Non-Hausdorff Groupoids,

Proper Actions and K-Theory

Jean-Louis Tu

Received: March 29, 2004 Revised: June 21, 2004

Communicated by Joachim Cuntz

Abstract. Let G be a (not necessarily Hausdorff) locally compact groupoid. We introduce a notion of properness for G, which is invariant under Morita-equivalence. We show that any generalized morphism between two locally compact groupoids which satisfies some properness conditions induces a C^{∗}-correspondence from C_{r}^{∗}(G_{2}) to C_{r}^{∗}(G_{1}), and thus two Morita equivalent groupoids have Moritaequivalent C^{∗}-algebras.

2000 Mathematics Subject Classification: 22A22 (Primary); 46L05, 46L80, 54D35 (Secondary). Keywords and Phrases: groupoid, C^{∗}-algebra, K-theory.

Introduction

Very often, groupoids that appear in geometry, such as holonomy groupoids of foliations, groupoids of inverse semigroups [15, 6] and the indicial algebra of a manifold with corners [10] are not Hausdorff. It is thus necessary to extend various basic notions to this broader setting, such as proper action and Morita equivalence. We also show that a generalized morphism from G_{2} to G_{1} satisfying certain properness conditions induces an element of KK(C_{r}^{∗}(G_{2}), C_{r}^{∗}(G_{1})).

In Section 2, we introduce the notion of proper groupoids and show that it is invariant under Morita-equivalence. Section 3 is a technical part of the paper in which from every locally compact topological space X is canonically constructed a locally compact Hausdorff space HX in which X is (not continuously) embedded. When G is a groupoid (locally compact, with Haar system, such that G^{(0)} is Hausdorff), the closure X′ of G^{(0)} in HG is endowed with a continuous action of G and plays an important technical rôle. In Section 4 we review basic properties of locally compact groupoids with Haar system and technical tools that are used later.

In Section 5 we construct, using tools of Section 3, a canonical C_{r}^{∗}(G)-Hilbert module E(G) for every (locally compact...) proper groupoid G. If G^{(0)}/G is compact, then there exists a projection p ∈ C_{r}^{∗}(G) such that E(G) is isomorphic to pC_{r}^{∗}(G). The projection p is given by p(g) = (c(s(g))c(r(g)))^{1/2}, where c: G^{(0)} → ℝ_{+} is a “cutoff” function (Section 6). Contrary to the Hausdorff case, the function c is not continuous, but it is the restriction to G^{(0)} of a continuous map X′ → ℝ_{+} (see above for the definition of X′). In Section 7, we examine the question of naturality G ↦ C_{r}^{∗}(G). Recall that if f: X → Y is a continuous map between two locally compact spaces, then f induces a map from C_{0}(Y) to C_{0}(X) if and only if f is proper. When G_{1} and G_{2} are groups, a morphism f: G_{1} → G_{2} does not induce a map C_{r}^{∗}(G_{2}) → C_{r}^{∗}(G_{1}) (when G_{1} ⊂ G_{2} is an inclusion of discrete groups there is a map in the other direction). When f: G_{1} → G_{2} is a groupoid morphism, we cannot expect to get more than a C^{∗}-correspondence from C_{r}^{∗}(G_{2}) to C_{r}^{∗}(G_{1}) when f satisfies certain properness assumptions: this was done in the Hausdorff situation by Macho-Stadler and O’Uchi ([11, Theorem 2.1], see also [7, 13, 17]), but the formulation of their theorem is somewhat complicated. In this paper, as a corollary of Theorem 7.8, we get that (in the Hausdorff situation), if the restriction of f to (G_{1})_{K}^{K} is proper for each compact set K ⊂ (G_{1})^{(0)} then f induces a correspondence E_{f} from C_{r}^{∗}(G_{2}) to C_{r}^{∗}(G_{1}). In fact we construct a C^{∗}-correspondence out of any groupoid generalized morphism ([5, 9]) which satisfies some properness conditions. As a corollary, if G_{1} and G_{2} are Morita equivalent then C_{r}^{∗}(G_{1}) and C_{r}^{∗}(G_{2}) are Morita-equivalent C^{∗}-algebras.

Finally, let us add that our original motivation was to extend Baum, Connes and Higson’s construction of the assembly map μ to non-Hausdorff groupoids; however, we couldn’t prove μ to be an isomorphism in any non-trivial case.

1. Preliminaries

1.1. Groupoids. Throughout, we will assume that the reader is familiar with basic definitions about groupoids (see [16, 15]). If G is a groupoid, we denote by G^{(0)} its set of units and by r: G → G^{(0)} and s: G → G^{(0)} its range and source maps respectively. We will use notations such as G_{x} = s^{−1}(x), G^{y} = r^{−1}(y), G_{x}^{y} = G_{x} ∩ G^{y}. Recall that a topological groupoid is said to be étale if r (and s) are local homeomorphisms.

For all sets X, Y, T and all maps f: X → T and g: Y → T, we denote by X ×_{f,g} Y, or by X ×_{T} Y if there is no ambiguity, the set {(x, y) ∈ X ×Y | f(x) = g(y)}.

Recall that a (right) action of G on a set Z is given by (a) a (“momentum”) map p: Z → G^{(0)};

(b) a map Z ×_{p,r} G → Z, denoted by (z, g) ↦ zg

with the following properties:

(i) p(zg) = s(g) for all (z, g) ∈ Z ×_{p,r} G;

(ii) z(gh) = (zg)h whenever p(z) = r(g) and s(g) = r(h);

(iii) zp(z) = z for all z ∈ Z. Then the crossed-product Z ⋊ G is the subgroupoid of (Z × Z) × G consisting of elements (z, z′, g) such that z′ = zg. Since the map Z ⋊ G → Z × G given by (z, z′, g) ↦ (z, g) is injective, the groupoid Z ⋊ G can also be considered as a subspace of Z × G, and this is what we will do most of the time.

1.2. Locally compact spaces. A topological space X is said to be quasicompact if every open cover of X admits a finite sub-cover. A space is compact if it is quasi-compact and Hausdorff. Let us recall a few basic facts about locally compact spaces.

Definition 1.1. A topological space X is said to be locally compact if every point x ∈ X has a compact neighborhood.

In particular, X is locally Hausdorff, thus every singleton subset of X is closed. Moreover, the diagonal in X × X is locally closed.

Proposition 1.2. Let X be a locally compact space. Then every locally closed subspace of X is locally compact.

Recall that A ⊂ X is locally closed if for every a ∈ A, there exists a neighborhood V of a in X such that V ∩ A is closed in V. Then A is locally closed if and only if it is of the form U ∩ F, with U open and F closed.

Proposition 1.3. Let X be a locally compact space. The following are equivalent:

(i) there exists a sequence (K_{n}) of compact subspaces such that X = ∪_{n∈ℕ}K_{n};

(ii) there exists a sequence (K_{n}) of quasi-compact subspaces such that X =

∪_{n∈ℕ}K_{n}; (iii) there exists a sequence (K_{n}) of quasi-compact subspaces such that X =

∪_{n∈ℕ}K_{n} and K_{n} ⊂ K̊_{n+1} for all n ∈ ℕ.

Such a space will be called σ-compact.

Proof. (i) ⟹ (ii) is obvious. The implications (ii) ⟹ (iii) ⟹ (i) follow easily from the fact that for every quasi-compact subspace K, there exists a finite family (K_{i})_{i∈I} of compact sets such that K ⊂ ∪_{i∈I} K̊_{i}. □

1.3. Proper maps.

Proposition 1.4. [2, Théorème I.10.2.1] Let X and Y be two topological spaces, and f: X → Y a continuous map. The following are equivalent:

(i) For every topological space Z, f × Id_{Z}: X × Z → Y × Z is closed; (ii) f is closed and for every y ∈ Y, f^{−1}(y) is quasi-compact.

A map which satisfies the equivalent properties of Proposition 1.4 is said to be proper.

Proposition 1.5. [2, Proposition I.10.2.6] Let X and Y be two topological spaces and let f: X → Y be a proper map. Then for every quasi-compact subspace K of Y, f^{−1}(K) is quasi-compact.

Proposition 1.6. Let X and Y be two topological spaces and let f: X → Y be a continuous map. Suppose Y is locally compact, then the following are equivalent:

(i) f is proper; (ii) for every quasi-compact subspace K of Y, f^{−1}(K) is quasi-compact; (iii) for every compact subspace K of Y, f^{−1}(K) is quasi-compact; (iv) for every y ∈ Y, there exists a compact neighborhood K_{y} of y such that

f^{−1}(K_{y}) is quasi-compact.

Proof. (i) ⟹ (ii) follows from Proposition 1.5. (ii) ⟹ (iii) ⟹ (iv) are obvious. Let us show (iv) ⟹ (i). Since f^{−1}(y) is closed, it is clear that f^{−1}(y) is quasi-compact for all y ∈ Y. It remains to prove that for every closed subspace F ⊂ X, f(F) is closed. Let y ∈ \overline{f(F)}. Let A = f^{−1}(K_{y}). Then A ∩ F is quasi-compact, so f(A ∩ F) is quasi-compact. As f(A ∩ F) ⊂ K_{y}, it is closed in K_{y}, i.e. K_{y} ∩ \overline{f(A ∩ F)} = K_{y} ∩ f(A ∩ F). We thus have y ∈ K_{y} ∩ \overline{f(A ∩ F)} = K_{y} ∩ f(A ∩ F) ⊂ f(F). It follows that f(F) is closed. □

2. Proper groupoids and proper actions

2.1. Locally compact groupoids.

Definition 2.1. A topological groupoid G is said to be locally compact (resp. σ-compact) if it is locally compact (resp. σ-compact) as a topological space.

Remark 2.2. The definition of a locally compact groupoid in [15] corresponds to our definition of a locally compact, σ-compact groupoid with Haar system whose unit space is Hausdorff, thanks to Propositions 2.5 and 2.8.

Example 2.3. Let Γ be a discrete group, H a closed normal subgroup and let G be the bundle of groups over [0, 1] such that G_{0} = Γ and G_{t} = Γ/H for all t > 0. We endow G with the quotient topology of ([0, 1] × Γ) / ((0, 1] × H). Then G is a non-Hausdorff locally compact groupoid such that (t, γ̄) converges to (0, γh) as t → 0, for all γ ∈ Γ and h ∈ H.

Example 2.4. Let Γ be a discrete group acting on a locally compact Hausdorff space X, and let G = (X ×Γ)/ ∼, where (x, γ) and (x, γ′) are identified if their germs are equal, i.e. there exists a neighborhood V of x such that yγ = yγ′ for all y ∈ V. Then G is locally compact, since the open sets V_{γ} = {[(x, γ)]| x ∈ X} are homeomorphic to X and cover G. Suppose that X is a manifold, M is a manifold such that π_{1}(M) = Γ, M̃ is the universal cover of M and V = (X × M̃)/Γ, then V is foliated by {[x, m̃]| m̃ ∈ M̃} and G is the restriction to a transversal of the holonomy groupoid of the above foliation.

Proposition 2.5. If G is a locally compact groupoid, then G^{(0)} is locally closed in G, hence locally compact. If furthermore G is σ-compact, then G^{(0)} is σcompact.

Proof. Let Δ be the diagonal in G × G. Since G is locally Hausdorff, Δ is locally closed. Then G^{(0)} = (Id, r)^{−1}(Δ) is locally closed in G. Suppose that G = ∪_{n∈ℕ}K_{n} with K_{n} quasi-compact, then s(K_{n}) is quasicompact and G^{(0)} = ∪_{n∈ℕ}s(K_{n}). □

Proposition 2.6. Let Z a locally compact space and G be a locally compact groupoid acting on Z. Then the crossed-product Z ⋊ G is locally compact.

Proof. Let p: Z → G^{(0)} be the momentum map of the action of G. From Proposition 2.5, the diagonal Δ ⊂ G^{(0)} × G^{(0)} is locally closed in G^{(0)} × G^{(0)}, hence Z ⋊ G = (p, r)^{−1}(Δ) is locally closed in Z × G. □

Let T be a space. Recall that there is a groupoid T × T with unit space T, and product (x, y)(y, z) = (x, z). Let G be a groupoid and T be a space. Let f: T → G^{(0)}, and let G[T] = {(t′, t, g) ∈ (T ×T)×G| g ∈ G_{f(t)}^{f(t′)}}. Then G[T] is a subgroupoid of (T ×T)×G.

Proposition 2.7. Let G be a topological groupoid with G^{(0)} locally Hausdorff, T a topological space and f: T → G^{(0)} a continuous map. Then G[T] is a locally closed subgroupoid of (T × T) × G. In particular, if T and G are locally compact, then G[T] is locally compact.

Proof. Let F ⊂ T ×G^{(0)} be the graph of f. Then F = (f ×Id)^{−1}(Δ), where Δ is the diagonal in G^{(0)} × G^{(0)}, thus it is locally closed. Let ρ: (t′, t, g) ↦ (t′, r(g)) and σ: (t′, t, g) ↦ (t, s(g)) be the range and source maps of (T × T) × G, then G[T] = (ρ, σ)^{−1}(F × F) is locally closed. □

Proposition 2.8. Let G be a locally compact groupoid such that G^{(0)} is Hausdorff. Then for every x ∈ G^{(0)}, G_{x} is Hausdorff.

Proof. Let Z = {(g, h) ∈ G_{x} × G_{x}| r(g) = r(h)}. Let φ: Z → G defined by φ(g, h) = g^{−1}h. Since {x} is closed in G, φ^{−1}(x) is closed in Z, and since G^{(0)} is Hausdorff, Z is closed in G_{x} × G_{x}. It follows that φ^{−1}(x), which is the diagonal of G_{x} × G_{x}, is closed in G_{x} × G_{x}. □

2.2. Proper groupoids.

Definition 2.9. A topological groupoid G is said to be proper if (r, s): G → G^{(0)} × G^{(0)} is proper.

Proposition 2.10. Let G be a topological groupoid such that G^{(0)} is locally compact. Consider the following assertions:

(i) G is proper; (ii) (r, s) is closed and for every x ∈ G^{(0)}, G_{x}^{x} is quasi-compact; (iii) for all quasi-compact subspaces K and L of G^{(0)}, G_{K}^{L} is quasi-compact; (iii)’ for all compact subspaces K and L of G^{(0)}, G_{K}^{L} is quasi-compact; (iv) for every quasi-compact subspace K of G^{(0)}, G_{K}^{K} is quasi-compact;

(v) ∀x, y ∈ G^{(0)}, ∃K_{x}, L_{y} compact neighborhoods of x and y such that G_{K_{x}}^{L_{y}} is quasi-compact.

Then (i) ⟺ (ii) ⟺ (iii) ⟺ (iii)’ ⟺ (v) ⟹ (iv). If G^{(0)} is Hausdorff, then (i)–(v) are equivalent.

Proof. (i) ⟺ (ii) follows from Proposition 1.4, and from the fact that G_{x}^{x} is homeomorphic to G_{x}^{y} if G_{x}^{y} ≠ ∅. (i) ⟹ (iii) and (v) ⟹ (i) follow Proposition 1.6 and the formula G_{K}^{L} = (r, s)^{−1}(L × K). (iii) ⟹ (iii)’ ⟹ (v) and (iii) ⟹ (iv) are obvious. If G^{(0)} is Hausdorff, then (iv) ⟹ (v) is obvious. □

Note that if G = G^{(0)} is a non-Hausdorff topological space, then G is not proper (since (r, s) is not closed), but satisfies property (iv).

Proposition 2.11. Let G be a topological groupoid. If r: G → G^{(0)} is open then the canonical mapping π: G^{(0)} → G^{(0)}/G is open.

Proof. Let V ⊂ G^{(0)} be an open subspace. If r is open, then r(s^{−1}(V)) = π^{−1}(π(V)) is open. Therefore, π(V) is open. □

Proposition 2.12. Let G be a topological groupoid such that G^{(0)} is locally compact and r: G → G^{(0)} is open. Suppose that (r, s)(G) is locally closed in G^{(0)} × G^{(0)}, then G^{(0)}/G is locally compact. Furthermore,

(a) if G^{(0)} is σ-compact, then G^{(0)}/G is σ-compact;

(b) if (r, s)(G) is closed (for instance if G is proper), then G^{(0)}/G is Hausdorff.

Proof. Let R = (r, s)(G). Let π: G^{(0)} → G^{(0)}/G be the canonical mapping. By Proposition 2.11, π is open, therefore G^{(0)}/G is locally quasi-compact. Let us show that it is locally Hausdorff. Let V be an open subspace of G^{(0)} such that (V × V) ∩ R is closed in V × V. Let Δ be the diagonal in π(V) × π(V). Then (π×π)^{−1}(Δ) = (V ×V)∩R is closed in V ×V. Since π×π: V ×V → π(V)×π(V) is continuous open surjective, it follows that Δ is closed in π(V) × π(V), hence π(V) is Hausdorff. This completes the proof that G^{(0)}/G is locally compact and of assertion (b). Assertion (a) follows from the fact that for every x ∈ G^{(0)} and every compact neighborhood K of x, π(K) is a quasi-compact neighborhood of π(x). □

2.3. Proper actions.

Definition 2.13. Let G be a topological groupoid. Let Z be a topological space endowed with an action of G. Then the action is said to be proper if Z ⋊ G is a proper groupoid. (We will also say that Z is a proper G-space.)

A subspace A of a topological space X is said to be relatively compact (resp. relatively quasi-compact) if it is included in a compact (resp. quasi-compact) subspace of X. This does not imply that \overline{A} is compact (resp. quasi-compact).

Proposition 2.14. Let G be a topological groupoid. Let Z be a topological space endowed with an action of G. Consider the following assertions:

(i) G acts properly on Z; (ii) (r, s): Z ⋊ G → Z × Z is closed and ∀z ∈ Z, the stabilizer of z is

quasi-compact; (iii) for all quasi-compact subspaces K and L of Z, {g ∈ G| Lg ∩ K ≠ ∅} is

quasi-compact; (iii)’ for all compact subspaces K and L of Z, {g ∈ G| Lg ∩ K ≠ ∅} is

quasi-compact; (iv) for every quasi-compact subspace K of Z, {g ∈ G| Kg ∩ K ≠ ∅} is

quasi-compact;

(v) there exists a family (A_{i})_{i∈I} of subspaces of Z such that Z = ∪_{i∈I} Å_{i} and {g ∈ G| A_{i}g ∩ A_{j} ≠ ∅} is relatively quasi-compact for all i, j ∈ I.

Then (i) ⟺ (ii) ⟹ (iii) ⟹ (iii)’ and (iii) ⟹ (iv). If Z is locally compact, then (iii)’ ⟹ (v) and (iv) ⟹ (v). If G^{(0)} is Hausdorff and Z is locally compact Hausdorff, then (i)–(v) are equivalent.

Proof. (i) ⟺ (ii) follows from Proposition 2.10[(i) ⟺ (ii)]. Implication (i) ⟹ (iii) follows from the fact that if (Z ⋊ G)_{K}^{L} is quasi-compact, then its image by the second projection Z ⋊ G → G is quasi-compact. (iii) ⟹ (iii)’ and (iii) ⟹ (iv) are obvious. Suppose that Z is locally compact. Take A_{i} ⊂ Z compact such that Z = ∪_{i∈I} Å_{i}. If (iii)’ is true, then {g ∈ G| A_{i}g ∩ A_{j} ≠ ∅} is quasi-compact, hence (v). If (iv) is true, then {g ∈ G| A_{i}g ∩ A_{j} ≠ ∅} is a subset of the quasi-compact set {g ∈ G| Kg ∩ K ≠ ∅}, where K = A_{i} ∪ A_{j}, hence (v). Suppose that Z is locally compact Hausdorff and that G^{(0)} is Hausdorff. Let us show (v) ⟹ (ii). Let C_{ij} be a quasi-compact set such that {g ∈ G| A_{i}g ∩A_{j} ≠ ∅} ⊂ C_{ij}. Let z ∈ Z. Choose i ∈ I such that z ∈ A_{i}. Since Z and G^{(0)} are Hausdorff, stab(z) is a closed subspace of C_{ii}, therefore it is quasi-compact. It remains to prove that the map Φ: Z ×_{G^{(0)}} G → Z × Z given by Φ(z, g) = (z, zg) is closed. Let F ⊂ Z ×_{G^{(0)}} G be a closed subspace, and (z, z′) ∈ \overline{Φ(F)}. Choose i and j such that z ∈ Å_{i} and z′ ∈ Å_{j}. Then (z, z′) ∈ \overline{Φ(F) ∩ (A_{i} × A_{j})} ⊂ \overline{Φ(F ∩ (A_{i} ×_{G^{(0)}} C_{ij}))} ⊂ \overline{Φ(F ∩ (Z ×_{G^{(0)}} C_{ij}))}. There exists a net (z_{λ}, g_{λ}) ∈ F ∩ (Z ×_{G^{(0)}} C_{ij}) such that (z, z′) is a limit point of (z_{λ}, z_{λ}g_{λ}). Since C_{ij} is quasi-compact, after passing to a universal subnet we may assume that g_{λ} converges to an element g ∈ C_{ij}. Since G^{(0)} is Hausdorff, F ∩ (Z ×_{G^{(0)}} C_{ij}) is closed in Z × C_{ij}, so (z, g) is an element of F ∩ (Z ×_{G^{(0)}} C_{ij}). Using the fact that Z is Hausdorff and Φ is continuous, we obtain (z, z′) = Φ(z, g) ∈ Φ(F). □

Remark 2.15. It is possible to define a notion of slice-proper action which implies properness in the above sense. The two notions are equivalent in many cases [1, 3].

Proposition 2.16. Let G be a locally compact groupoid. Then G acts properly on itself if and only if G^{(0)} is Hausdorff. In particular, a locally compact space is proper if and only if it is Hausdorff.

Proof. It is clear from Proposition 2.10(ii) that G acts properly on itself if and only if the product φ: G^{(2)} → G × G is closed. Since φ factors through the homeomorphism G^{(2)} → G ×_{r,r} G, (g, h) ↦ (g, gh), G acts properly on itself if and only if G ×_{r,r} G is a closed subset of G × G. If G^{(0)} is Hausdorff, then clearly G ×_{r,r} G is closed in G × G. Conversely, if G^{(0)} is not Hausdorff, then there exists (x, y) ∈ G^{(0)} × G^{(0)} such that x ≠ y and (x, y) is in the closure of the diagonal of G^{(0)} × G^{(0)}. It follows that (x, y) is in the closure of G ×_{r,r} G, but (x, y) ∉ G ×_{r,r} G, therefore G ×_{r,r} G is not closed. □

2.4. Permanence properties.

Proposition 2.17. If G_{1} and G_{2} are proper topological groupoids, then G_{1}×G_{2} is proper.

Proof. Follows from the fact that the product of two proper maps is proper [2, Corollaire I.10.2.3]. □

Proposition 2.18. Let G_{1} and G_{2} be two topological groupoids such that G_{1}^{(0)} is Hausdorff and G_{2} is proper. Suppose that f: G_{1} → G_{2} is a proper morphism. Then G_{1} is proper.

Proof. Denote by r_{i} and s_{i} the range and source maps of G_{i} (i = 1, 2). Let f̄ be the map G_{1}^{(0)}×G_{1}^{(0)} → G_{2}^{(0)}×G_{2}^{(0)} induced from f. Since f̄∘(r_{1}, s_{1}) = (r_{2}, s_{2})∘f is proper and G_{1}^{(0)} is Hausdorff, it follows from [2, Proposition I.10.1.5] that (r_{1}, s_{1}) is proper. □

Proposition 2.19. Let G_{1} and G_{2} be two topological groupoids such that G_{1} is proper. Suppose that f: G_{1} → G_{2} is a surjective morphism such that the induced map f′: G_{1}^{(0)} → G_{2}^{(0)} is proper. Then G_{2} is proper.

Proof. Denote by r_{i} and s_{i} the range and source maps of G_{i} (i = 1, 2). Let F_{2} ⊂ G_{2} be a closed subspace, and F_{1} = f^{−1}(F_{2}). Since G_{1} is proper, (r_{1}, s_{1})(F_{1}) is closed, and since f′ × f′ is proper, (f′ × f′) ∘ (r_{1}, s_{1})(F_{1}) is closed. By surjectivity of f, we have (r_{2}, s_{2})(F_{2}) = (f′ × f′) ∘ (r_{1}, s_{1})(F_{1}). This proves that (r_{2}, s_{2}) is closed. Since for every topological space T, the assumptions of the proposition are also true for the morphism f × 1: G_{1} × T → G_{2} × T, the above shows that (r_{2}, s_{2}) × 1_{T} is closed. Therefore, (r_{2}, s_{2}) is proper. □

Proposition 2.20. Let G be a topological groupoid with G^{(0)} Hausdorff, acting on two spaces Y and Z. Suppose that the action of G on Z is proper, and that Y is Hausdorff. Then G acts properly on Y ×_{G^{(0)}} Z.

Proof. The groupoid (Y ×_{G^{(0)}} Z) ⋊ G is isomorphic to the subgroupoid Γ = {(y, y′, z, g) ∈ (Y × Y) × (Z ⋊ G)| p(y) = r(g), y′ = yg} of the proper groupoid (Y × Y) × (Z ⋊ G). Since Y and G^{(0)} are Hausdorff, Γ is closed in (Y × Y) × (Z ⋊ G), hence by Proposition 2.10(ii), (Y ×_{G^{(0)}} Z) ⋊ G is proper. □

Corollary 2.21. Let G be a proper topological groupoid with G^{(0)} Hausdorff. Then any action of G on a Hausdorff space is proper.

Proof. Follows from Proposition 2.20 with Z = G^{(0)}. □

Proposition 2.22. Let G be a topological groupoid and f: T → G^{(0)} be a continuous map.

(a) If G is proper, then G[T] is proper.

(ii) If G[T] is proper and f is open surjective, then G is proper.

Proof. Let us prove (a). Suppose first that T is a subspace of G^{(0)} and that f is the inclusion. Then G[T] = G_{T}^{T}. Since (r_{T} , s_{T}) is the restriction to (r, s)^{−1}(T × T) of (r, s), and (r, s) is proper, it follows that (r_{T} , s_{T}) is proper. In the general case, let Γ = (T ×T)×G and let T′ ⊂ T ×G^{(0)} be the graph of f. Then Γ is a proper groupoid (since it is the product of two proper groupoids), and G[T] = Γ[T′].

Let us prove (b). The only difficulty is to show that (r, s) is closed. Let F ⊂ G be a closed subspace and (y, x) ∈ \overline{(r, s)(F)}. Let F̃ = G[T] ∩ (T × T) × F. Choose (t′, t) ∈ T × T such that f(t′) = y and f(t) = x. Denote by r̃ and s̃ the range and source maps of G[T]. Then (t′, t) ∈ \overline{(r̃, s̃)(F̃)}. Indeed, let Ω ∋ (t′, t) be an open set, and Ω′ = (f × f)(Ω). Then Ω′ is an open neighborhood of (y, x), so Ω′ ∩ (r, s)(F) ≠ ∅. It follows that Ω ∩ (r̃, s̃)(F̃) ≠ ∅. We have proved that (t′, t) ∈ \overline{(r̃, s̃)(F̃)} = (r̃, s̃)(F̃), so (y, x) ∈ (r, s)(F). □

Corollary 2.23. Let G be a groupoid acting properly on a topological space Z, and let Z_{1} be a saturated subspace. Then G acts properly on Z_{1}.

Proof. Use the fact that Z_{1} ⋊ G = (Z ⋊ G)[Z_{1}]. □

2.5. Invariance by Morita-equivalence. In this section, we will only consider groupoids whose range maps are open. We thus need a stability lemma:

Lemma 2.24. Let G be a topological groupoid whose range map is open. Let Z be a G space and f: T → G^{(0)} be a continuous open map. Then the range maps for Z ⋊ G and G[T] are open.

To prove Lemma 2.24 we need a preliminary result:

Lemma 2.25. Let X, Y, T be topological spaces, g: Y → T an open map and f: X → T continuous. Let Z = X ×_{T} Y. Then the first projection pr_{1} : X ×_{T} Y → X is open.

Proof. Let Ω ⊂ Z open. There exists an open subspace Ω′ of X × Y such that Ω = Ω′ ∩ Z. Let Δ be the diagonal in X × X. One easily checks that (pr_{1}, pr_{1})(Ω) = (1 × f)^{−1}(1 × g)(Ω′) ∩ Δ, therefore (pr_{1}, pr_{1})(Ω) is open in Δ. This implies that pr_{1}(Ω) is open in X. □

Proof of Lemma 2.24. This is clear for Z ⋊ G = Z ×_{G^{(0)}} G using Lemma 2.25. For G[T], first use Lemma 2.25 to prove that T ×_{f,s} G ⟶^{pr_{2}} G is open. Since the range map is open by assumption, the composition T ×_{f,s} G ⟶^{pr_{2}} G ⟶^{r} G^{(0)} is open. Using again Lemma 2.25, G[T] ≃ T ×_{f,r∘pr_{2}} (T ×_{f,s} G) ⟶^{pr_{1}} T is open. □

In order to define the notion of Morita-equivalence for topological groupoids, we introduce some terminology:

Definition 2.26. Let G be a topological groupoid. Let T be a topological space and ρ: G^{(0)} → T be a G-invariant map. Then G is said to be ρ-proper if the map (r, s): G → G^{(0)} ×_{T} G^{(0)} is proper. If G acts on a space Z and ρ: Z → T is G-invariant, then the action is said to be ρ-proper if Z ⋊ G is ρ-proper.

It is clear that properness implies ρ-properness. There is a partial converse:

Proposition 2.27. Let G be a topological groupoid, T a topological space, ρ: G^{(0)} → T a G-invariant map. If G is ρ-proper and T is Hausdorff, then G is proper.

Proof. Since T is Hausdorff, G^{(0)} ×_{T} G^{(0)} is a closed subspace of G^{(0)} × G^{(0)}, therefore (r, s), being the composition of the two proper maps G → G^{(0)} ×_{T} G^{(0)} → G^{(0)} × G^{(0)}, is proper. □

Remark 2.28. When T is locally Hausdorff, one easily shows that G is ρ-proper iff for every Hausdorff open subspace V of T, G_{ρ^{−1}(V)}^{ρ^{−1}(V)} is proper.

Proposition 2.29. [14] Let G_{1} and G_{2} be two topological (resp. locally compact) groupoids. Let r_{i}, s_{i} (i = 1, 2) be the range and source maps of G_{i}, and suppose that r_{i} are open. The following are equivalent:

(i) there exist a topological (resp. locally compact) space T and f_{i}: T → G_{i}^{(0)} open surjective such that G_{1}[T] and G_{2}[T] are isomorphic; (ii) there exists a topological (resp. locally compact) space Z, two continuous maps ρ: Z → G_{1}^{(0)} and σ: Z → G_{2}^{(0)}, a left action of G_{1} on Z with momentum map ρ and a right action of G_{2} on Z with momentum map σ such that

(a) the actions commute and are free, the action of G_{2} is ρ-proper and the action of G_{1} is σ-proper;

(b) the natural maps Z/G_{2} → G_{1}^{(0)} and G_{1}\Z → G_{2}^{(0)} induced from ρ and σ are homeomorphisms.

Moreover, one may replace (b) by

(b)’ ρ and σ are open and induce bijections Z/G_{2} → G_{1}^{(0)} and G_{1}\Z → G_{2}^{(0)}. In (i), if T is locally compact then it may be assumed Hausdorff.

If G_{1} and G_{2} satisfy the equivalent conditions in Proposition 2.29, then they are said to be Morita-equivalent. Note that if G_{i}^{(0)} are Hausdorff, then by Proposition 2.27, one may replace “ρ-proper” and “σ-proper” by “proper”. To prove Proposition 2.29, we need preliminary lemmas:

Lemma 2.30. Let G be a topological groupoid. The following are equivalent: (i) r: G → G^{(0)} is open; (ii) for every G-space Z, the canonical mapping π: Z → Z/G is open.

Proof. To show (ii) ⟹ (i), take Z = G: the canonical mapping π: G → G/G is open. Therefore, for every open subspace U of G, r(U) = G^{(0)} ∩ π^{−1}(π(U)) is open. Let us show (i) ⟹ (ii). By Lemma 2.24, the range map r: Z ⋊G → Z is open. The conclusion follows from Proposition 2.11. □

Lemma 2.31. Let G be a topological groupoid such that the range map r: G → G^{(0)} is open. Let X be a topological space endowed with an action of G and T a topological space. Then the canonical map

f: (X × T)/G → (X/G) × T

is an isomorphism.

Proof. Let π: X → X/G and π′: X × T → (X × T)/G be the canonical mappings. Since π is open (Lemma 2.30), f ∘ π′ = π × 1 is open. Since π′ is continuous surjective, it follows that f is open. □

Lemma 2.32. Let G be a topological groupoid whose range map is open and f: Y → Z a proper, G-equivariant map between two G-spaces. Then the induced map f̄: Y/G → Z/G is proper.

Proof. We first show that f̄ is closed. Let π: Y → Y/G and π′: Z → Z/G be the canonical mappings. Let A ⊂ Y/G be a closed subspace. Since f is closed and π is continuous, (π′)^{−1}(f̄(A)) = f(π^{−1}(A)) is closed. Therefore, f̄(A) is closed. Applying this to f × 1, we see that for every topological space T, (Y × T)/G → (Z × T)/G is closed. By Lemma 2.31, f̄× 1_{T} is closed. □

Lemma 2.33. Let G_{2} and G_{3} be topological groupoids whose range maps are open. Let Z_{1}, Z_{2} and X be topological spaces. Suppose there are maps

X ⟵^{ρ_{1}} Z_{1} ⟶^{σ_{1}} G_{2}^{(0)} ⟵^{ρ_{2}} Z_{2} ⟶^{σ_{2}} G_{3}^{(0)},

a right action of G_{2} on Z_{1} with momentum map σ_{1}, such that ρ_{1} is G_{2}-invariant and the action of G_{2} is ρ_{1}-proper, a left action of G_{2} on Z_{2} with momentum map ρ_{2} and a right ρ_{2}-proper action of G_{3} on Z_{2} with momentum map σ_{2} which commutes with the G_{2}-action. Then the action of G_{3} on Z = Z_{1} ×_{G_{2}} Z_{2} is ρ_{1}-proper.

Proof. Let φ: Z_{2} ⋊ G_{3} → Z_{2} ×_{G_{2}^{(0)}} Z_{2} be the map (z_{2}, γ) ↦ (z_{2}, z_{2}γ). By assumption, φ is proper, therefore 1_{Z_{1}} × φ is proper. Let F = {(z_{1}, z_{2}, z′_{2}) ∈ Z_{1} × Z_{2} × Z_{2}| σ_{1}(z_{1}) = ρ_{2}(z_{2}) = ρ_{2}(z′_{2})}. Then 1_{Z_{1}} × φ: (1 × φ)^{−1}(F) → F is proper, i.e. Z_{1} ×_{G_{2}^{(0)}} (Z_{2} ⋊ G_{3}) → Z_{1} ×_{G_{2}^{(0)}} (Z_{2} ×_{G_{2}^{(0)}} Z_{2}) is proper. By Lemma 2.32, taking the quotient by G_{2}, we get that the map

α: Z ⋊ G_{3} → Z_{1} ×_{G_{2}} (Z_{2} ×_{G_{2}^{(0)}} Z_{2})

defined by (z_{1}, z_{2}, γ) ↦ (z_{1}, z_{2}, z_{2}γ) is proper. By assumption, the map Z_{1} ⋊ G_{2} → Z_{1} ×_{X} Z_{1} given by (z_{1}, g) ↦ (z_{1}, z_{1}g) is proper. Endow Z_{1} ⋊ G_{2} with the following right action of G_{2} × G_{2}: (z_{1}, g) · (g′, g′′) = (z_{1}g′, (g′)^{−1}gg′′). Using again Lemma 2.32, the map

β: Z_{1} ×_{G_{2}} (Z_{2} ×_{G_{2}^{(0)}} Z_{2}) = (Z_{1} ⋊ G_{2}) ×_{G_{2}×G_{2}} (Z_{2} × Z_{2})
→ (Z_{1} ×_{X} Z_{1}) ×_{G_{2}×G_{2}} (Z_{2} × Z_{2}) ≃ Z ×_{X} Z

is proper. By composition, β ∘ α: Z ⋊ G_{3} → Z ×_{X} Z is proper. □

Proof of Proposition 2.29. Let us treat the case of topological groupoids. Assertion (b’) follows from the fact that the canonical mappings Z → Z/G_{2} and Z → G_{1}\Z are open (Lemma 2.30). Let us first show that (ii) is an equivalence relation. Reflexivity is clear (taking Z = G, ρ = r, σ = s), and symmetry is obvious. Suppose that (Z_{1}, ρ_{1}, σ_{2}) and (Z_{2}, ρ_{2}, σ_{2}) are equivalences between G_{1} and G_{2}, and G_{2} and G_{3} respectively. Let Z = Z_{1} ×_{G_{2}} Z_{2} be the quotient of Z_{1} ×_{G_{2}^{(0)}} Z_{2} by the action (z_{1}, z_{2}) · γ =

(z_{1}γ, γ^{−1}z_{2}) of G_{2}. Denote by ρ: Z → G_{1}^{(0)} and σ: Z → G_{3}^{(0)} the maps induced from ρ_{1} ×1 and 1×σ_{2}. By Lemma 2.25, the first projection pr_{1} : Z_{1} ×_{G_{2}^{(0)}} Z_{2} → Z_{1} is open, therefore ρ = ρ_{1} ∘ pr_{1} is open. Similarly, σ is open. It remains to show that the actions of G_{3} and G_{1} are ρ-proper and σ-proper respectively. For G_{3}, this follows from Lemma 2.33 and the proof for G_{1} is similar. This proves that (ii) is an equivalence relation. Now, let us prove that (i) and (ii) are equivalent. Suppose (ii). Let Γ = G_{1} ⋉ Z ⋊ G_{2} and T = Z. The maps ρ: T → G_{1}^{(0)} and σ: T → G_{2}^{(0)} are open surjective by assumption. Since G_{1} ⋉Z ≃ Z ×_{G_{2}^{(0)}} Z and Z ⋊G_{2} ≃ Z ×_{G_{1}^{(0)}} Z, we have G_{2}[T] = (T ×T)×_{G_{2}^{(0)}×G_{2}^{(0)}} G_{2} ≃ (Z ⋊G_{2})×_{s∘pr_{2},σ} Z ≃ (Z ×_{G_{1}^{(0)}} Z) ×_{σ∘pr_{2},σ} Z = Z ×_{G_{1}^{(0)}} (Z ×_{G_{2}^{(0)}} Z) ≃ Z ×_{G_{1}^{(0)}} (G_{1} ⋉ Z) ≃ G_{1} ⋉ (Z ×_{G_{1}^{(0)}} Z) ≃ G_{1} ⋉ (Z ⋊ G_{2}) = Γ. Similarly, Γ ≃ G_{1}[T], hence (i).

Conversely, to prove (i) ⟹ (ii) it suffices to show that if f: T → G^{(0)} is open surjective, then G and G[T] are equivalent in the sense (ii), since we know that (ii) is an equivalence relation. Let Z = T ×_{r,f} G. Let us check that the action of G is pr_{1}-proper. Write Z ⋊ G = {(t, g, h) ∈ T × G × G| f(t) = r(g) and s(g) = r(h)}. One needs to check that the map Z ⋊ G → (T ×_{f,r} G)^{2} defined by (t, g, h) ↦ (t, g, t, h) is a homeomorphism onto its image. This follows easily from the facts that the diagonal map T → T × T and the map G^{(2)} → G × G, (g, h) ↦ (g, gh) are homeomorphisms onto their images. Let us check that the action of G[T] is s∘pr_{2}-proper. One easily checks that the groupoid G′ = G[T] ⋉ (T ×_{f,r} G) is isomorphic to a subgroupoid of the trivial groupoid (T × T) × (G × G). It follows that if r′ and s′ denote the range and source maps of G′, the map (r′, s′) is a homeomorphism of G′ onto its image.

Let us now treat the case of locally compact groupoids. In the proof that (ii) is a transitive relation, it just remains to show that Z is locally compact. Let U_{3} be a Hausdorff open subspace of G_{3}^{(0)}. We show that σ^{−1}(U_{3}) is locally compact. Replacing G_{3} by (G_{3})_{U_{3}}^{U_{3}}, we may assume that G_{2} acts freely and properly on Z_{2}. Let Γ be the groupoid (Z_{1} ×_{G_{2}^{(0)}} Z_{2}) ⋊ G_{2}, and R = (r, s)(Γ) ⊂ (Z_{1} ×_{G_{2}^{(0)}} Z_{2})^{2}. Since the action of G_{2} on Z_{2} is free and proper, there exists a continuous map φ: Z_{2} ×_{G_{3}^{(0)}} Z_{2} → G_{2} such that z_{2} = φ(z_{2}, z′_{2})z′_{2}. Then R = {(z_{1}, z_{2}, z′_{1}, z′_{2}) ∈ (Z_{1} ×_{G_{2}^{(0)}} Z_{2})^{2}; z′_{1} = z_{1}φ(z_{2}, z′_{2})} is locally closed. By Proposition 2.12, Z = (Z_{1} ×_{G_{2}^{(0)}} Z_{2})/G is locally compact. Finally, if (i) holds with T = ∪_{i}V_{i} with V_{i} open Hausdorff, let T′ = ⨿V_{i}. It is clear that G_{1}[T′] ≃ G_{2}[T′]. □

Let us examine standard examples of Morita-equivalences:

Example 2.34. Let G be a topological groupoid whose range map is open. Let (U_{i})_{i∈I} be an open cover of G^{(0)} and U = ⨿_{i∈I}U_{i}. Then G[U] is Moritaequivalent to G.

Example 2.35. Let G be a topological groupoid, and let H_{1}, H_{2} be subgroupoids such that the range maps r_{i}: H_{i} → H_{i}^{(0)} are open. Then (H_{1}\G_{s(H_{2})}^{s(H_{1})})⋊H_{2} and

H_{1} ⋉ (G_{s(H_{2})}^{s(H_{1})}/H_{2}) are Morita-equivalent.

Proof. Take Z = G_{s(H_{2})}^{s(H_{1})} and let ρ: Z → Z/H_{2} and σ: H_{1}\Z be the canonical mappings. The fact that these maps are open follows from Lemma 2.30. □

The following proposition is an immediate consequence of Proposition 2.22.

Proposition 2.36. Let G and G′ be two topological groupoids such that the range maps of G and G′ are open. Suppose that G and G′ are Morita-equivalent. Then G is proper if and only if G′ is proper.

Corollary 2.37. With the notations of Example 2.34, G is proper if and only if G[U] is proper.

3. A topological construction

Let X be a locally compact space. Since X is not necessarily Hausdorff, a filter^{1} F on X may have more than one limit. Let S be the set of limits of a convergent filter F. The goal of this section is to construct a Hausdorff space HX in which X is (not continuously) embedded, and such that F converges to S in HX.

1or a net; we will use indifferently the two equivalent approaches

3.1. The space HX.

Lemma 3.1. Let X be a topological space, and S ⊂ X. The following are equivalent:

(i) for every family (V_{s})_{s∈S} of open sets such that s ∈ V_{s}, and V_{s} = X except perhaps for finitely many s’s, one has ∩_{s∈S}V_{s} ≠ ∅;

(ii) for every finite family (V_{i})_{i∈I} of open sets such that S ∩ V_{i} ≠ ∅ for all i, one has ∩_{i∈I}V_{i} ≠ ∅.

Proof. (i) ⟹ (ii): let (V_{i})_{i∈I} as in (ii). For all i, choose s(i) ∈ S ∩ V_{i}. Put W_{s} = ∩_{s=s(i)}V_{i}, with the convention that an empty intersection is X. Then by (i), ∅ ≠ ∩_{s∈S}W_{s} = ∩_{i∈I}V_{i}. (ii) ⟹ (i): let (V_{s})_{s∈S} as in (i), and let I = {s ∈ S| V_{s} ≠ X}. Then ∩_{s∈S}V_{s} = ∩_{i∈I}V_{i} ≠ ∅. □

We shall denote by HX the set of non-empty subspaces S of X which satisfy the equivalent conditions of Lemma 3.1, and ĤX = HX ∪ {∅}.

Lemma 3.2. Let X be a locally Hausdorff space. Then every S ∈ HX is locally finite. More precisely, if V is a Hausdorff open subspace of X, then V ∩ S has at most one element.

Proof. Suppose a ≠ b and {a, b} ⊂ V ∩S. Then there exist V_{a}, V_{b} open disjoint neighborhoods of a and b respectively; this contradicts Lemma 3.1(ii). □

Suppose that X is locally compact. We endow ĤX with a topology. Let us introduce the notations Ω_{V} = {S ∈ HX| V ∩ S ≠ ∅} and Ω^{Q} = {S ∈ HX| Q ∩ S = ∅}. The topology on ĤX is generated by the Ω_{V} ’s and Ω^{Q}’s (V open and Q quasi-compact). More explicitly, a set is open if and only if it is a union of sets of the form Ω_{(V_{i})_{i∈I}}^{Q} = Ω^{Q} ∩ (∩_{i∈I}Ω_{V_{i}}) where (V_{i})_{i∈I} is a finite family of open Hausdorff sets and Q is quasi-compact.

Proposition 3.3. For every locally compact space X, the space ĤX is Hausdorff.

Proof. Suppose S ⊄ S′ and S, S′ ∈ ĤX. Let s ∈ S − S′. Since S′ is locally finite and since every singleton subspace of X is closed, there exist V open and K compact such that s ∈ V ⊂ K and K ∩ S′ = ∅. Then Ω_{V} and Ω^{K} are disjoint neighborhoods of S and S′ respectively. □

For every filter F on ĤX, let

(1)  L(F) = {a ∈ X| ∀V ∋ a open, Ω_{V} ∈ F}.

Lemma 3.4. Let X be a locally compact space. Let F be a filter on ĤX. Then F converges to S ∈ ĤX if and only if properties (a) and (b) below hold:

(a) ∀V open, V ∩ S ≠ ∅ ⟹ Ω_{V} ∈ F;

(b) ∀Q quasi-compact, Q ∩ S = ∅ ⟹ Ω^{Q} ∈ F.

If F is convergent, then L(F) is its limit.

Proof. The first statement is obvious, since every open set in ĤX is a union of finite intersections of Ω_{V} ’s and Ω^{Q}’s. Let us prove the second statement. It is clear from (a) that S ⊂ L(F). Conversely, suppose there exists a ∈ L(F) − S. Since S is locally finite and every singleton subspace of X is closed, there exists a compact neighborhood K of a such that K ∩ S = ∅. Then a ∈ L(F) implies Ω_{K} ∈ F, and condition (b) implies Ω^{K} ∈ F, thus ∅ = Ω^{K} ∩ Ω_{K} ∈ F, which is impossible: we have proved the reverse inclusion L(F) ⊂ S. □

Remark 3.5. This means that if S_{λ} → S, then a ∈ S if and only if ∀λ there exists s_{λ} ∈ S_{λ} such that s_{λ} → a.

Example 3.6. Consider Example 2.3 with Γ = ℤ_{2} and H = {0}. Then HG = G ∪ {S} where S = {(0, 0), (0, 1)}. The sequence (1/n, 0) ∈ G converges to S in HG, and (0, 0) and (0, 1) are two isolated points in HG.

Proposition 3.7. Let X be a locally compact space and K ⊂ X quasi-compact. Then L = {S ∈ HX| S ∩K ≠ ∅} is compact. The space HX is locally compact, and it is σ-compact if X is σ-compact.

Proof. We show that L is compact, and the two remaining assertions follow easily. Let F be a ultrafilter on L. Let S_{0} = L(F). Let us show that S_{0}∩K ≠ ∅: for every S ∈ L, choose a point φ(S) ∈ K ∩ S. By quasi-compactness, φ(F) converges to a point a ∈ K, and it is not hard to see that a ∈ S_{0}. Let us show S_{0} ∈ HX: let (V_{s}) (s ∈ S_{0}) be a family of open subspaces of X such that s ∈ V_{s} for all s ∈ S_{0}, and V_{s} = X for every s ∉ S_{1} (S_{1} ⊂ S_{0} finite). By definition of S_{0}, Ω_{(V_{s})_{s∈S_{1}}} = ∩_{s∈S_{1}}Ω_{V_{s}} belongs to F, hence it is non-empty. Choose S ∈ Ω_{(V_{s})_{s∈S_{1}}} , then S ∩ V_{s} ≠ ∅ for all s ∈ S_{1}. By Lemma 3.1(ii), ∩_{s∈S_{1}}V_{s} ≠ ∅. This shows that S_{0} ∈ HX. Now, let us show that F converges to S_{0}.

• If V is open Hausdorff such that S_{0} ∈ Ω_{V}, then by definition Ω_{V} ∈ F.

• If Q is quasi-compact and S_{0} ∈ Ω^{Q}, then Ω^{Q} ∈ F, otherwise one would have {S ∈ HX| S ∩ Q ≠ ∅} ∈ F, which would imply as above that S_{0} ∩ Q ≠ ∅, a contradiction.

From Lemma 3.4, F converges to S_{0}. □

Proposition 3.8. Let X be a locally compact space. Then ĤX is the one-point compactification of HX.

Proof. It suffices to prove that ĤX is compact. The proof is almost the same as in Proposition 3.7. □

Remark 3.9. If f: X → Y is a continuous map from a locally compact space X to any Hausdorff space Y, then f induces a continuous map Hf: HX → Y. Indeed, for every open subspace V of Y, (Hf)^{−1}(V) = Ω_{f^{−1}(V)} is open.

Proposition 3.10. Let G be a topological groupoid such that G^{(0)} is Hausdorff, and r: G → G^{(0)} is open. Let Z be a locally compact space endowed with a continuous action of G. Then HZ is endowed with a continuous action of G which extends the one on Z.

Proof. Let p: Z → G^{(0)} such that G acts on Z with momentum map p. Since p has a continuous extension Hp: HZ → G^{(0)}, for all S ∈ HZ, there exists x ∈ G^{(0)} such that S ⊂ p^{−1}(x). For all g ∈ G^{x}, write Sg = {sg| s ∈ S}. Let us show that Sg ∈ HZ. Let V_{s} (s ∈ S) be open sets such that sg ∈ V_{s}. By continuity, there exist open sets W_{s} ∋ s and W_{g} ∋ g such that for all (z, h) ∈ W_{s} ×_{G^{(0)}} W_{g}, zh ∈ V_{s}. Let V′_{s} = W_{s} ∩ p^{−1}(r(W_{g})). Then V′_{s} is an open neighborhood of s, so there exists z ∈ ∩_{s∈S}V′_{s}. Since p(z) ∈ r(W_{g}), there exists h ∈ W_{g} such that p(z) = r(h). It follows that zh ∈ ∩_{s∈S}V_{s}. This shows that Sg ∈ HZ. Let us show that the action defined above is continuous. Let Φ: HZ ×_{G^{(0)}} G → HZ be the action of G on HZ. Suppose that (S_{λ}, g_{λ}) → (S, g) and let S′ = L((S_{λ}, g_{λ})). Then for all a ∈ S there exists s_{λ} ∈ S_{λ} such that s_{λ} → a. This implies s_{λ}g_{λ} → ag, thus ag ∈ S′. The converse may be proved in a similar fashion, hence Sg = S′. Applying this to any universal net (S_{λ}, g_{λ}) converging to (S, g) and knowing from Proposition 3.8 that Φ(S_{λ}, g_{λ}) is convergent in ĤZ, we find that Φ(S_{λ}, g_{λ}) converges to Φ(S, g). This shows that Φ is continuous in (S, g).

□

3.2. The space H′X. Let X be a locally compact space. Let Ω′_{V} = {S ∈ HX| S ⊂ V }. Let H′X be HX as a set, with the coarsest topology such that the identity map H′X → HX is continuous, and Ω′_{V} is open for every relatively quasi-compact open set V. The space H′X is Hausdorff since HX is Hausdorff, but it is usually not locally compact.

Lemma 3.11. Let X be a locally compact space. Then the map

H′X → ℕ^{∗} ∪ {∞}, S ↦ #S

is upper semi-continuous.

Proof. Let S ∈ H′X such that #S < ∞. Let V_{s} (s ∈ S) be open relatively compact Hausdorff sets such that s ∈ V_{s}, and let W = ∪_{s∈S}V_{s}. Then S′ ∈ H′X implies #(S′ ∩ V_{s}) ≤ 1, therefore S′ ∈ Ω′_{W} implies #S′ ≤ #S. □

Proposition 3.12. Let X be a locally compact space such that the closure of every quasi-compact subspace is quasi-compact. Then

(a) the natural map H′X → HX is a homeomorphism,

(b) for every compact subspace K ⊂ X, there exists C_{K} > 0 such that

∀S ∈ HX, S ∩ K ≠ ∅ ⟹ #S ≤ C_{K},

(c) If G is a locally compact proper groupoid with G^{(0)} Hausdorff then G satisfies the above properties.

Proof. To prove (b), let K_{1} be a quasi-compact neighborhood of K and let K′ = \overline{K}_{1}. Let a ∈ K ∩ S and suppose there exists b ∈ S − K′. Then K̊_{1} and X − K′ are disjoint neighborhoods of a and b respectively, which is impossible. We deduce that S ⊂ K′. Now, let (V_{i})_{i∈I} be a finite cover of K′ by open Hausdorff sets. For all b ∈ S, let I_{b} = {i ∈ I| b ∈ V_{i}}. By Lemma 3.2, the I_{b}’s (b ∈ S) are disjoint, whence one may take C_{K} = #I.

To prove (a), denote by Δ ⊂ X × X the diagonal. Let us first show that pr_{1} : \overline{Δ} → X × X is proper. Let K ⊂ X compact. Let L ⊂ X quasi-compact such that K ⊂ L̊. If (a, b) ∈ \overline{Δ} ∩ (K × X), then b ∈ \overline{L}: otherwise, L × L^{c} would be a neighborhood of (a, b) whose intersection with Δ is empty. Therefore, pr_{1}^{−1}(K) = \overline{Δ} ∩ (K × \overline{L}) is quasi-compact, which shows that pr_{1} is proper. It remains to prove that Ω′_{V} is open in HX for every relatively quasi-compact open set V ⊂ X. Let S ∈ Ω′_{V}, a ∈ S and K a compact neighborhood of a. Let L = pr_{2}(\overline{Δ} ∩ (K × X)). Then Q = L − V is quasi-compact, and S ∈ ΩQ ; K̊ ⊂ Ω′_{V}, therefore Ω′_{V} is a neighborhood of each of its points.

To prove (c), let K ⊂ G be a quasi-compact subspace. Then L = r(K) ∪ s(K) is quasi-compact, thus G_{L}^{L} is also quasi-compact. But \overline{K} is closed and \overline{K} ⊂ G_{L}^{L}, therefore \overline{K} is quasi-compact. □

4. Haar systems

4.1. The space C_{c}(X). For every locally compact space X, C_{c}(X)_{0} will denote the set of functions f ∈ C_{c}(V) (V open Hausdorff), extended by 0 outside V. Let C_{c}(X) be the linear span of C_{c}(X)_{0}. Note that functions in C_{c}(X) are not necessarily continuous.

Proposition 4.1. Let X be a locally compact space, and let f: X → ℂ. The following are equivalent:

(i) f ∈ C_{c}(X); (ii) f^{−1}(ℂ^{∗}) is relatively quasi-compact, and for every filter F on X, let F̃ = i(F), where i: X → HX is the canonical inclusion; if F̃ converges to S ∈ HX, then lim_{F} f = ∑_{s∈S} f(s).

Proof. Let us show (i) ⟹ (ii). By linearity, it is enough to consider the case f ∈ C_{c}(V), where V ⊂ X is open Hausdorff. Let K be the compact set \overline{f^{−1}(ℂ^{∗})} ∩ V. Then f^{−1}(ℂ^{∗}) ⊂ K. Let F and S as in (ii). If S ∩ V = ∅, then S ∈ Ω^{K}, hence Ω^{K} ∈ F̃, i.e. X − K ∈ F. Therefore, lim_{F} f = 0 = ∑_{s∈S} f(s). If S ∩ V = {a}, then a is a limit point of F, therefore lim_{F} f = f(a) = ∑_{s∈S} f(s). Let us show (ii) ⟹ (i) by induction on n ∈ ℕ^{∗} such that there exist V_{1}, … V_{n} open Hausdorff and K quasi-compact satisfying f^{−1}(ℂ^{∗}) ⊂ K ⊂ V_{1} ∪ ⋯ ∪ V_{n}.

For n = 1, for every x ∈ V_{1}, let F be a ultrafilter convergent to x. By Proposition 3.8, F̃ is convergent; let S be its limit, then lim_{F} f = ∑_{s∈S} f(s) = f(x), thus f_{|V_{1}} is continuous. Now assume the implication is true for n − 1 (n ≥ 2) and let us prove it for n. Since K is quasi-compact, there exist V′_{1}, …, V′_{n} open sets, K_{1} …, K_{n} compact such that K ⊂ V′_{1} ∪ ⋯ ∪ V′_{n} and V′_{i} ⊂ K_{i} ⊂ V_{i}. Let F = (V′_{1} ∪ ⋯ ∪ V′_{n}) − (V′_{1} ∪ ⋯ ∪ V′_{n−1}). Then F is closed in V′_{n} and f_{|F} is continuous. Moreover, f_{|F} = 0 outside K′ = K − (V′_{1} ∪ ⋯ ∪ V′_{n−1}) which is closed in K, hence quasicompact, and Hausdorff, since K′ ⊂ V′_{n}. Therefore, f_{|F} ∈ C_{c}(F). It follows that there exists an extension h ∈ C_{c}(V′_{n}) of f_{|F}. By considering f − h, we may assume that f = 0 on F, so f = 0 outside K′ = K_{1} ∪ ⋯ ∪ K_{n−1}. But K′ ⊂ V_{1} ∪ ⋯ ∪ V_{n−1}, hence by induction hypothesis, f ∈ C_{c}(X). □

Corollary 4.2. Let X be a locally compact space, f: X → ℂ, f_{n} ∈ C_{c}(X). Suppose that there exists fixed quasi-compact set Q ⊂ X such that f_{n}^{−1}(ℂ^{∗}) ⊂ Q for all n, and f_{n} converges uniformly to f. Then f ∈ C_{c}(X).

Lemma 4.3. Let X be a locally compact space. Let (U_{i})_{i∈I} be an open cover of X by Hausdorff subspaces. Then every f ∈ C_{c}(X) is a finite sum f = ∑ f_{i}, where f_{i} ∈ C_{c}(U_{i}).

Proof. See [6, Lemma 1.3]. □

Lemma 4.4. Let X and Y be locally compact spaces. Let f ∈ C_{c}(X × Y). Let V and W be open subspaces of X and Y such that f^{−1}(ℂ^{∗}) ⊂ Q ⊂ V × W for some quasi-compact set Q. Then there exists a sequence f_{n} ∈ C_{c}(V) ⊗ C_{c}(W) such that lim_{n→∞} ∥f − f_{n}∥_{∞} = 0.

Proof. We may assume that X = V and Y = W. Let (U_{i}) (resp. (V_{j})) be an open cover of X (resp. Y) by Hausdorff subspaces. Then every element of C_{c}(X ×Y) is a linear combination of elements of C_{c}(U_{i} ×V_{j}) (Lemma 4.3). The conclusion follows from the fact that the image of C_{c}(U_{i})⊗C_{c}(V_{j}) → C_{c}(U_{i}×V_{j}) is dense. □

Lemma 4.5. Let X be a locally compact space and Y ⊂ X a closed subspace. Then the restriction map C_{c}(X) → C_{c}(Y) is well-defined and surjective.

Proof. Let (U_{i})_{i∈I} be a cover of X by Hausdorff open subspaces. The map C_{c}(U_{i}) → C_{c}(U_{i} ∩ Y) is surjective (since Y is closed), and ⊕_{i∈I}C_{c}(U_{i} ∩ Y) → C_{c}(Y) is surjective (Lemma 4.3). Therefore, the map ⊕_{i∈I}C_{c}(U_{i}) → C_{c}(Y) is surjective. Since it is also the composition of the surjective map ⊕_{i∈I}C_{c}(U_{i}) → C_{c}(X) and of the restriction map C_{c}(X) → C_{c}(Y), the conclusion follows. □

4.2. Haar systems. Let G be a locally compact proper groupoid with Haar system (see definition below) such that G^{(0)} is Hausdorff. If G is Hausdorff, then C_{c}(G^{(0)}) is endowed with the C_{r}^{∗}(G)-valued scalar product ⟨ξ, η⟩(g) = \overline{ξ(r(g))}η(s(g)). Its completion is a C_{r}^{∗}(G)-Hilbert module. However, if G is not Hausdorff, the function g ↦ \overline{ξ(r(g))}η(s(g)) does not necessarily belong to C_{c}(G), therefore we need a different construction in order to obtain a C_{r}^{∗}(G)- module.

Definition 4.6. [16, pp. 16-17] Let G be a locally compact groupoid such that G^{x} is Hausdorff for every x ∈ G^{(0)}. A Haar system is a family of positive measures λ = {λ^{x}| x ∈ G^{(0)}} such that ∀x, y ∈ G^{(0)}, ∀φ ∈ C_{c}(G),

(i) supp(λ^{x}) = G^{x}; (ii) λ(φ): x ↦ ∫_{g∈G^{x}} φ(g) λ^{x}(dg) ∈ C_{c}(G^{(0)}); (iii) ∫_{h∈G^{x}} φ(gh) λ^{x}(dh) = ∫_{h∈G^{y}} φ(h) λ^{y}(dh).

Note that G^{x} is automatically Hausdorff if G^{(0)} is Hausdorff (Prop. 2.8). Recall also [15, p. 36] that the range map for G is open.

Lemma 4.7. Let G be a locally compact groupoid with Haar system. Then for every quasi-compact subspace K of G, sup_{x∈G^{(0)}} λ^{x}(K ∩ G^{x}) < ∞.

Proof. It is easy to show that there exists f ∈ C_{c}(G) such that 1_{K} ≤ f. Since sup_{x∈G^{(0)}} λ(f)(x) < ∞, the conclusion follows. □

Lemma 4.8. Let G be a locally compact groupoid with Haar system such that G^{(0)} is Hausdorff. Suppose that Z is a locally compact space and that p: Z → G^{(0)} is continuous. Then for every f ∈ C_{c}(Z ×_{p,r} G), λ(f): z ↦ ∫_{g∈G^{p(z)}} f(z, g) λ^{p(z)}(dg) belongs to C_{c}(Z).

Proof. By Lemma 4.5, f is the restriction of an element of C_{c}(Z × G). If f(z, g) = f_{1}(z)f_{2}(g), then ψ(x) = ∫_{g∈G^{x}} f_{2}(g) λ^{x}(dg) belongs to C_{c}(G^{(0)}), therefore ψ ∘ p ∈ C_{b}(Z). It follows that λ(f) = f_{1}(ψ ∘ p) belongs to C_{c}(Z). By linearity, if f ∈ C_{c}(Z) ⊗ C_{c}(G), then λ(f) ∈ C_{c}(Z). Now, for every f ∈ C_{c}(Z × G), there exist relatively quasi-compact open subspaces V and W of Z and G and a sequence f_{n} ∈ C_{c}(V) ⊗ C_{c}(W) such that f_{n} converges uniformly to f. From Lemma 4.7, λ(f_{n}) converges uniformly to λ(f), and λ(f_{n}) ∈ C_{c}(Z). From Corollary 4.2, λ(f) ∈ C_{c}(Z). □

Proposition 4.9. Let G be a locally compact groupoid with Haar system such that G^{(0)} is Hausdorff. If G acts on a locally compact space Z with momentum map p: Z → G^{(0)}, then (λ^{p(z)})_{z∈Z} is a Haar system on Z ⋊ G.

Proof. Results immediately from Lemma 4.8. □

5. The Hilbert module of a proper groupoid

5.1. The space X′. Before we construct a Hilbert module associated to a proper groupoid, we need some preliminaries. Let G be a locally compact groupoid such that G^{(0)} is Hausdorff. Denote by X′ the closure of G^{(0)} in HG.

Lemma 5.1. Let G be a locally compact groupoid such that G^{(0)} is Hausdorff. Then for all S ∈ X′, S is a subgroup of G.

Proof. Since r and s: G → G^{(0)} extend continuously to maps HG → G^{(0)}, and since r = s on G^{(0)}, one has Hr = Hs on X′, i.e. ∃x_{0} ∈ G^{(0)}, S ⊂ G_{x_{0}}^{x_{0}}. Let F be a filter on G^{(0)} whose limit is S. Then a ∈ S if and only if a is a limit point of F. Since for every x ∈ G^{(0)} we have x^{−1}x = x, it follows that for every a, b ∈ S one has a^{−1}b ∈ S, whence S is a subgroup of G_{x_{0}}^{x_{0}}. □

Denote by q: X′ → G^{(0)} the map such that S ⊂ G_{q(S)}^{q(S)}. The map q is continuous since it is the restriction to X′ of Hr.

Lemma 5.2. Let G be a locally compact proper groupoid such that G^{(0)} is Hausdorff. Let F be a filter on X′, convergent to S. Suppose that q(F) converges to S_{0} ∈ X′. Then S_{0} is a normal subgroup of S, and there exists Ω ∈ F such that ∀S′ ∈ Ω, S′ is group-isomorphic to S/S_{0}. In particular, {S′ ∈ X′| #S = #S_{0}#S′} ∈ F.

Proof. Using Proposition 3.12, we see that S is finite. We shall use the notation Ω̃_{(V_{i})_{i∈I}} = Ω_{(V_{i})_{i∈I}} ∩ Ω′_{∪_{i∈I}V_{i}}. Let V′_{s} ⊂ V_{s} (s ∈ S) be Hausdorff, open neighborhoods of s, chosen small enough so that for some Ω ∈ F,

(a) Ω ⊂ Ω̃_{(V′_{s})_{s∈S}};

(b) V′_{s_{1}}V′_{s_{2}} ⊂ V_{s_{1}s_{2}}, ∀s_{1}, s_{2} ∈ S.
(c) ∀s ∈ S − S_{0}, ∀S′ ∈ Ω, q(S′) ∉ V_{s};
(d) q(Ω) ⊂ Ω̃_{(V_{s})_{s∈S_{0}}} ;

Let S′ ∈ Ω. Let φ: S → S′ such that {φ(s)} = S′ ∩ V′_{s}. Then φ is well-defined since S′ ∩ V′_{s} ≠ ∅ (see (a)) and V′_{s} is Hausdorff. If s_{1}, s_{2} ∈ S then φ(s_{i}) ∈ S′ ∩ V′_{s_{i}}. By (b), φ(s_{1})φ(s_{2}) ∈ S′ ∩ V_{s_{1}s_{2}}. Since V_{s_{1}s_{2}} is Hausdorff and also contains φ(s_{1}s_{2}) ∈ S′, we have φ(s_{1}s_{2}) = φ(s_{1})φ(s_{2}). This shows that φ is a group morphism. The map φ is surjective, since S′ ⊂ ∪_{s∈S}V′_{s} (see (a)). By (c), ker(φ) ⊂ S_{0} and by (d), S_{0} ⊂ ker(φ). □

Suppose now that the range map r: G → G^{(0)} is open. Then X′ is endowed with an action of G (Prop. 3.10) defined by S · g = g^{−1}Sg = {g^{−1}sg| s ∈ S}.

5.2. Construction of the Hilbert module. Now, let G be a locally compact, proper groupoid. Assume that G is endowed with a Haar system, and that G^{(0)} is Hausdorff. Let

E^{0} = \{f ∈ C_{c}(X′)| f(S) = \sqrt{#S}f(q(S)) ∀S ∈ X′\}.

(q(S) ∈ G^{(0)} is identified to {q(S)} ∈ X′.) Define, for all ξ, η ∈ E^{0} and f ∈ C_{c}(G): ⟨ξ, η⟩(g) = \overline{ξ(r(g))}η(s(g)) and

(ξf)(S) = ∫_{g∈G^{q(S)}} ξ(g^{−1}Sg)f(g^{−1}) λ^{x}(dg).

Proposition 5.3. With the above assumptions, the completion E(G) of E^{0} with respect to the norm ∥ξ∥ = ∥⟨ξ, ξ⟩∥^{1/2} is a C_{r}^{∗}(G)-Hilbert module.

We won’t give the direct proof here since this is a particular case of Theorem 7.8 (see Example 7.7(c)).

6. Cutoff functions

If G is a locally compact Hausdorff proper groupoid with Haar system. Assume for simplicity that G^{(0)}/G is compact. Then there exists a so-called “cutoff” function c ∈ C_{c}(G^{(0)})_{+} such that for every x ∈ G^{(0)}, ∫_{g∈G^{x}} c(s(g)) λ^{x}(dg) = 1, and the function g ↦ \sqrt{c(r(g))c(s(g))} defines projection in C_{r}^{∗}(G). However, if G is not Hausdorff, then the above function does not belong to C_{c}(G) is general, thus we need another definition of a cutoff function.

Let X′_{≥k} = {S ∈ X′| #S ≥ k}. By Lemma 3.11, X′_{≥k} is closed.

Lemma 6.1. Let G be a locally compact, proper groupoid with G^{(0)} Hausdorff. Let X_{≥k} = q(X′_{≥k}). Then X_{≥k} is closed in G^{(0)}.

Proof. It suffices to show that for every compact subspace K of G^{(0)}, X_{≥k} ∩ K is closed. Let K′ = G_{K}^{K}. Then K′ is quasi-compact, and from Proposition 3.7, K′′ = {S ∈ HG| S ∩ K′ ≠ ∅} is compact. The set q^{−1}(K) ∩ X′_{≥k} = K′′ ∩ X′_{≥k} is closed in K′′, hence compact; its image by q is X_{≥k} ∩ K. □

Lemma 6.2. Let G be a locally compact, proper groupoid, with G^{(0)} Hausdorff. Let α ∈ ℝ. For every compact set K ⊂ G^{(0)}, there exists f: X′_{K} → ℝ_{+}^{∗} continuous, where X′_{K} = q^{−1}(K) ⊂ X′, such that

∀S ∈ X′_{K}, f(S) = f(q(S))(#S)^{α}.

Proof. Let K′ = G_{K}^{K}. It is closed and quasi-compact. From Proposition 3.7, X′_{K} is quasi-compact. For every S ∈ X′_{K}, we have S ⊂ K′. By Proposition 3.12, there exists n ∈ ℕ^{∗} such that X′_{≥n+1}∩X′_{K} = ∅. We can thus proceed by reverse induction: suppose constructed f_{k+1} : X′_{K} ∩q^{−1}(X_{≥k+1}) → ℝ_{+}^{∗} continuous such that f_{k+1}(S) = f_{k+1}(q(S))(#S)^{α} for all S ∈ X′_{K} ∩ q^{−1}(X_{≥k+1}). Since X′_{K} ∩ q^{−1}(X_{≥k+1}) is closed in the compact set X′_{K} ∩ q^{−1}(X_{≥k}), there exists a continuous extension h: X′_{K} ∩ q^{−1}(X_{≥k}) → ℝ of f_{k+1}. Replacing h(x) by sup(h(x), inf f_{k+1}), we may assume that h(X′_{K} ∩ q^{−1}(X_{≥k})) ⊂ ℝ_{+}^{∗}. Put f_{k}(S) = h(q(S))(#S)^{α}. Let us show that f_{k} is continuous. Let F be a ultrafilter on X′_{K} ∩ q^{−1}(X_{≥k}), and let S be its limit. Since q(F) is a ultrafilter on K, it has a limit S_{0} ∈ X′_{K}. For every S_{1} ∈ q^{−1}(X_{≥k}), choose ψ(S_{1}) ∈ X′_{≥k} such that q(S_{1}) = q(ψ(S_{1})). Let S′ ∈ X′_{K} ∩ X′_{≥k} be the limit of ψ(F). From Lemma 5.2, Ω_{1} = {S_{1} ∈ X′_{K} ∩ q^{−1}(X_{≥k})| #S = #S_{0}#S_{1}} is an element of F, and Ω_{2} = {S_{2} ∈ X′_{≥k}| #S′ = #S_{0}#S_{2}} is an element of ψ(F).

• If #S_{0} > 1, then S′ ∈ X_{≥k+1}, so S and S_{0} belong to q^{−1}(X_{≥k+1}). Therefore, f_{k}(S_{1}) = (#S_{1})^{α}h(q(S_{1})) converges with respect to F to

\frac{(#S)^{α}}{(#S_{0})^{α}} h(S_{0}) = \frac{(#S)^{α}}{(#S_{0})^{α}} f_{k+1}(S_{0}) = f_{k+1}(S)

= f_{k+1}(q(S))(#S)^{α} = h(q(S))(#S)^{α} = f_{k}(S).

• If S_{0} = {q(S)}, then f_{k}(S_{1}) = (#S_{1})^{α}h(q(S_{1})) converges with respect

to F to (#S)^{α}h(q(S)) = f_{k}(S).

Therefore, f_{k} is a continuous extension of f_{k+1}. □

Theorem 6.3. Let G be a locally compact, proper groupoid such that G^{(0)} is Hausdorff and G^{(0)}/G is σ-compact. Let π: G^{(0)} → G^{(0)}/G be the canonical mapping. Then there exists c: X′ → ℝ_{+} continuous such that

(a) c(S) = c(q(S))#S for all S ∈ X′;

(b) ∀α ∈ G^{(0)}/G, ∃x ∈ π^{−1}(α), c(x) ≠ 0;

(c) ∀K ⊂ G^{(0)} compact, supp(c) ∩ q^{−1}(F) is compact, where F = s(G^{K}). If moreover G admits a Haar system, then there exists c: X′ → ℝ_{+} continuous satisfying (a), (b), (c) and

(d) ∀x ∈ G^{(0)}, ∫_{g∈G^{x}} c(s(g)) λ^{x}(dg) = 1.

Proof. There exists a locally finite cover (V_{i}) of G^{(0)}/G by relatively compact open subspaces. Since π is open and G^{(0)} is locally compact, there exists K_{i} ⊂ G^{(0)} compact such that π(K_{i}) ⊃ V_{i}. Let (φ_{i}) be a partition of unity associated to the cover (V_{i}). For every i, from Lemma 6.2, there exists c_{i}: X′_{K_{i}} → ℝ_{+}^{∗} continuous such that c_{i}(S) = c_{i}(q(S))#S for all S ∈ X′_{K_{i}}. Let

c(S) = ∑_{i} c_{i}(S)φ_{i}(π(q(S))).

It is clear that c is continuous from X′ to ℝ_{+}, and that c(S) = c(q(S))#S. Let us prove (b): let x_{0} ∈ G^{(0)}. There exists i such that φ_{i}(π(x_{0})) ≠ 0. Choose x ∈ K_{i} such that π(x) = π(x_{0}), then c(x) ≥ c_{i}(x)φ_{i}(π(x_{0})) > 0. Let us show (c). Note that F = π^{−1}(π(K)) is closed, so q^{−1}(F) is closed. Let K_{1} be a compact neighborhood of K and F_{1} = π^{−1}(π(K_{1})). Let J = {i| V_{i} ∩ π(K_{1}) ≠ ∅}. Then for all i ∉ J, c_{i}(φ_{i} ∘ π ∘ q) = 0 on q^{−1}(F_{1}), therefore c = ∑_{j∈J} c_{j}(φ_{j} ∘ π ∘ q) in a neighborhood of q^{−1}(F). Since for all i, supp(c_{i}(φ_{i} ∘ π ∘ q)) is compact and since J is finite, supp(c) ∩ q^{−1}(F) ⊂ ∪_{i∈J}supp(c_{i}(φ_{i} ∘ π ∘ q)) is compact. Let us show the last assertion. Let φ(g) = c(s(g)). Let F be a filter on G convergent in HG to A ⊂ G. Choose a ∈ A and let S = a^{−1}A. Then s(F) converges to S in HG, hence

lim_{F} φ = #Sc(s(a)) = ∑_{g∈S} c(s(g)) = ∑_{g∈S} φ(g).

For every compact set K ⊂ G^{(0)},

{g ∈ G| r(g) ∈ K and φ(g) ≠ 0}
⊂ {g ∈ G| r(g) ∈ K and s(g) ∈ supp(c)}
⊂ G_{q(supp(c)∩q^{−1}(F))}^{K},

so G^{K} ∩ {g ∈ G| φ(g) ≠ 0} is included in a quasi-compact set. Therefore, for every l ∈ C_{c}(G^{(0)}), g ↦ l(r(g))φ(g) belongs to C_{c}(G). It follows that h(x) = ∫_{g∈G^{x}} φ(g) λ^{x}(dg) is a continuous function. Moreover, for every x ∈ G^{(0)} there exists g ∈ G^{x} such that φ(g) ≠ 0, so h(x) > 0 ∀x ∈ G^{(0)}. It thus suffices to replace c(x) by c(x)/h(x). □

Example 6.4. In Example 2.3 with Γ = ℤ_{n} and H = {0}, the cutoff function is the unique continuous extension to X′ of the function c(x) = 1 for x ∈ (0, 1], and c(0) = 1/n.

Proposition 6.5. Let G be a locally compact, proper groupoid with Haar system such that G^{(0)} is Hausdorff and G^{(0)}/G is compact. Let c be a cutoff function. Then the function p(g) = \sqrt{c(r(g))c(s(g))} defines a selfadjoint projection p ∈ C_{r}^{∗}(G), and E(G) is isomorphic to pC_{r}^{∗}(G).

Proof. Let ξ_{0}(x) = \sqrt{c(x)}. Then one easily checks that ξ_{0} ∈ E^{0}, ⟨ξ_{0}, ξ_{0}⟩ = p and ξ_{0}⟨ξ_{0}, ξ_{0}⟩ = ξ_{0}, therefore p is a selfadjoint projection in C_{r}^{∗}(G). The maps

E(G) → pC_{r}^{∗}(G),  ξ ↦ ⟨ξ_{0}, ξ⟩ = p⟨ξ_{0}, ξ⟩
pC_{r}^{∗}(G) → E(G),  a ↦ ξ_{0}a = ξ_{0}pa

are inverses from each other. □

7. Generalized morphisms and C^{∗}-algebra correspondences

Until the end of the paper, all groupoids are assumed locally compact, with open range map. In this section, we introduce a notion of generalized morphism for locally compact groupoids which are not necessarily Hausdorff, and a notion of locally proper generalized morphism. Then, we show that a locally proper generalized morphism from G_{1} to G_{2} which satisfies an additional condition induces a C_{r}^{∗}(G_{1})-module E and a ∗-morphism C_{r}^{∗}(G_{2}) → K(E), hence an element of KK(C_{r}^{∗}(G_{2}), C_{r}^{∗}(G_{1})).

7.1. Generalized morphisms.

Definition 7.1. [4, 5, 8, 9, 12, 14] Let G_{1} and G_{2} be two groupoids. A generalized morphism from G_{1} to G_{2} is a triple (Z, ρ, σ) where

G_{1}^{(0)} ⟵^{ρ} Z ⟶^{σ} G_{2}^{(0)},

Z is endowed with a left action of G_{1} with momentum map ρ and a right action of G_{2} with momentum map σ which commute, such that

(a) the action of G_{2} is free and ρ-proper,

(b) ρ induces a homeomorphism Z/G_{2} ≃ G_{1}^{(0)}.

In Definition 7.1, one may replace (b) by (b)’ or (b)” below:

(b)’ ρ is open and induces a bijection Z/G_{2} → G_{1}^{(0)}.

(b)” the map Z ⋊ G_{2} → Z ×_{G_{1}^{(0)}} Z defined by (z, γ) ↦ (z, zγ) is a homeomorphism.

Example 7.2. Let G_{1} and G_{2} be two groupoids.If f: G_{1} → G_{2} is a groupoid morphism, let Z = G_{1}^{(0)} ×_{f,r} G_{2}, ρ(x, γ) = x and σ(x, γ) = s(γ). Define the actions of G_{1} and G_{2} by g · (x, γ) · γ′ = (r(g), f(g)γγ′). Then (Z, ρ, σ) is a generalized morphism from G_{1} to G_{2}.

That ρ is open follows from the fact that the range map G_{2} → G_{2}^{(0)} is open and from Lemma 2.25. The other properties in Definition 7.1 are easy to check.

7.2. Locally proper generalized morphisms.

Definition 7.3. Let G_{1} and G_{2} be two groupoidsA generalized morphism from G_{1} to G_{2} is said to be locally proper if the action of G_{1} on Z is σ-proper.

Our terminology is justified by the following proposition:

Proposition 7.4. Let G_{1} and G_{2} be two groupoids such that G_{2}^{(0)} is Hausdorff. Let f: G_{1} → G_{2} be a groupoid morphism. Then the associated generalized groupoid morphism is locally proper if and only if the map (f, r, s): G_{1} → G_{2} × G_{1}^{(0)} × G_{1}^{(0)} is proper.

Proof. Let φ: G_{1} ×_{f∘s,r} G_{2} → (G_{2} ×_{s,s} G_{2}) ×_{r×r,f×f} (G_{1}^{(0)} × G_{1}^{(0)}) defined by φ(g_{1}, g_{2}) = (f(g_{1})g_{2}, g_{2}, r(g_{1}), s(g_{1})). By definition, the action of G_{1} on Z is proper if and only if φ is a proper map. Consider θ: G_{2} ×_{s,s} G_{2} → G_{2}^{(2)} given by (γ, γ′) = (γ(γ′)^{−1}, γ′). Let ψ = (θ × 1) ∘ φ. Since θ is a homeomorphism, the action of G_{1} on Z is proper if and only if ψ is proper. Suppose that (f, r, s) is proper. Let f′ = (f, r, s) × 1: G_{1} × G_{2} → G_{2} × G_{1}^{(0)} × G_{1}^{(0)} × G_{2}. Then f′ is proper. Let F = {(γ, x, x′, γ′) ∈ G_{2} × G_{1}^{(0)} × G_{1}^{(0)} × G_{2}| s(γ) = r(γ′) = f(x′), r(γ) = f(x)}. Then f′: (f′)^{−1}(F) → F is proper, i.e. ψ is proper. Conversely, suppose that ψ is proper. Let F′ = {(γ, y, x, x′) ∈ G_{2} × G_{2}^{(0)} × G_{1}^{(0)} × G_{1}^{(0)}| s(γ) = y}. Then ψ: ψ^{−1}(F′) → F′ is proper, therefore (f, r, s) is proper. □

Our objective is now to show the

Proposition 7.5. Let G_{1}, G_{2}, G_{3} be groupoidsLet (Z_{1}, ρ_{1}, σ_{1}) and (Z_{2}, ρ_{2}, σ_{2}) be two generalized groupoid morphisms from G_{1} to G_{2} and from G_{2} to G_{3} respectively. Then (Z, ρ, σ) = (Z_{1}×_{G_{2}} Z_{2}, ρ_{1}×1, 1×σ_{2}) is a generalized groupoid morphism. If (Z_{1}, ρ_{1}, σ_{1}) and (Z_{2}, ρ_{2}, σ_{2}) are locally proper, then (Z, ρ, σ) is locally proper.

Proposition 7.5 shows that groupoids form a category whose arrows are generalized morphisms, and that two groupoids are isomorphic in that category if and only if they are Morita-equivalent. Moreover, the same conclusions hold for the category whose arrows are locally proper generalized morphisms. In particular, local properness of generalized morphisms is invariant under Moritaequivalence. All the assertions of Proposition 7.5 follow from Lemma 2.33.

7.3. Proper generalized morphisms.

Definition 7.6. Let G_{1} and G_{2} be groupoids. A generalized morphism (Z, ρ, σ) from G_{1} to G_{2} is said to be proper if it is locally proper, and if for every quasicompact subspace K of G_{2}^{(0)}, σ^{−1}(K) is G_{1}-compact.

Examples 7.7. (a) Let X and Y be locally compact spaces and f: X → Y a continuous map. Then the generalized morphism (X, Id, f) is proper if and only if f is proper.

(b) Let f: G_{1} → G_{2} be a continuous morphism between two locally compact groups. Let p: G_{2} → {∗}. Then (G_{2}, p, p) is proper if and only if f is proper and f(G_{1}) is co-compact in G_{2}.

(c) Let G be a locally compact proper groupoid with Haar system such that G^{(0)} is Hausdorff, and let π: G^{(0)} → G^{(0)}/G be the canonical mapping. Then (G^{(0)}, Id, π) is a proper generalized morphism from G to G^{(0)}/G.

7.4. Construction of a C^{∗}-correspondence. Until the end of the section, our goal is to prove:

Theorem 7.8. Let G_{1} and G_{2} be locally compact groupoids with Haar system such that G_{1}^{(0)} and G_{2}^{(0)} are Hausdorff, and (Z, ρ, σ) a locally proper generalized morphism from G_{1} to G_{2}. Then one can construct a C_{r}^{∗}(G_{1})-Hilbert module E_{Z} and a map π: C_{r}^{∗}(G_{2}) → L(E_{Z}). Moreover, if (Z, ρ, σ) is proper, then π maps to K(E_{Z}). Therefore, it gives an element of KK(C_{r}^{∗}(G_{2}), C_{r}^{∗}(G_{1})).

Corollary 7.9. (see [14]) Let G_{1} and G_{2} be locally compact groupoids with Haar system such that G_{1}^{(0)} and G_{2}^{(0)} are Hausdorff. If G_{1} and G_{2} are Moritaequivalent, then C_{r}^{∗}(G_{1}) and C_{r}^{∗}(G_{2}) are Morita-equivalent.

Corollary 7.10. Let f: G_{1} → G_{2} be morphism between two locally compact groupoids with Haar system such that G_{1}^{(0)} and G_{2}^{(0)} are Hausdorff. If the restriction of f to (G_{1})_{K}^{K} is proper for each compact set K ⊂ (G_{1})^{(0)} then f induces a correspondence E_{f} from C_{r}^{∗}(G_{2}) to C_{r}^{∗}(G_{1}). If in addition for every compact set K ⊂ G_{2}^{(0)} the quotient of G_{1}^{(0)} ×_{f,r} (G_{2})_{K} by the diagonal action of G_{1} is compact, then C_{r}^{∗}(G_{2}) maps to K(E_{f}) and thus f defines a KK-element [f] ∈ KK(C_{r}^{∗}(G_{2}), C_{r}^{∗}(G_{1})).

Proof. See Proposition 7.4 and Definition 7.6 applied to the generalized morphism Z_{f} = G_{1}^{(0)} ×_{f,r} G_{2} as in Example 7.2 □

The rest of the section is devoted to proving Theorem 7.8.

Let us first recall the construction of the correspondence when the groupoids are Hausdorff [11]. It is the closure of C_{c}(Z) with the C_{r}^{∗}(G_{1})-valued scalar product

(2)  ⟨ξ, η⟩(g) = ∫_{γ∈(G_{2})^{σ(z)}} \overline{ξ(zγ)}η(g^{−1}zγ) λ^{σ(z)}(dγ),

where z is an arbitrary element of Z such that ρ(z) = r(g). The right C_{r}^{∗}(G_{1})- module structure is defined ∀ξ ∈ C_{c}(Z), ∀a ∈ C_{c}(G_{1}) by

(3)  (ξa)(z) = ∫_{g∈(G_{1})^{ρ(z)}} ξ(g^{−1}z)a(g^{−1}) λ^{ρ(z)}(dg),

and the left action of C_{r}^{∗}(G_{2}) is

(4)  (bξ)(z) = ∫_{γ∈(G_{2})^{σ(z)}} b(γ)ξ(zγ) λ^{σ(z)}(dγ)

for all b ∈ C_{c}(G_{2}).

We now come back to non-Hausdorff groupoids. For every open Hausdorff set V ⊂ Z, denote by V′ its closure in H((G_{1} ⋉ Z)_{V}^{V}), where z ∈ V is identified to (ρ(z), z) ∈ H((G_{1} ⋉ Z)_{V}^{V}). Let E_{V}^{0} be the set of ξ ∈ C_{c}(V′) such that

ξ(z) = \frac{ξ(S × \{z\})}{\sqrt{#S}} for all S × \{z\} ∈ V′.

Lemma 7.11. The space E_{Z}^{0} = ∑_{i∈I} E_{V_{i}}^{0} is independent of the choice of the cover (V_{i}) of Z by Hausdorff open subspaces.

Proof. It suffices to show that for every open Hausdorff subspace V of Z, one has E_{V}^{0} ⊂ ∑_{i∈I} E_{V_{i}}^{0}. Let ξ ∈ E_{V}^{0} . Denote by q_{V} : V′ → V the canonical map defined by q_{V} (S × {z}) = z. Let K ⊂ V compact such that supp(ξ) ⊂ q_{V}^{−1}(K). There exists J ⊂ I finite such that K ⊂ ∪_{j∈J}V_{j}. Let (φ_{j})_{j∈J} be a partition of unity associated to that cover, and ξ_{j} = ξ.(φ_{j} ∘ q_{V}). One easily checks that ξ_{j} ∈ E_{V_{j}}^{0} and that ξ = ∑_{j∈J} ξ_{j}. □

We now define a C_{r}^{∗}(G_{1})-valued scalar product on E_{Z}^{0} by Eqn. (2) where z is an arbitrary element of Z such that ρ(z) = r(g). Our definition is independent of the choice of z, since if z′ is another element, there exists γ′ ∈ G_{2} such that z′ = zγ′, and the Haar system on G_{2} is left-invariant. Moreover, the integral is convergent for all g ∈ G_{1} because the action of G_{2} on Z is proper. Let us show that ⟨ξ, η⟩ ∈ C_{c}(G_{1}) for all ξ, η ∈ E_{Z}^{0}. We need a preliminary lemma:

Lemma 7.12. Let X and Y be two topological spaces such that X is locally compact and f: X → Y proper. Let F be a ultrafilter such that f converges to y ∈ Y with respect to F. Then there exists x ∈ X such that f(x) = y and F converges to x.

Proof. Let Q = f^{−1}(y). Since f is proper, Q is quasi-compact. Suppose that for all x ∈ Q, F does not converge to x. Then there exists an open neighborhood V_{x} of x such that V_{x}^{c} ∈ F. Extracting a finite cover (V_{1}, …, V_{n}) of Q, there exists an open neighborhood V of Q such that V^{c} ∈ F. Since f is closed, f(V^{c})^{c} is a neighborhood of y. By assumption, f(V^{c})^{c} ∈ f(F), i.e. ∃A ∈ F, f(A) ⊂ f(V^{c})^{c}. This implies that A ⊂ V, therefore V ∈ F: this contradicts V^{c} ∈ F. Consequently, there exists x ∈ Q such that F converges to x. □

To show that ⟨ξ, η⟩ ∈ C_{c}(G_{1}), we can suppose that ξ ∈ E_{U}^{0} and η ∈ E_{V}^{0} , where U and V are open Hausdorff. Let F (g, z) = \overline{ξ(z)}η(g^{−1}z), defined on Γ = G_{1} ×_{r,ρ} Z. Since the action of G_{1} on Z is proper, F is quasi-compactly supported. Let us show that F ∈ C_{c}(Γ). Let F be a ultrafilter on Γ, convergent in HΓ. Since G_{1}^{(0)} is Hausdorff, its limit has the form S = S′g_{0} × S′′ where S′ ⊂ (G_{1})_{r(g_{0})}^{r(g_{0})}, S′′ ⊂ ρ^{−1}(r(g_{0})). Moreover,

S′ is a subgroup of (G_{1})_{r(g)}^{r(g)} by the proof of Lemma 5.1. Suppose that there exist z_{0}, z_{1} ∈ S′′ and g_{1} ∈ S′g_{0} such that z_{0} ∈ U and g_{1}^{−1}z_{1} ∈ V. By Lemma 7.12 applied to the proper map G_{1} ⋊ Z → Z × Z, there exists s_{0} ∈ S′ such that z_{0} = s_{0}z_{1}. We may assume that g_{0} = s_{0}g_{1}. Then ∑_{s∈S} F (s) = ∑_{s′∈S′} \overline{ξ(z_{0})}η(g_{0}^{−1}(s′)^{−1}z_{0}). If s′ ∉ stab(z_{0}), then g_{0}^{−1}(s′)^{−1}z_{0} ∉ V since g_{0}^{−1}z_{0} and g_{0}^{−1}(s′)^{−1}z_{0} are distinct limits of (g, z) ↦ g^{−1}z with respect to F and V is Hausdorff. Therefore,

∑_{s∈S} F (s) = #(stab(z_{0}) ∩ S′)\overline{ξ(z_{0})}η(g_{0}^{−1}z_{0})

= \overline{\sqrt{#(stab(z_{0}) ∩ S′)}ξ(z_{0})}\sqrt{#(stab(g_{0}^{−1}z_{0}) ∩ (g_{0}^{−1}S′g_{0}))}η(z_{0})
= lim_{F} \overline{ξ(z)}η(g^{−1}z) = lim_{F} F (g, z).

If for all z_{0}, z_{1} ∈ S′′ and all g_{1} ∈ S′g_{0}, (z_{0}, g_{1}^{−1}z_{1}) ∉ U ×V, then ∑_{s∈S} F (g, z) = 0 = lim_{F} F (g, z). By Proposition 4.1, F ∈ C_{c}(Γ). Since ⟨ξ, η⟩(g) = ∫_{γ∈(G_{2})^{σ(z)}} F (g, zγ) λ^{σ(z)}(dγ), to prove that ⟨ξ, η⟩ ∈ C_{c}(G_{1}) it suffices to show:

Lemma 7.13. Let G_{1} and G_{2} be two locally compact groupoids with Haar system such that G_{i}^{(0)} are Hausdorff. Let (Z, ρ, σ) be a generalized morphism from G_{1} to G_{2}. Let Γ = G_{1} ×_{r,ρ} Z. Then for every F ∈ C_{c}(Γ), the function

g ↦ ∫_{γ∈(G_{2})^{σ(z)}} F (g, zγ) λ^{σ(z)}(dγ),

where z ∈ Z is an arbitrary element such that ρ(z) = r(g), belongs to C_{c}(G_{1}).

Proof. Suppose first that F (g, z) = f(g)h(z), where f ∈ C_{c}(G_{1}) and h ∈ C_{c}(Z). Let H(z) = ∫_{γ∈(G_{2})^{σ(z)}} h(zγ) λ^{σ(z)}(dγ). By Lemma 7.14 below (applied to the groupoid Z ⋊ G_{2}), H is continuous. It is obviously G_{2}-invariant, therefore H ∈ C_{c}(Z/G_{2}). Let H̃ ∈ C_{c}(G_{1}^{(0)}) ≃ C_{c}(Z/G_{2}) correspond to H. The map

g ↦ ∫_{γ∈(G_{2})^{σ(z)}} F (g, zγ) λ^{σ(z)}(dγ) = f(g)H̃(s(g))

thus belongs to C_{c}(G_{1}). By linearity, the lemma is true for F ∈ C_{c}(G_{1}) ⊗ C_{c}(Z). By Lemma 4.4 and Lemma 4.5, F is the uniform limit of functions F_{n} ∈ C_{c}(G_{1}) ⊗ C_{c}(Z) which are supported in a fixed quasi-compact set Q = Q_{1} × Q_{2} ⊂ G_{1} × Z. Let Q′ ⊂ Z quasi-compact such that ρ(Q′) ⊃ r(Q_{1}). Since the action of G_{2} on Z is proper, K = {γ ∈ G_{2}| Q′γ ∩ Q_{2} ≠ ∅} is quasi-compact. Using the fact that G_{1}^{(0)} ≃ Z/G_{2}, it is easy to see that

sup_{(g,z)∈Γ} ∫_{γ∈(G_{2})^{σ(z)}} 1_{Q}(g, zγ) λ^{σ(z)}(dγ) ≤ sup_{z∈Q′} ∫_{γ∈G_{2}^{σ(z)}} 1_{Q_{2}}(zγ)λ^{σ(z)}(dγ)

≤ sup_{x∈G_{2}^{(0)}} ∫_{γ∈G_{2}^{x}} 1_{K}(γ)λ^{x}(dγ) < ∞

by Lemma 4.7. Therefore,

lim_{n→∞} sup_{g∈G_{1}} |∫_{γ∈G_{2}^{σ(z)}} F (g, zγ) − F_{n}(g, zγ) λ^{σ(z)}(dγ)| = 0.

The conclusion follows from Corollary 4.2. □

In the proof of Lemma 7.13 we used the

Lemma 7.14. Let G be a locally compact, proper groupoid with Haar system, such that G^{x} is Hausdorff for all x ∈ G^{(0)}, and G_{x}^{x} = {x} for all x ∈ G^{(0)}. We do not assume G^{(0)} to be Hausdorff. Then ∀f ∈ C_{c}(G^{(0)}),

φ: G^{(0)} → ℂ, x ↦ ∫_{g∈G^{x}} f(s(g)) λ^{x}(dg)

is continuous.

Proof. Let V be an open, Hausdorff subspace of G^{(0)}. Let h ∈ C_{c}(V). Since (r, s): G → G^{(0)} × G^{(0)} is a homeomorphism from G onto a closed subspace of G^{(0)} × G^{(0)}, and (x, y) ↦ h(x)f(y) belongs to C_{c}(G^{(0)} × G^{(0)}), the map g ↦ h(r(g))f(s(g)) belongs to C_{c}(G), therefore by definition of a Haar system, x ↦ ∫_{g∈G^{x}} h(r(g))f(s(g)) λ^{x}(dg) = h(x)φ(x) belongs to C_{c}(G^{(0)}). Since h ∈ C_{c}(V) is arbitrary, this shows that φ_{|V} is continuous, hence φ is continuous on G^{(0)}. □

Now, let us show the positivity of the scalar product. Recall that for all x ∈ G_{1}^{(0)} there is a representation π_{G_{1},x}: C^{∗}(G_{1}) → L(L^{2}(G_{1}^{x})) such that for all a ∈ C_{c}(G_{1}) and all η ∈ C_{c}(G_{1}^{x}),

(π_{G_{1},x}(a)η)(g) = ∫_{h∈G_{1}^{s(g)}} a(h)η(gh) λ^{s(g)}(dh).

By definition, ∥a∥_{C_{r}^{∗}(G_{1})} = sup_{x∈G_{1}^{(0)}} ∥π_{G_{1},x}(a)∥.

⟨η, π_{G_{1},x}(a)η⟩ = ∫_{g∈G_{1}^{x}, h∈G_{1}^{s(g)}} \overline{η(g)}a(h)η(gh) λ^{s(g)}(dh)λ^{x}(dg)
= ∫_{g∈G_{1}^{x}, h∈G^{s(g)}} \overline{η(g)}a(g^{−1}h)η(h) λ^{x}(dg)λ^{x}(dh).

Fix z ∈ Z such that ρ(z) = x. Replacing a(g^{−1}h) by

⟨ξ, ξ⟩(g^{−1}h) = ∫_{γ∈G_{2}^{σ(z)}} \overline{ξ(g^{−1}zγ)}ξ(h^{−1}zγ) λ^{σ(z)}(dγ),

we get

(5) ⟨η, π_{G_{1},x}(⟨ξ, ξ⟩)η⟩ = ∫_{γ∈G_{2}^{σ(z)}} λ^{σ(z)}(dγ) |∫_{g∈G^{x}} η(g)ξ(g^{−1}zγ) λ^{x}(dg)|^{2} .

It follows that π_{G_{1},x}(⟨ξ, ξ⟩) ≥ 0 for all x ∈ G_{1}^{(0)}, so ⟨ξ, ξ⟩ ≥ 0 in C_{r}^{∗}(G_{1}).

Now, let us define a C_{r}^{∗}(G_{1})-module structure on E_{Z}^{0} by Eqn.(3) for all ξ ∈ E_{Z}^{0} and a ∈ C_{c}(G_{1}). Let us show that ξa ∈ E_{Z}^{0}. We need a preliminary lemma:

Lemma 7.15. Let X and Y be quasi-compact spaces, (Ω_{k}) an open cover of X ×Y. Then there exist finite open covers (X_{i}) and (Y_{j}) of X and Y such that ∀i, j ∃k, X_{i} × Y_{j} ⊂ Ω_{k}.

Proof. For all (x, y) ∈ X ×Y choose open neighborhoods U_{x,y} and V_{x,y} of x and y such that U_{x,y} ×V_{x,y} ⊂ Ω_{k} for some k. For y fixed, there exist x_{1}, …, x_{n} such that (U_{x_{i},y})_{1≤i≤n} covers X. Let V_{y} = ∩_{i=1}^{n}U_{x_{i},y}. Then for all (x, y) ∈ X × Y, there exists an open neighborhood U′_{x,y} of x and k such that U′_{x,y} × V_{y} ⊂ Ω_{k}. Let (V_{1}, …, V_{m}) = (V_{y_{1}}, …, V_{y_{m}}) such that ∪_{1≤j≤m}V_{j} = Y. For all x ∈ X, let U′_{x} = ∩_{j=1}^{m}U′_{x,y_{j}}. Let (U_{1}, …, U_{p}) be a finite sub-cover of (U′_{x})_{x∈X}. Then for all i and for all j, there exists k such that U_{i} × V_{j} ⊂ Ω_{k}. □

Let Q_{1} and Q_{2} be quasi-compact subspaces of G_{1} of Z respectively such that a^{−1}(ℂ^{∗}) ⊂ Q_{1} and ξ^{−1}(ℂ^{∗}) ⊂ Q_{2}. Let Q be a quasi-compact subspace of Z such that ∀g ∈ Q_{1}, ∀z ∈ Q_{2}, g^{−1}z ∈ Q. Let (U_{k}) be a finite cover of Q by Hausdorff open subspaces of Z. Let Q′ = Q_{1} ×_{r,ρ} Q_{2}. Then Q′ is a closed subspace of Q_{1} × Q_{2}. Let Ω′_{k} = {(g, z) ∈ Q′| g^{−1}z ∈ U_{k}}. Then (Ω′_{k}) is a finite open cover of Q′. Let Ω_{k} be an open subspace of Q_{1} × Q_{2} such that Ω′_{k} = Ω_{k} ∩ Q′. Then {Q_{1} × Q_{2} − Q′} ∪ {Ω_{k}} is an open cover of Q_{1} × Q_{2}. Using Lemma 7.15, there exist finite families of Hausdorff open sets (W_{i}) and (V_{j}) which cover Q_{1} and Q_{2}, such that for all i, j and for all (g, z) ∈ W_{i} ×_{G_{1}^{(0)}} V_{j}, there exists k such that g^{−1}z ∈ U_{k}. Thus, we can assume by linearity and by Lemmas 4.3 and 7.11 that ξ ∈ E_{V}^{0} , a ∈ C_{c}(W), U = W^{−1}V, and U, V and W are open and Hausdorff.

Let Ω = {(g, S) ∈ W^{−1} × U′| g^{−1}q_{U}(S) ∈ V }. Then the map (g, S) ↦ (g^{−1}, g^{−1}S) is a homeomorphism from Ω onto W ×_{r,ρ∘q_{V}} V′. Therefore, the map (g, z) ↦ ξ(g^{−1}z)a(g^{−1}) belongs to C_{c}(Ω) ⊂ C_{c}(G_{1} ×_{r,ρ∘q_{V}} U′). By Lemma 4.8,

S ↦ (ξa)(S) = ∫_{g∈G_{1}^{ρ∘q_{V} (S)}} ξ(g^{−1}S)a(g^{−1}) λ^{ρ∘q_{V} (S)}(dg)

belongs to C_{c}(U′). It is immediate that (ξa)(S) = \sqrt{#S}(ξa)(q(S)) for all S ∈ U′, therefore ξa ∈ E_{U}^{0}. This completes the proof that ξa ∈ E_{Z}^{0}.

Finally, it is not hard to check that ⟨ξ, ηa⟩ = ⟨ξ, η⟩∗a. Therefore, the completion E_{Z} of E_{Z}^{0} with respect to the norm ∥ξ∥ = ∥⟨ξ, ξ⟩∥^{1/2} is a C_{r}^{∗}(G_{1})-Hilbert module.

Let us now construct a morphism π: C_{r}^{∗}(G_{2}) → L(E_{Z}). For every ξ ∈ E_{Z}^{0} and every b ∈ C_{c}(G_{2}), define bξ by Eqn.(4). Let us check that bξ ∈ E_{Z}^{0}. As above, by linearity we may assume that ξ ∈ E_{V}^{0} , b ∈ C_{c}(W) and V W^{−1} ⊂ U, where V ⊂ Z, U ⊂ Z and W ⊂ G_{2} are open and Hausdorff. Let Φ(S, γ) = (Sγ, γ). Then Φ is a homeomorphism from Ω = {(S, γ) ∈ U′ ×_{σ∘q_{U} ,r} W | q_{U}(S)γ ∈ V } onto V′ ×_{σ∘q_{V} ,s} W. Let F (z, γ) = b(γ)ξ(zγ). Since F = (ξ ⊗ b) ∘ Φ, F is an element of C_{c}(Ω) ⊂ C_{c}(U′ ×_{σ∘q_{U} ,r} W). By Lemma 4.8, bξ ∈ C_{c}(U′). It is immediate that (bξ)(S) = \sqrt{#S}(bξ)(q(S)). Therefore, bξ ∈ E_{U}^{0} ⊂ E_{Z}^{0}. Let us prove that ∥bξ∥ ≤ ∥b∥ ∥ξ∥. Let

ζ(γ) = ∫_{g∈G_{1}^{x}} η(g)ξ(g^{−1}zγ) λ^{x}(dg),

where z ∈ Z such that ρ(z) = r(g) is arbitrary. From (5),

⟨η, π_{G_{1},x}(⟨ξ, ξ⟩)η⟩ = ∥ζ∥_{L^{2}(G_{2}^{σ(z)})}^{2}.

A similar calculation shows that

⟨η, π_{G_{1},x}(⟨bξ, bξ⟩)η⟩ = ∫_{γ∈G_{2}^{σ(z)}} λ^{σ(z)}(dγ) |∫_{g∈G_{1}^{x}} η(g)ξ(g^{−1}zγγ′)b(γ′) λ^{s(γ)}(dγ′)|^{2}

= ⟨bζ, bζ⟩ ≤ ∥b∥^{2}∥ζ∥^{2}.

By density of C_{c}(G_{2}^{x}) in L^{2}(G_{2}^{x}), ∥π_{G_{1},x}(⟨bξ, bξ⟩)∥ ≤ ∥b∥^{2}∥π_{G_{1},x}(⟨ξ, ξ⟩)∥. Taking the supremum over x ∈ G_{1}^{(0)}, we get ∥bξ∥ ≤ ∥b∥ ∥ξ∥. It follows that b ↦ (ξ ↦ bξ) extends to a ∗-morphism π: C_{r}^{∗}(G_{2}) → L(E_{Z}).

Finally, suppose now that (Z, ρ, σ) is proper, and let us show that C_{r}^{∗}(G_{2}) maps to K(E_{Z}). For every η, ζ ∈ E_{Z}^{0}, denote by T_{η,ζ} the operator T_{η,ζ}(ξ) = η⟨ζ, ξ⟩. Compact operators are elements of the closed linear span of T_{η,ζ}’s. Let us write an explicit formula for T_{η,ζ}:

T_{η,ζ}(ξ)(z) = ∫_{g∈G_{1}^{ρ(z)}} η(g^{−1}z)⟨ζ, ξ⟩(g^{−1}) λ^{ρ(z)}(dg)
= ∫_{g∈G_{1}^{ρ(z)}} η(g^{−1}z) ∫_{γ∈G_{2}^{σ(z)}} \overline{ζ(g^{−1}zγ)}ξ(zγ) λ^{σ(z)}(dγ)λ^{ρ(z)}(dg).

Let b ∈ C_{c}(G_{2}), let us show that π(b) ∈ K(E_{Z}). Let K be a quasi-compact subspace of G_{2} such that b^{−1}(ℂ^{∗}) ⊂ K. Since (Z, ρ, σ) is a proper generalized morphism, there exists a quasi-compact subspace Q of Z such that σ^{−1}(r(K)) ⊂ G_{1}Q̊. Before we proceed, we need a lemma:

Lemma 7.16. Let G_{2} be a locally compact groupoid acting freely and properly on a locally compact space Z with momentum map σ: Z → G_{2}^{(0)}. Then for every (z_{0}, γ_{0}) ∈ Z ⋊ G_{2}, there exists a Hausdorff open neighborhood Ω_{z_{0},γ_{0}} of (z_{0}, γ_{0}) such that

• U = {z_{1}γ_{1}| (z_{1}, γ_{1}) ∈ Ω_{z_{0},γ_{0}}} is Hausdorff;

• there exists a Hausdorff open neighborhood W of γ_{0} such that ∀γ ∈ G_{2}, ∀z ∈ pr_{1}(Ω_{z_{0},γ_{0}}), ∀z′ ∈ U, z′ = zγ ⟹ γ ∈ W.

Proof. Let R = {(z, z′) ∈ Z × Z| ∃γ ∈ G_{2}, z′ = zγ}. Since the G_{2}-action is free and proper, there exists a continuous function ϕ: R → G_{2} such that ϕ(z, zγ) = γ. Let W be an open Hausdorff neighborhood of γ_{0}. By continuity of ϕ, there exist open Hausdorff neighborhoods V and U_{0} of z_{0} and z_{0}γ_{0} such that for all (z, z′) ∈ R ∩ (V × U_{0}), ϕ(z, z′) ∈ W. By continuity of the action, there exists an open neighborhood Ω_{z_{0},γ_{0}} of (z_{0}, γ_{0}) such that ∀(z_{1}, γ_{1}) ∈ Ω_{z_{0},γ_{0}}, z_{1}γ_{1} ∈ U_{0} and z_{1} ∈ V. □

By Lemma 7.15, there exist finite covers (V_{i}) of Q and (W_{j}) of K such that for every i, j, (Z ×_{G_{2}^{(0)}} G_{2}) ∩ (V_{i} × W_{j}) ⊂ Ω_{z_{0},γ_{0}} for some (z_{0}, γ_{0}).

By Lemma 6.2 applied to the groupoid (G_{1} ⋉ Z)_{V_{i}}^{V_{i}}, for all i there exists c′_{i} ∈ C_{c}(V′_{i})_{+} such that c′_{i}(S) = (#S)c′_{i}(q_{V_{i}}(S)) for all S ∈ V′_{i}, and such that ∑_{i} c′_{i} ≥ 1 on Q. Let

f_{i}(z) = ∫_{g∈G_{1}^{ρ(z)}} c′_{i}(g^{−1}z) λ^{ρ(z)}(dg)

and let f = ∑_{i} f_{i}. As in the proof of Theorem 6.3, one can show that for every Hausdorff open subspace V of Z and every h ∈ C_{c}(V), (g, z) ↦ h(z)c′_{i}(g^{−1}z) belongs to C_{c}(G⋉Z), therefore hf_{i} is continuous on V. Since h is arbitrary, it follows that f_{i} is continuous, thus f is continuous. Moreover, f is G_{1}-equivariant, nonnegative, and inf_{Q} f > 0. Therefore, there exists f_{1} ∈ C_{c}(G_{1}\Z) such that f_{1}(z) = 1/f(z) for all z ∈ Q. Let c_{i}(z) = f_{1}(z)c′_{i}(z). Let

T_{i}(ξ)(z) = ∫_{g∈G_{1}^{ρ(z)}} ∫_{γ∈G_{2}^{σ(z)}} c_{i}(g^{−1}z)b(γ)ξ(zγ) λ^{ρ(z)}(dg)λ^{σ(z)}(dγ).

Then π(b) = ∑_{i} T_{i}, therefore it suffices to show that T_{i} is a compact operator for all i. By linearity and by Lemma 4.3, one may assume that b ∈ C_{c}(W_{j}) for some j. Then, by construction of V_{i} (see Lemma 7.16), there exist open Hausdorff sets U ⊂ Z and W ⊂ G_{2} such that {γ ∈ G_{2}| ∃(z, z′) ∈ V_{i} × U, z′ = zγ} ⊂ W, and {zγ| (z, γ) ∈ V_{i} ×_{σ,r} W } ⊂ U. The map (z, zγ) ↦ c(z)b(γ) defines an element of C_{c}(V′_{i} × U). Let L_{1} × L_{2} ⊂ V_{i} × U compact such that (z, zγ) ↦ c(z)b(γ) is supported on q_{V_{i}}^{−1}(L_{1}) × L_{2}.

By Lemma 6.2 applied to the groupoids (G_{1} ⋉ Z)_{V_{i}}^{V_{i}} and (G_{1} ⋉ Z)_{U}^{U}, there exist d_{1} ∈ C_{c}(V′_{i})_{+} and d_{2} ∈ C_{c}(U′)_{+} such that d_{1} > 0 on L_{1} and d_{2} > 0 on L_{2}, d_{1}(S) = \sqrt{#S}d_{1}(q_{V_{i}}(S)) for all S ∈ V′_{i}, and d_{2}(S) = \sqrt{#S}d_{2}(q_{U}(S)) for all S ∈ U′. Let

f(z, zγ) = \frac{c(z)b(γ)}{d_{1}(z)d_{2}(zγ)}.

Then f ∈ C_{c}(V_{i} ×_{G_{1}^{(0)}} U). Therefore, f is the uniform limit of a sequence

f_{n} = ∑ α_{n,k} ⊗ \overline{β_{n,k}} in C_{c}(V_{i}) ⊗ C_{c}(U) such that all the f_{n} are supported in a fixed compact set. Then T_{i} is the norm-limit of ∑_{k} T_{d_{1}α_{n,k},d_{2}β_{n,k}}, therefore it is compact.

Remark 7.17. The construction in Theorem 7.8 is functorial with respect to the composition of generalized morphisms and of correspondences. We don’t include a proof of this fact, as it is tedious but elementary. It is an easy exercise when G_{1} and G_{2} are Hausdorff.

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Jean-Louis Tu University Paris VI Institut de Mathématiques 175, rue du Chevaleret 75013 Paris, France.

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Documenta Mathematica 9 (2004)