈ G 0 , the tangent space T G x to the t-fiber G x is equipped with the associated Riemannian structure as discussed in Sect. 2.1. Let t * : T G → T G 0 be the differential of the target map and define T t G := ker(t * ), a subbundle of T G. Let T * t G be the dual bundle to T t G; in other words, T * t G is the smooth bundle over G whose restriction to each t-fiber G x is the cotangent bundle T * G x . Note that G acts freely and properly on the right of T * t G, and the quotient identifies canonically with A * ; on the other hand, A * also identifies canonically with the restriction T * t G| G 0 of T * t G to the unit space. Let t : T * t G → G 0 be the map that takes each cotangent bundle T * G x to x, and note that t is a smooth submersion as t : G → G 0 is. Let F be the integrable subbundle of T T * t G corresponding to the regular foliation of T * t G by the fibers of t; in other words, each restriction
Let now W 1 , W 2 be smooth vector bundles over A * equipped with Euclidean metrics, and let σ : W 1 → W 2 be a bundle endomorphism. Using that the quotient of T * t G for the G-action is A * , we can lift this data to G-equivariant bundles and a G-equivariant bundle map σ : W 1 → W 2 over T * t G; note that the Euclidean metrics on W 1 and W 2 also lift to Euclidean metrics on W 1 and W 2 . Let (V ) denote the space of smooth sections of a vector bundle V . For each x, note that T * G x inherits a Riemannian structure from G x , and let ∇ x be the associated Levi-Civita connection, so in particular ∇ x defines a map
induced by combining all the maps ∇ x . We use ∇ k σ to denote the kth-order directional derivatives of the symbol σ , which is a bundle map
Let S(⊗ k F ) denote the unit sphere bundle of ⊗ k F , and for m ∈ N define
where the norm on Hom( W 1 , W 2 ) is induced by the G-invariant Euclidean metrics on W 1 and W 2 .
We now define the symbol data we will use. Let A * E denote the Gelfand spectrum of the commutative C * -algebra C e (A * ), which is a locally compact Hausdorff space. Note that C e (A * ) contains C 0 (A * ) as an essential ideal, and therefore A * E contains A * as a dense open subset. Let π A * : A * → G 0 denote the bundle projection for A * . Abusing notation, if V is a bundle over G 0 , we will write V for the G-equivariant bundle π * A * V over T * t G defined above.
σ ) that satisfies the following conditions.
1. V 1 and V 2 are smooth complex vector bundles over G 0 such that the pullbacks π * A * V 1 and π * A * V 2 are restrictions of bundles over the one point compactification of A * E .
and τ are both bundle endomorphisms that come as restrictions of bundle endomorphisms on the one-point compactification of A * E . 3. Thinking of σ and τ as bundle endomorphisms on the one-point compactification of A * E , the compositions σ • τ and τ • σ are equal to the identity in some neighbourhood of the point at infinity. 4. The norms N m ( σ ) and N m ( τ ) are finite for all m.
We will need the following remarks about these symbol classes.
1. Using standard techniques in topological K -theory (compare for example [24, Proposition (A. I)] or the discussion in [2, pp. 491-492]), we see that any smooth symbol
)). We do not know if any class in K 0 (C e (A * )) can be represented by a smooth symbol, but this seems unlikely. 2. The choice of 'partial inverse' τ does not affect the resulting K -theory class. 3. As σ and τ are mutually inverse on some neighbourhood of the point at infinity, they are mutually inverse outside some neighbourhood of the zero section in A * which has compact closure in the one-point compactification of A * E . Replacing σ and τ with functions that are homogeneous of degree zero on the fibers of A * outside of this neighbourhood does not affect the resulting K -theory class (compare [2, pp. 491-492] again); we will sometimes assume homogeneity of this sort. 4. Let G 0 denote the closure of G 0 in the one-point compactification of A * E , so G 0 is some compact Hausdorff space. Writing V for V 1 or V 2 , we know that π * A * V is the restriction of some bundle on the one-point compactification of A * E , and therefore V = π * A * V | G 0 is the restriction of some bundle on the compact space G 0 . In particular, there is a (smooth) bundle W on G 0 with V ⊕ W trivializable.
) by the idea of quantization.
We will need to make the following assumptions on the family {G x } x∈G 0 of Riemannian manifolds, which will be in force throughout this subsection. As the Riemannian metrics on this family are determined by the Euclidean metric on A, they are (indirectly) just assumptions about the latter structure. Assumption 4.3 1. The associated metric spaces (G x , d x ) are proper for every x ∈ G 0 . 2. There is ι > 0 such that for all x ∈ G 0 , the injectivity radius of G x is bounded from below by ι.
x be the Riemannian curvature tensor associated to the Riemannian metric on G x . For any k ≥ 0, there is κ k > 0 such that kth-order directional derivatives of R ∇
x are bounded by κ k for all x ∈ G 0 , i.e.
(note that this implies via the discussion of Example 3.4 that the Haar system on G has bounded geometry in the sense of Definition 3.3).
The above Assumption 4.3 on uniformly bounded derivatives of the Riemannian curvature tensor implies that there is κ > 0 such that for all x ∈ G 0 , all sectional curvatures of G x are in [-κ, κ].
Choose R > 0 and r ∈ (0, ι) such that r + R < ι; these constants will be fixed throughout. Lemma 6.1 and Assumption 4.3 (more precisely the boundedness of the sectional curvature) imply that there is an integer m > 0 such that for each x ∈ G 0 there is an open cover U x of G x by balls of radius r such that for any U 0 ∈ U x there are at most m balls U ∈ U x with
For each x ∈ G 0 , consider the exponential map exp x : A x → G x with respect to the Riemannian metric , x . Choose R with ι > R > R > 0. We can find a smooth function
By the exponential map exp, χ lifts to a smooth function χ on G with the property that for any x ∈ G 0 , χ| G x is supported inside the R-ball centered at x in the t-fiber G x . Let (T G x ) denote the sections of T G x and let
be the usual directional derivative operator on the Riemannian manifold G x ; recalling that
be defined by combining the various ∇ x . Then if d is the dimension of G x (equivalently, the rank of A * ), we can use the fact that R and R are fixed constants and Assumption 4.3 on uniformly bounded derivatives of the Riemannian curvature tensor, to assume that ∇ k χ is uniformly bounded on the unit sphere S(⊗ k T t G) of T t G for k = 1, . . . , 2d + 1.
Given a smooth section ξ of A * , define e ξ : G → C by
By the definition of χ , e ξ is a smooth function on G such that for each x ∈ G 0 , the restriction
) be a smooth symbol as is defined in Definition 4.1. With the above preparation, we are ready to define the quantization Q λ (σ ). Recall first the definitions of the metric and measure structures on T G from Sect. 3.2, which depend on λ. We will write , x,λ for the Euclidean metric on the fiber A x,λ of the Lie algebroid of T G over (x, λ) ∈ G 0 ×[0, 1], and μ x,λ for the measure on the t-fiber T G λ,x . Note that this t-fiber T G x,λ is naturally a diffeomorphic copy of the corresponding t-fiber G x of G and we will often use this identification; however, the identification is not isometric (or volume preserving).
The quantization of a smooth symbol (V 1 , V 2 , σ ) is defined as follows. For i = 1, 2, let H x,λ i be the space of L 2 -sections of the bundle s * V i | G x on G x , defined with respect to the measure μ x,λ .
For a function f ∈ C E (G), define a bounded operator ρ x,λ i ( f
where ξ is in H x,λ i and g ∈ G x (the same proof as of Lemma 3.8 above shows that this is well-defined and bounded independently of x and λ).
2 by the formula
1 , dξ is the volume element on A * s(g),λ associated to the metric , x,λ on A s(g),λ , and e λ ξ is defined in the same way as Eq. ( 7) with the λ-scaled metric (we will see below that the formula does indeed define a bounded operator).
In the rest of this section, we will prove that the analytic index of (V 1 , V 2 , σ ) defined by Definition 3.20 agrees with the one defined by {Q x,1 (σ )} x∈G 0 . We achieve this in the following three steps.
1. The operators {Q x,λ (σ )} x∈G 0 ,0∈(0,1] are uniformly bounded for all λ and x. (Lemma 4.5, Proposition 4.6).
). (Theorem 4.8, Corollary 4.9).
To study the norm properties of the operators Q x,λ (σ ), we will need the following result from [26, Lemma 3.9].
where k = [d/2] + 1. Let N(k, k, d + 1)( p) be the norm of the function p defined as follows
In the above definition of N k,k,d+1 ( p), the term
Let Op λ ( p) be a linear operator on L 2 (R d ) defined by
The constant C only depends on the dimension d.
i.e. Q x,λ (σ ) is uniformly bounded for all λ and x. Proof Remark 4.2 Part (4) tells us that each of V 1 , V 2 is complemented; using these complements, we may assume that we are working with trivial bundles. For notational simplicity, however, we assume that V 1 and V 2 are the trivial line bundle on G 0 ; the straightforward generalization of the following proof to matrix valued versions of C e (A * ), C * E (T G), and C * E (G) leads to the result for general V 1 , V 2 ; we leave the details to the reader. Moreover, using Remark 4.2 Part (3) we may assume that σ is a C-valued function on A * that is homogeneous of degree zero outside some neighborhood of the zero section. Fix now
is the trivial line bundle, the Hilbert spaces H x,λ 1 and H x,λ 2 both identify with L 2 (G x , μ x,λ ).
According to its definition, Q x,λ (σ ) has the Schwartz kernel K(σ )(g, h) defined for g, h ∈ G x by
e λ -ξ (hg -1 )σ (s(g), ξ )dξ.
From its expression, we can see that Q x,λ (σ ) is a pseudodifferential operator on H with symbol σ (s(g), ξ ). As e λ ξ (-) is supported within the R-neighborhood of x in G x for the scaled metric, the kernel K(σ
Using Assumption 4.3 and Lemma 6.1, there is an open cover U x of G x with the properties listed there with respect to the constants r and R; note that U x is necessarily countable. Let {ϕ i } i∈N be a smooth partition of unity on G x subordinate to the cover U x . Let u ∈ L 2 (G x , μ x,λ ) and write u = i ϕ i u, where ϕ i u ∈ L 2 (G x , μ x,λ ). As 0 ≤ ϕ ≤ 1,
Therefore,
123
The conditions in Lemma 6.1 imply that given U i ∈ U x there are at most m other open sets U j in U x that intersects U i nontrivially. Accordingly, we compute
From the above estimate, we see that
Note that Q x,λ (σ ) is an order zero pseudodifferential operator on G x , and therefore is a bounded linear operator on L 2 (G x , μ x,λ ) by essentially the same argument as used in the proof of [12,Theorem 18.1.11]. Hence, for u, v ∈ L 2 (G x , μ x,λ ), we can compute Q x,λ (σ )(u) by
and so we have that Q
The inner product Q x,λ (σ ) (ϕ i u) , ϕ j v is non-zero only when the supports of Q x,λ (σ ) (ϕ i u) and ϕ j v have non-trivial intersection. We continue the above computation by
for any L > 0.
We have assumed that ∇ k χ is uniformly bounded on the unit sphere S(⊗ k T G x ) of ⊗ k T G x for 0 ≤ k ≤ 2d + 1. Furthermore, we have assumed that ∇ k σ is uniformly bounded on S(T T * G x ) for 0 ≤ k ≤ 2d +1. It is not hard to check from this that the function σ (g, h, ξ) := χ(hg -1 ) σ (s(g), ξ ) satisfies the assumption of Lemma 4.5, once we restrict its support to any suitably small ball diffeomorphic to Euclidean space. The support of ϕ i u is contained inside some ball U i of radius r with center z i . As the function χ(gh -1 ) is zero if d(g, h) ≥ R, the support of Q x,λ (σ )(ϕ i u) is inside the geodesic ball centered at z i with radius at most r + R < ι. We can then use Lemma 4.5 (plus a change of variables to take into account the dependence of the metric and volume on λ) to conclude that
where M is an absolute constant, depending only on the size of ∇ k σ and ∇ k χ , and the kth-order derivatives of the Riemannian curvature tensor and dimension of G x (and not on λ). Hence we can bound L Q x,λ (σ )(ϕ i u) 2 + 1 L ϕ j v 2 from above by
By the assumption on the cover U x , the support supp(Q x,λ (σ )(ϕ i u)) (and supp(ϕ j v) ) intersects at most m of the supports supp(ϕ j v) (and supp(Q x,λ (σ )(ϕ i u))). We can continue the above estimate of
by the following inequalities
In summary, we have proved that for fixed λ,
where M and m are constants independent of λ, and L > 0. This implies by taking L = m,
and we are done.
Our next goal is to patch these operators Q x,λ together in a suitable sense to give a globally defined multiplier of C * E (T G). For notational simplicity, we will keep to the case where V 1 and V 2 are trivial, leaving the (minor) extra details necessary in the matricial case to the reader.
Define
and let
be the direct sum of the representations defining the norm on C E (T G) (see Definition 3.9), so ρ extends to a faithful representation of the Roe algebra C * E (T G). Let Q(σ ) be the operator on H which acts as Q x,λ (σ ) on each summand L 2 (G x , μ x,λ ), and by fiberwise convolution by the fiberwise Fourier transform F (σ x ) of σ on each L 2 (A x ). Combining Proposition 4.6 with basic estimates on the norms of the convolution operators, we see that Q(σ ) is a well-defined, bounded operator on H . Lemma 4. 7 We have following inclusions,
Proof For simplicity, we just look at the product Q(σ )ρ( f ) for f ∈ C E (T G); the other case is similar. We now start computing. First consider λ > 0, and let
e λ -ξ (hg -1 )σ (s(g), ξ )dξ denote the Schwartz kernel of Q x,λ (σ ); here the function e λ -ξ (-) is defined in the same way as the function e -ξ (-) in Eq. ( 7), but using the λ-scaled metric on A. Then the Schwartz kernel of the operator Q(σ )ρ( f ) acting on L 2 (G x , μ x,λ ) is given by
e λ -ξ (kg -1 )σ (s(g), ξ )dξ.
Set k = kh -1 , allowing us to rewrite the above integral as
e λ -ξ (k g -1 )σ (s(g), ξ )dξ.
At this point, we know that the restriction of Q(σ )ρ( f ) to each L 2 (G x , μ x,λ ) has Schwartz kernel (g, h) → λ (hg -1 ). We claim that λ actually belongs to C E (G), which follows from the next two observations. 1. As f | T G x,λ is supported in a compact set, the Fourier transform
is a function of ξ decaying rapidly at infinity. Therefore,
is a continuous (actually, smooth) function of g. 2. As f (k ) is supported within an S(> 0) tube of the unit space, when g is outside the S + R tube of the unit space, e λ -ξ (k g -1 ) f (k ) = 0. Therefore λ is supported within the S + R tube of G 0 .
Define now : T G → C by stipulating that the restriction λ of to the fiber over λ is given by
where F σ (x) is the fiberwise Fourier transform of σ (x, -). From our work so far, if can be shown to be continuous, then it is in C E (T G) and we will have Q(σ )ρ( f ) = ρ( ). The continuity of | G 0 ×(0,1] and G 0 ×{0} follow directly from the formulas and our work above, so we are left to check the compatibility of these two cases. Let {g λ } λ∈(0,1] be a family in T G such that g λ → (x, v) ∈ A as λ → 0, where A is identified with the fiber of T G over 0. More explicitly, this means that as λ → 0, g λ → x, and exp -1 t (g λ ) (g λ ) → v (it is important here that the exponential maps are defined with respect to the λ-scaled metrics). We want to show that λ (g λ ) converges to 0 (x, v). Indeed,
Write k = exp t (g λ ) (w), and let J be the Jacobian of this exponential exponential map. We can rewrite λ (g λ ) as
From the above estimate, we can conclude λ (g λ ) → 0 (x, v), λ → 0, completing the proof.
Proof Following the proofs of Proposition 4.6 and Lemma 4.7, we assume for notational simplicity that V 1 and V 2 are the trivial line bundle on G 0 and σ is a C-valued function on A * that is homogeneous of degree zero and invertible outside of some neighborhood of the zero section. Lemma 4.7 combined with boundedness of Q(σ ) shows that Q(σ ) identifies canonically with an element of the multiplier algebra of C * E (T G) (see for example [17,Sect. 3.12]). Now, say τ is the partial inverse to σ that appears in the definition of a smooth symbol. Consider the operators Q(σ ) • Q(τ ) -I and Q(τ ) • Q(σ ) -I . Using the symbolic calculus of pseudodifferential operator theory, c.f. [12, Theorem 18.1.23], [18], given any x ∈ G 0 and λ ∈ (0, 1], Q (x,λ) (σ ) • Q (x,λ) (τ ) -I is a pseudodifferential operator on G x of order 0 with principal symbol the appropriate restriction of σ τ -1. By Definition 4.1, σ τ -1 is the Fourier transform of a function supported in some tube around the zero section of uniform width. From this, together with similar (and simpler) arguments when λ = 0, we can derive that both Q
σ ] follows easily from the definition of Q(σ ) and of the latter class.
Observe from Theorem 4.8 that at λ
, having identified G with the fiber of T G for λ = 1. We conclude this section with the following corollary, which says that the analytic index of a [V 1 , V 2 , σ ] with smooth symbol is given by the (higher, or K -theoretic) index of the associated pseudodifferential operator in the classical sense.
Remark 4.10 When the unit space G 0 is closed, i.e. compact without boundary, Assumption 4.3 holds automatically. Furthermore, every element in K 0 (C e (A * )) has a representative that is a smooth symbol as in Definition 4.1. Therefore, Corollary 4.9 applies to equate the analytic index of a K -theory class σ of A * with the index of its quantization Q(σ ). In this section, we have extended this result to groupoids with noncompact unit space under Assumption 4.3 on bounded geometry.
In this section, we apply the theory developed above to study (regular) foliations with leafwise positive scalar metrics. We will discuss some specific instances of regular foliations; nonetheless, we develop the theory in a more general setting than this as the extra generality does not cause any extra difficulties. We assume throughout that G is an s-connected Lie groupoid with a Lie algebroid A, and that A is equipped with a Euclidean metric g A that satisfies Assumption 4.3. We first consider two natural examples with this structure that arise from regular foliations on closed manifolds.
Let M be a closed (compact without boundary) manifold with a regular foliation F . In this subsection, we discuss two examples of groupoids naturally associated to F with Riemannian metrics on the Lie algebroids satisfying the above properties.
1. Let π : M → M be the universal covering space. As π is a local diffeomorphism, F lifts to a regular foliation F on M. Let H F and H F be the holonomy groupoids on M and M defined by holonomy equivalent paths along leaves of F and F . Let g be a Riemannian metric on M. Accordingly, g induces Euclidean metrics g F and g F on F and F . By Proposition 2.10, H F is equipped with a right-invariant system of metrics {d x } x∈ M . As M is compact, on every leaf of F and also of F , directional derivatives of the Riemannian curvature tensor of any order are uniformly bounded and injectivity radii are uniformly bounded away from zero. For every x ∈ M, by the right invariance of the metrics on H F , H x F is an isometric covering space of the leaf L x of F containing x. As leaves of F have Riemannian curvature tensors with uniformly bounded directional derivatives and injectivity radii uniformly bounded below, (H F , {d x }) x∈ M satisfies Assumption 4.3. 2. Let N F = T M/F be the normal bundle of F in T M. Define a fiber bundle π : M → M by stipulating that the fiber π -1 (x) over x ∈ M is the space of euclidean metrics on the normal space N F | x . The normal bundle N F is equipped with the Bott connection, which is a flat connection along the direction of the foliation F . The Bott connection induces a partial Ehresmann connection on the fiber bundle M along the direction of F . As the Bott connection is flat along F , the Ehresmann connection on the fiber bundle M defines a foliation F on M, where every leaf of the foliation F is a covering space of the corresponding leaf of F on M with π the covering map.
Choose a Riemannian metric g on M, which induces Riemannian metrics g F and g F on F and F . The leaves of F have uniformly bounded directional derivative (of any order) Riemannian curvature tensor and injectivity radii uniformly bounded from below.
Since the restriction of π to each leaf of F is a covering map, the leaves of F also have uniformly bounded directional derivative Riemannian curvature tensor and injectivity radii uniformly bounded from below. Let H F be the holonomy groupoid of the regular foliation F on M. For every x ∈ M, each t-fiber H x F of H F is an isometric covering space of the corresponding leaf L x on M. Therefore, the t-fibers of H F have Riemannian curvature tensors with uniformly bounded directional derivatives and injectivity radii uniformly bounded below.
In either case, as the leaves of F are all complete, the leaves of F and F , which are covering spaces of the leaves of F , are also complete.
In this subsection, we study the analytic index of a spin Dirac operator / D. Let g A be a metric on the Lie algebroid A. The right translation of the groupoid G on itself is proper and free. The right translation of the metric g A defines a G-invariant Riemannian metric on each t-fiber G x for x ∈ X . The Levi-Civita connection ∇ on each G x is invariant under the G-action. The associated curvature of ∇ is also G-invariant. Therefore, the scalar curvature k is a smooth function on G and G-invariant. Therefore the scalar curvature k is the pullback of a smooth function on X .
We now restrict attention to a Lie groupoid G whose t-fibers {G x } are even dimensional and equipped with G-invariant spin structure. Moreover, we assume that there is a constant > 0 satisfying k g A (x) > for every x ∈ G 0 . Let S ± x be the even and odd parts of the spinor bundles on G x for x ∈ X (the spinor bundle splits into even and odd parts by our even-dimensionality assumption). On each G x , consider the leafwise spin Dirac operator / D +
x : (S + x ) → (S - x ) and its (formal) adjoint / D - x : (S - x ) → (S + x ). The Lichnerowicz formula combined with our assumption on scalar curvature implies that
for all x ∈ X . It follows that ( / D + x / D - x ) -1/2 makes sense, and that P + x := / D + x ( / D - x / D + x ) -1/2 is an invertible order zero operator from the Hilbert space completion H x + of S x + to the Hilbert space completion H x -of S x -. Note that all this structure is G-invariant, and therefore in particular S + and S -are pullbacks of bundles on X . Abusing notation, write σ ( / D + x ) for the symbol of P + x . It is straightforward to check using the pseudodifferential calculus, the standard description of the symbol of the Dirac operator and Assumption 4. (where we abuse notation by omitting the bundles involved).
Proof As was explained before, if F is equipped with a spin structure with positive scalar curvature k ≥ > 0, Assumptions 4.3 holds on H F and H F . We conclude from the above discussion that all the assumptions of Theorem 5.1 hold for both / D + and / D + , so the desired vanishing property follows from Theorem 5.1.
If G is a Lie groupoid, the family should be assumed smooth: just replace continuous functions by smooth functions everywhere.
It is a norm rather than a semi-norm as the measures μ x have full support.
AppendixIn this Appendix, we prove the following lemma. Such a result is known to experts, but we could not find a suitable reference so include a proof here for the reader's convenience.