Download NON-HAUSDORFF GROUPOIDS, PROPER ACTIONS AND K

NON-HAUSDORFF GROUPOIDS, PROPER ACTIONS AND
K-THEORY
JEAN-LOUIS TU
Abstract. Let G be a (not necessarily Hausdorff) locally compact groupoid.
We introduce a notion of properness for G, which is invariant under Moritaequivalence. We show that any generalized morphism between two locally
compact groupoids which satisfies some properness conditions induces a C ∗ correspondence from Cr∗ (G1 ) to Cr∗ (G2 ), and thus two Morita equivalent
groupoids have Morita-equivalent C ∗ -algebras. Finally, we attempt to define an assembly map à la Baum-Connes-Higson.
Key words: groupoid, C ∗ -algebra, K-theory.
Mathematics Subjects Classification (2000): 22A22 (Primary); 46L05, 46L80,
54D35 (Secondary).
Contents
Introduction
1. Preliminaries
1.1. Groupoids
1.2. Locally compact spaces
1.3. Proper maps
2. Proper groupoids and proper actions
2.1. Locally compact groupoids
2.2. Proper groupoids
2.3. Proper actions
2.4. Permanence properties
2.5. Invariance by Morita-equivalence
2.6. The slice property
3. A topological construction
3.1. The space HX
3.2. The space H0 X
3.3. Property (P)
4. Haar systems
4.1. The space Cc (X)
4.2. Haar systems
5. The Hilbert module of a proper groupoid
5.1. The space X 0
5.2. Construction of the Hilbert module
6. Cutoff functions
7. Generalized homomorphisms and C ∗ -algebra correspondences
7.1. Generalized homomorphisms
7.2. Locally proper generalized homomorphisms
7.3. Proper generalized morphisms
7.4. Construction of a C ∗ -correspondence
8. The Hilbert module L2 (G)
9. The classifying space for proper actions
10. The assembly map
10.1. A few lemmas
10.2. Topological K-theory
Appendix A. Existence of a Hausdorff model for EG
Appendix B. The classifying space for proper actions on Hausdorff spaces
References
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Introduction
Very often, groupoids that appear in geometry, such as holonomy groupoids
of foliations, groupoids of inverse semigroups [23, 14] and the indicial algebra
of a manifold with corners [17] are not Hausdorff. For Hausdorff groupoids,
Baum, Connes and Higson [3, 29] define a topological K-theory group K∗top (G)
and an assembly map µ : K∗top (G) → K∗ (Cr∗ (G)) (see also [25, 27]). When G
is a group, the map µ is known to be an isomorphism in many cases, such as
almost connected groups, linear p-adic groups [7], a-T-menable groups [10] and
hyperbolic discrete groups [22].
For groupoids, the situation is, unexpectedly, somewhat different: Higson,
Lafforgue and Skandalis [11] constructed a Hausdorff, r-discrete groupoid such
that µ is not surjective. The construction uses a residually finite group (such
as SLn (Z), n ≥ 2) whose successive quotients constitute an expander sequence [18]. They also obtained foliation counterexamples, where failure of
the Baum–Connes map to be an isomorphism depends in an essential way on
non-Hausdorffness and on the use of a non-amenable group.
In fact, it seems very hard to modify the left-hand side of the Baum–Connes
assembly map in order to obtain an isomorphism in the cases mentioned above,
∗
(G), or wouldn’t
since either the new assembly map wouldn’t factor through Cmax
be natural with respect to restrictions to a closed, saturated space [11]. Thus,
our original motivation was somewhat more modest: namely, try to define an
assembly map in the spirit of [3], and hope that it is an isomorphism in particular cases. For instance, the counterexamples of [11] make an essential use
of non-amenability, therefore one might hope that the assembly map should be
an isomorphism for amenable groupoids.
Certainly the first step is to define the notion of proper groupoid (since
an action of a groupoid G on a space Z is proper if and only if the crossedproduct groupoid Z o G is proper). Our definition is as follows: a topological
groupoid G is proper if the map (r, s) : G → G(0) × G(0) is proper in the sense
of Bourbaki [5]. Properness is shown in Section 2 to be invariant under Moritaequivalence, which gives us confidence that our definition is the right one.
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. Roughly speaking, if (xi ) is a convergent sequence (or filter/net)
in X, then the set S of limit points may have more than one element. The
space HX is constructed so that X is (not continuously) embedded in HX and
(xi ) converges to S in HX. When G is a groupoid (locally compact, with Haar
system, such that G(0) is Hausdorff), the closure X 0 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 Cr∗ (G)-Hilbert
module E(G) for every (locally compact...) proper groupoid G. If G(0) /G is
compact, then there exists a projection p ∈ Cr∗ (G) such that E(G) is isomorphic
to pCr∗ (G). The projection p is given by p(g) = (c(s(g))c(r(g)))1/2 , where
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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 0 → R+ (see above for the definition of X 0 ). The Hilbert
module E(G) is one of the ingredients in the definition of the assembly map, as
it defines an element of KK-theory λG ∈ KKG (C0 (G(0) /G), Cr∗ (G)).
In Section 7, we examine the question of naturality G 7→ Cr∗ (G). Recall that
if f : X → Y is a continuous map between two locally compact spaces, then f
induces a map from C0 (Y ) to C0 (X) if and only if f is proper. When G1 and G2
are groups, a morphism f : G1 → G2 does not induce a map Cr∗ (G2 ) → Cr∗ (G1 )
(when G1 ⊂ G2 is an inclusion of discrete groups there is a map in the other
direction). When f : G1 → G2 is a groupoid morphism, we cannot expect to get
more than a C ∗ -correspondence from Cr∗ (G2 ) to Cr∗ (G1 ) when f satisfies certain
properness assumptions: this was done in the Hausdorff situation by MachoStadler and O’Uchi ([19, Theorem 2.1]), but the formulation of their theorem
is somewhat complicated. In this paper, as a corollary of Theorem 7.11, we get
that (in the Hausdorff situation), if the restriction of f to (G1 )K
K is proper for
(0)
each compact set K ⊂ (G1 ) then f induces a correspondence Ef from Cr∗ (G2 )
to Cr∗ (G1 ). In fact we construct a C ∗ -correspondence out of any groupoid
generalized morphism ([12, 16]) which satisfies some properness conditions. As
a corollary, if G1 and G2 are Morita equivalent then Cr∗ (G1 ) and Cr∗ (G2 ) are
Morita-equivalent C ∗ -algebras.
In Section 8, we give a construction of a Hilbert C0 (X 0 )-module, denoted by
on which Cr∗ (G) acts faithfully; our construction is similar to the one of
Khoskam and Skandalis [14]. We show that, like in the Hausdorff case, L2 (G)
contains C0 (X 0 ) as a direct factor when G is proper. This fact is used, for
Hausdorff G, to prove the Baum–Connes conjecture for proper groupoids [25].
In the final sections 9 and 10, we attempt to define a topological K-theory
K∗top (G) and an assembly map K∗top (G) → K∗ (Cr∗ (G)) like in [3], and explain
where difficulties arise. In particular, the existence of a classifying space for
proper actions is not clear, and a more generalized version of KKG -theory [16]
is necessary. Moreover, the method used in [10, 26] to prove the Baum–Connes
conjecture for Hausdorff amenable groupoids does not (apparently) work. In
fact, we don’t really have positive results on the conjecture for non-Hausdorff
groupoids but we at least hope that this paper disproves the belief that known
facts about Hausdorff groupoids easily generalize to the non-Hausdorff case.
L2 (G),
Acknowledgments: we would like to thank Siegfried Echterhoff, PierreYves Le Gall and Georges Skandalis for useful discussions and remarks.
1. Preliminaries
1.1. Groupoids. Throughout, we will assume that the reader is familiar with
basic definitions about groupoids (see [24, 23, 2]). 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 Gx = s−1 (x), Gy = r−1 (y),
Gyx = Gx ∩ Gy . Recall that a topological groupoid is said to be étale if r (and
s) are local homeomorphisms.
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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) 7→ 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 o G is the subgroupoid of (Z × Z) × G consisting
of elements (z, z 0 , g) such that z 0 = zg. Since the map Z o G → Z × G given
by (z, z 0 , g) 7→ (z, g) is injective, the groupoid Z o 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 (Kn ) of compact subspaces such that X = ∪n∈N Kn ;
(ii) there exists a sequence (Kn ) of quasi-compact subspaces such that X =
∪n∈N Kn ;
(iii) there exists a sequence (Kn ) of quasi-compact subspaces such that X =
∪n∈N Kn and Kn ⊂ K̊n+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 (Ki )i∈I of compact sets such that K ⊂ ∪i∈I K̊i .
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1.3. Proper maps.
Proposition 1.4. [5, 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 × IdZ : 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. [5, 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 Ky of y such that
f −1 (Ky ) 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 ∈ f (F ). Let Ky be a compact neighborhood of y such that A = f −1 (Ky ) is
quasi-compact. Then A ∩ F is quasi-compact, so f (A ∩ F ) is quasi-compact.
As f (A ∩ F ) ⊂ Ky , it is closed in Ky , i.e. Ky ∩ f (A ∩ F ) = Ky ∩ f (A ∩ F ).
Since Ky ∩ f (F ) 6= ∅, we have y ∈ Ky ∩ f (A ∩ F ) = Ky ∩ 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.
locally compact σ-compact) if it is locally compact (resp. locally compact σcompact) as a topological space.
Remark 2.2. The definition of a locally compact groupoid in [23] 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 G0 = Γ and Gt = 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.
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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 0 ) are identified if their
germs are equal, i.e. there exists a neighborhood V of x such that yγ = yγ 0 for
all y ∈ V . Then G is locally compact, since the open sets Vγ = {[(x, γ)]| x ∈ X}
are homeomorphic to X and cover G.
Suppose that X is a manifold, M is a manifold such that π1 (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 ∆ be the diagonal in G×G. Since G is locally Hausdorff, ∆ is locally
closed. Then G(0) = (Id, r)−1 (∆) is locally closed in G.
Suppose that G = ∪n∈N Kn with Kn quasi-compact, then s(Kn ) is quasicompact and G(0) = ∪n∈N s(Kn ).
Proposition 2.6. Let Z a locally compact space and G be a locally compact
groupoid acting on Z. Then the crossed-product Z o 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 ∆ ⊂ G(0) × G(0) is locally closed in G(0) × G(0) ,
hence Z o G = (p, r)−1 (∆) 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 ] =
f (t0 )
{(t0 , t, g) ∈ (T ×T )×G| g ∈ Gf (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 (∆), where ∆ is
the diagonal in G(0) × G(0) , thus it is locally closed. Let ρ : (t0 , t, g) 7→ (t0 , r(g))
and σ : (t0 , t, g) 7→ (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) , Gx is Hausdorff.
Proof. Let Z = {(g, h) ∈ Gx × Gx | 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 Gx × Gx . It follows that ϕ−1 (x), which is the diagonal
of Gx × Gx , is closed in Gx × Gx .
NON-HAUSDORFF GROUPOIDS
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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) , Gxx is quasi-compact;
(iii) for all quasi-compact subspaces K and L of G(0) , GL
K is quasi-compact;
(iii)’ for all compact subspaces K and L of G(0) , GL
is
quasi-compact;
K
(iv) for every quasi-compact subspace K of G(0) , GK
K is quasi-compact;
L
(0)
(v) ∀x, y ∈ G , ∃Kx , Ly compact neighborhoods of x and y such that GKyx
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 Gxx is
homeomorphic to Gyx if Gyx 6= ∅. (i) =⇒ (iii) and (v) =⇒ (i) follow Proposi−1
tion 1.6 and the formula GL
K = (r, s) (L × K). (iii) =⇒ (iii)’ =⇒ (v) and
(0)
(iii) =⇒ (iv) are obvious. If G 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. Suppose that r is open. Let V ⊂ G(0) be an open subspace. Since s is
continuous and r is open, 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 ∆ be the diagonal in π(V ) × π(V ). Then
(π×π)−1 (∆) = (V ×V )∩R is closed in V ×V . Since π×π : V ×V → π(V )×π(V )
is continuous open surjective, it follows that ∆ 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).
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Recall that if G is a topological groupoid with G(0) locally compact, and if
V
(Vi ) is a cover of G(0) by open Hausdorff sets, then G0 = qGVji is a topological
groupoid such that G0(0) is locally compact and Hausdorff. By characterization
(v) in Proposition 2.10, the groupoid G is proper if and only if G0 is proper.
More generally, we will prove below that if G and G0 are Morita-equivalent
topological groupoids, then G is proper if and only if G0 is proper.
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 o 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 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 o 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 6= ∅} is
quasi-compact;
(iii)’ for all compact subspaces K and L of Z, {g ∈ G| Lg ∩ K 6= ∅} is
quasi-compact;
(iv) for every quasi-compact subspace K of Z, {g ∈ G| Kg ∩ K 6= ∅} is
quasi-compact;
(v) there exists a family (Ai )i∈I of subspaces of Z such that Z = ∪i∈I Åi
and {g ∈ G| Ai g ∩ Aj 6= ∅} 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 o G)L
K is quasi-compact, then
its image by the second projection Z o G → G is quasi-compact. (iii) =⇒ (iii)’
and (iii) =⇒ (iv) are obvious.
Suppose that Z is locally compact. Take Ai ⊂ Z compact such that Z =
∪i∈I Åi . If (iii)’ is true, then {g ∈ G| Ai g ∩ Aj 6= ∅} is quasi-compact, hence (v).
If (iv) is true, then {g ∈ G| Ai g ∩ Aj 6= ∅} is a subset of the quasi-compact set
{g ∈ G| Kg ∩ K 6= ∅}, where K = Ai ∪ Aj , hence (v).
Suppose that Z is locally compact Hausdorff and that G(0) is Hausdorff.
Let us show (v) =⇒ (ii). Let Cij be a quasi-compact set such that {g ∈
G| Ai g ∩ Aj 6= ∅} ⊂ Cij .
Let z ∈ Z. Choose i ∈ I such that z ∈ Ai . Since Z and G(0) are Hausdorff,
stab(z) is a closed subspace of Cii , therefore it is quasi-compact.
NON-HAUSDORFF GROUPOIDS
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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 0 ) ∈ Φ(F ).
Choose i and j such that z ∈ Åi and z 0 ∈ Åj . Then (z, z 0 ) ∈ Φ(F ) ∩ (Ai × Aj ) ⊂
Φ(F ∩ (Ai ×G(0) Cij )) ⊂ Φ(F ∩ (Z ×G(0) Cij )). There exists a net (zλ , gλ ) ∈
F ∩ (Z ×G(0) Cij ) such that (z, z 0 ) is a limit point of (zλ , zλ gλ ). Since Cij
is quasi-compact, after passing to a universal subnet we may assume that gλ
converges to an element g ∈ Cij . Since G(0) is Hausdorff, F ∩ (Z ×G(0) Cij ) is
closed in Z × Cij , so (z, g) is an element of F ∩ (Z ×G(0) Cij ). Using the fact that
Z is Hausdorff and Φ is continuous, we obtain (z, z 0 ) = Φ(z, g) ∈ Φ(F ).
Proposition 2.15. Let G be a topological groupoid such that G(0) is Hausdorff.
Suppose that G acts properly on a topological space Z with momentum map
p : Z → G(0) . Then for every x ∈ G(0) , p−1 (x) is Hausdorff.
Proof. Let Zx = p−1 (x). Let ϕ : Z ×G(0) G → Z ×Z defined by ϕ(z, g) = (z, zg).
By assumption, ϕ is closed. Since G(0) is Hausdorff, Zx × {x} is closed in
Z ×G(0) G, hence the diagonal of Zx × Zx , which is ϕ(Zx × {x}), is closed in
Z × Z. Since Zx × Zx is closed in Z × Z, it follows that Zx is Hausdorff.
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) 7→ (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 6= 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 G1 and G2 are proper topological groupoids, then G1 ×G2
is proper.
Proof. Follows from the fact that the product of two proper maps is proper [5,
Corollaire I.10.2.3].
(0)
Proposition 2.18. Let G1 and G2 be two topological groupoids such that G1
is Hausdorff and G2 is proper. Suppose that f : G1 → G2 is a proper morphism.
Then G1 is proper.
Proof. Denote by ri and si the range and source maps of Gi (i = 1, 2). Let f¯ be
(0)
(0)
(0)
(0)
the map G1 ×G1 → G2 ×G2 induced from f . Since f¯◦(r1 , s1 ) = (r2 , s2 )◦f
(0)
is proper and G1 is Hausdorff, it follows from [5, Proposition I.10.1.5] that
(r1 , s1 ) is proper.
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JEAN-LOUIS TU
Proposition 2.19. Let G1 and G2 be two topological groupoids such that G1
is proper. Suppose that f : G1 → G2 is a surjective morphism such that the
(0)
(0)
induced map f 0 : G1 → G2 is proper. Then G2 is proper.
Proof. Denote by ri and si the range and source maps of Gi (i = 1, 2). Let F2 ⊂
G2 be a closed subspace, and F1 = f −1 (F2 ). Since G1 is proper, (r1 , s1 )(F1 ) is
closed, and since f 0 ×f 0 is proper, (f 0 ×f 0 )◦(r1 , s1 )(F1 ) is closed. By surjectivity
of f , we have (r2 , s2 )(F2 ) = (f 0 × f 0 ) ◦ (r1 , s1 )(F1 ). This proves that (r2 , s2 ) is
closed. Since for every topological space T , the assumptions of the proposition
are also true for the morphism f × 1 : G1 × T → G2 × T , the above shows that
(r2 , s2 ) × 1T is closed. Therefore, (r2 , s2 ) 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) o G is isomorphic to the subgroupoid Γ =
{(y, y 0 , z, g) ∈ (Y × Y ) × (Z o G)| p(y) = r(g), y 0 = yg} of the proper groupoid
(Y × Y ) × (Z o G). Since Y and G(0) are Hausdorff, Γ is closed in (Y × Y ) ×
(Z o G), hence by Proposition 2.10(ii), (Y ×G(0) Z) o 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 ] = GTT . Since (rT , sT ) is the restriction to (r, s)−1 (T ×
T ) of (r, s), and (r, s) is closed, it follows that (rT , sT ) is closed.
In the general case, let Γ = (T × T ) × G and let T 0 ⊂ 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 0 ].
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) ∈ (r, s)(F ). Let F̃ = G[T ] ∩ (T × T ) × F .
Choose (t0 , t) ∈ T × T such that f (t0 ) = y and f (t) = x. Denote by r̃ and s̃ the
range and source maps of G[T ]. Then (t0 , t) ∈ (r̃, s̃)(F̃ ). Indeed, let Ω 3 (t0 , t)
be an open set, and Ω0 = (f × f )(Ω). Then Ω0 is an open neighborhood of
(y, x), so Ω0 ∩ (r, s)(F ) 6= ∅. It follows that Ω ∩ (r̃, s̃)(F̃ ) 6= ∅.
We have proved that (t0 , t) ∈ (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 Z1 be a saturated subspace. Then G acts properly on Z1 .
Proof. Use the fact that Z1 o G = (Z o G)[Z1 ].
NON-HAUSDORFF GROUPOIDS
11
2.5. Invariance by Morita-equivalence. Recall the algebraic notion of equivalence of groupoids:
Proposition 2.24. Let G1 and G2 be two groupoids. Let ri , si (i = 1, 2) be
the range and source maps of Gi . The following are equivalent:
(0)
(i) there exist a set T and fi : T → Gi surjective such that G1 [T ] and
G2 [T ] are isomorphic;
(0)
(0)
(ii) there exists a set Z, two maps ρ : Z → G1 and σ : Z → G2 , a left
action of G1 on Z with momentum map ρ and a right action of G2 on
Z with momentum map σ such that the actions commute and are free,
(0)
(0)
and the natural maps Z/G2 → G1 and G1 \Z → G2 induced from ρ
and σ are bijective.
Proof. Let us first note that (i) and (ii) are indeed equivalence relations. For
(i), if G1 [T ] ' G2 [T ] and G2 [T 0 ] ' G3 [T 0 ], then G1 (T ×G(0) T 0 ) ' G2 (T ×G(0)
2
2
T 0 ) ' G3 (T ×G(0) T 0 ). For (ii), if (Z1 , ρ1 , σ1 ) defines an equivalence between
2
G1 and G2 and (Z2 , ρ2 , σ2 ) an equivalence relation between G2 and G3 , let
Z = Z1 ×G2 Z2 be the quotient of Z1 ×G(0) Z2 by the action of G2 defined by
2
(z1 , z2 ) · g2 = (z1 g2 , g2−1 z2 ). Let ρ(z1 , z2 ) = ρ1 (z1 ) and σ(z1 , z2 ) = σ2 (z2 ). Then
(Z, ρ, σ) is an equivalence between G1 and G3 .
To prove that (i) =⇒ (ii), it suffices to show that if G is a groupoid and
f : T → G(0) is surjective, that G[T ] is equivalent to G in the sense (ii). Let
Z = T ×f,r G. Let ρ(t, g) = t and σ(t, g) = s(g). Define the actions of G[T ] and
G by (t0 , t, γ) · (t, g) = (t0 , γg) and (t, g) · γ = (t, gγ). One easily checks that the
two actions commute, and, using the fact that f is surjective, that the natural
maps Z/G → T and G[T ]\Z → G(0) are bijective.
To prove (ii) =⇒ (i), let Γ = G1 n Z o G2 = {(z 0 , z, g1 , g2 ) ∈ (Z × Z) × G1 ×
G2 | g1 z = z 0 g2 }. Then the maps (z 0 , z, g1 , g2 ) 7→ (z 0 , z, g1 ) and (z 0 , z, g1 , g2 ) 7→
(z 0 , z, g2 ) are isomorphisms between Γ and G1 (Z), and between Γ and G2 (Z)
respectively.
Let us examine Morita-equivalence for topological groupoids. The analogue
(0)
of Proposition 2.24 works nicely when ri : Gi → Gi are open. Thus we need a
stability lemma:
Lemma 2.25. 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 o G and G[T ] are open.
To prove Lemma 2.25 we need a preliminary result:
Lemma 2.26. 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 pr1 : X ×T
Y → X is open.
Proof. Let Ω ⊂ Z open. There exists an open subspace Ω0 of X × Y such
that Ω = Ω0 ∩ Z. Let ∆ be the diagonal in X × X. One easily checks that
(pr1 , pr1 )(Ω) = (1 × f )−1 (1 × g)(Ω0 ) ∩ ∆, therefore (pr1 , pr1 )(Ω) is open in ∆.
This implies that pr1 (Ω) is open in X.
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Proof of Lemma 2.25. The proof that the range map is open for ZoG = Z×G(0)
G is immediate from Lemma 2.26.
pr2
For G[T ], first use Lemma 2.26 to prove that T ×f,s G −−→ G is open. Since
pr2
r
the range map is open by assumption, the composition T ×f,s G −−→ G −
→ G(0) is
pr1
open. Using again Lemma 2.26, G[T ] ' T ×f,r◦pr2 (T ×f,s G) −−→ T is open. In order to define the notion of Morita-equivalence for topological groupoids,
we introduce some terminology:
Definition 2.27. 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 o G is
ρ-proper.
It is clear that properness implies ρ-properness. There is a partial converse:
Proposition 2.28. 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.
To understand the notion of ρ-properness better, let us consider the case
where T is locally Hausdorff:
Proposition 2.29. Let G be a topological groupoid, T a locally Hausdorff topological space, ρ : G(0) → T a G-invariant map. Then G is ρ-proper if and only
ρ−1 (V )
if for every Hausdorff open subspace V of T , Gρ−1 (V ) is proper.
Proof. Suppose that G is ρ-proper. Let V ⊂ T be a Hausdorff open subspace
and let W = ρ−1 (V ). Then (r, s) : G ∩ (r, s)−1 (W × W ) → (G(0) ×T G(0) ) ∩
(W × W ) is proper, i.e. (r, s) : GW
W → W ×V W is proper. Since V is Hausdorff,
W
W ×V W is closed in W × W , so (r, s) : GW
W → W × W is proper, i.e. GW is
proper.
Conversely, suppose that for every open Hausdorff subspace V of T and
W = ρ−1 (V ), GW
W is proper. Let F be a closed subspace of G and a ∈ (r, s)(F ).
Choose V such that ρ ◦ pr1 (a) ∈ V , then a ∈ (r, s)(F ∩ GW
W ) ∩ (W × W ) =
(r, s)(F ∩ GW
)
⊂
(r,
s)(F
),
therefore
(r,
s)
is
closed.
The
same
proof shows that
W
0
0
(r, s) × 1T is closed for any topological space T , therefore G is proper.
Proposition 2.30. Let G1 and G2 be two topological groupoids. Let ri , si
(i = 1, 2) be the range and source maps of Gi , and suppose that ri are open.
The following are equivalent:
(0)
(i) there exist a topological space T and fi : T → Gi
that G1 [T ] and G2 [T ] are isomorphic;
open surjective such
NON-HAUSDORFF GROUPOIDS
13
(0)
(ii) there exists a space Z, two continuous maps ρ : Z → G1 and σ : Z →
(0)
G2 , a left action of G1 on Z with momentum map ρ and a right action
of G2 on Z with momentum map σ such that
(a) the actions commute and are free, the action of G2 is ρ-proper and
the action of G1 is σ-proper;
(0)
(0)
(b) the natural maps Z/G2 → G1 and G1 \Z → G2 induced from ρ
and σ are homeomorphisms.
Moreover, one may replace (b) by
(0)
(0)
(b)’ ρ and σ are open and induce bijections Z/G2 → G1 and G1 \Z → G2 .
If G1 and G2 satisfy the equivalent conditions in Proposition 2.30, then they
(0)
are said to be Morita-equivalent. Note that if Gi are Hausdorff, then by
Proposition 2.28, one may replace “ρ-proper” and “σ-proper” by “proper”.
To prove Proposition 2.30, we need a lemma:
Lemma 2.31. Let X, Y and T be topological spaces, g : Y → T an open map
and f : X → T continuous. Let R be an equivalence relation on X such that
the canonical mapping π : X → X/R is open and such that f factors through
X/R. Then the canonical map
(X ×T Y )/(R × 1) → (X/R) ×T Y
is an isomorphism.
Proof. By Lemma 2.26, the first projection pr1 : X ×T Y → X is open. By
composition, π ◦ pr1 : X ×T Y → X/R is open. Let U ⊂ X and V ⊂ Y be
open subspaces. Then (π × 1)(U ×T V ) = π(U ) ×T V is an open subspace
of (X/R) ×T Y . Therefore, π × 1 : X ×T Y → (X/R) ×T Y is open. Since
the canonical mapping X ×T Y → (X ×T Y )/(R × 1) is surjective, the map
(X ×T Y )/(R × 1) → (X/R) ×T Y is open, hence an isomorphism.
Proof of Proposition 2.30. The last assertion follows from the fact that the
canonical mappings Z → Z/G2 and Z → G1 \Z are open (Lemma 2.34).
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
(Z1 , ρ1 , σ2 ) and (Z2 , ρ2 , σ2 ) are equivalences between G1 and G2 , and G2 and
G3 respectively. As in Proposition 2.24, let Z = Z1 ×G2 Z2 be the quotient of
(0)
Z1 ×G(0) Z2 by the action (z1 , z2 )·γ = (z1 γ, γ −1 z2 ) of G2 . Denote by ρ : Z → G1
2
(0)
and σ : Z → G3 the maps induced from ρ1 × 1 and 1 × σ2 . By Lemma 2.26,
the first projection pr1 : Z1 ×G(0) Z2 → Z1 is open, therefore ρ = ρ1 ◦ pr1 is
2
open. Similarly, σ is open. It remains to show that the actions of G3 and G1
are ρ-proper and σ-proper respectively. We show it for G3 , the proof for G1
being similar.
We need to show that the map Z o G3 → Z ×G(0) Z is a homeomorphism,
1
i.e. that the map Z ×G(0) Z → G3 defined by (z, zγ) 7→ γ is the unique γ ∈ G3
1
such that z 0 = zγ, is continuous.
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Since the action of G2 on Z1 is free and ρ1 -proper, there is an isomorphism
(Z1 ×G(0) Z2 ) ×G(0) (Z1 ×G(0) Z2 ) ' (Z1 o G2 ) ×G(0) ×G(0) (Z2 × Z2 ).
2
1
2
2
2
Taking the quotient by the action of G2 on the second factor (Z1 ×G(0) Z2 ) of
2
the left-hand side, and using Lemma 2.31, one gets
(Z1 ×G(0) Z2 ) ×G(0) (Z1 ×G2 Z2 ) ' Z1 ×G(0) (Z2 ×G(0) Z2 ).
2
2
1
2
Taking once again the quotient by G2 and using Lemma 2.31, we obtain
Z ×G(0) Z ' Z1 ×G2 (Z2 ×G(0) Z2 ).
2
1
Since the action of G3 on Z2 is free and ρ2 -proper, we have Z2 ×G(0) Z2 '
2
Z2 o G3 , therefore Z ×G(0) Z ' Z1 ×G2 (Z2 o G3 ). Now, the third projection
1
Z1 ×G(0) (Z2 o G3 ) → G3 is continuous and passes through the quotient by G2 ,
2
hence Z1 ×G2 (Z2 o G3 ) → G3 is continuous.
This proves that (ii) is an equivalence relation. Now, we proceed as in Proposition 2.24.
(0)
Suppose (ii). Let Γ = G1 n Z o G2 and T = Z. The maps ρ : T → G1 and
(0)
σ : T → G2 are open surjective by assumption. Since G1 n Z ' Z ×G(0) Z and
2
Z o G2 ' Z ×G(0) Z, we have G2 [T ] = (T × T ) ×G(0) ×G(0) G2 ' (Z o G2 ) ×s◦pr2 ,σ
1
2
2
Z ' (Z ×G(0) Z) ×σ◦pr2 ,σ Z = Z ×G(0) (Z ×G(0) Z) ' Z ×G(0) (G1 n Z) '
1
1
2
1
G1 n (Z ×G(0) Z) ' G1 n (Z o G2 ) = Γ. Similarly, Γ ' G1 [T ], hence (i).
1
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 pr1 -proper. Write Z o G = {(t, g, h) ∈
T × G × G| f (t) = r(g) and s(g) = r(h)}. One needs to check that the map
Z o G → (T ×f,r G)2 defined by (t, g, h) 7→ (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) 7→ (g, gh) are homeomorphisms onto their
images.
Let us check that the action of G[T ] is s ◦ pr2 -proper. One easily checks
that the groupoid G0 = G[T ] n (T ×f,r G) is isomorphic to a subgroupoid of
the trivial groupoid (T × T ) × (G × G). It follows that if r0 and s0 denote the
range and source maps of G0 , the map (r0 , s0 ) is a homeomorphism of G0 onto
its image.
Let us examine standard examples of Morita-equivalences:
Example 2.32. Let G be a topological groupoid whose range map is open.
Let (Ui )i∈I be an open cover of G(0) and U = qi∈I Ui . Then G[U] is Moritaequivalent to G.
Example 2.33. Let G be a topological groupoid, and let H1 , H2 be sub(0)
s(H )
groupoids such that the range maps ri : Hi → Hi are open. Then (H1 \Gs(H12 ) )o
s(H )
H2 and H1 n (Gs(H12 ) /H2 ) are Morita-equivalent.
NON-HAUSDORFF GROUPOIDS
15
s(H )
Proof. Take Z = Gs(H12 ) and let ρ : Z → Z/H2 and σ : H1 \Z be the canonical
mappings. The fact that these maps are open follows from Lemma 2.34 below.
Lemma 2.34. 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 2.25, the range map r : Z o G → Z is
open. The conclusion follows from Proposition 2.11.
The following proposition is an immediate consequence of Proposition 2.22.
Proposition 2.35. Let G and G0 be two topological groupoids such that the
range maps of G and G0 are open. Suppose that G and G0 are Morita-equivalent.
Then G is proper if and only if G0 is proper.
We now examine the notion of Morita-equivalence for locally compact groupoids.
Proposition 2.36. Let G1 and G2 be two locally compact groupoids. Let ri ,
si (i = 1, 2) be the range and source maps of Gi , and suppose that ri are open.
The following are equivalent:
(0)
(i) there exist a locally compact space T and fi : T → Gi open surjective
such that G1 [T ] and G2 [T ] are isomorphic;
(0)
(i)’ there exist a locally compact Hausdorff space T and fi : T → Gi open
surjective such that G1 [T ] and G2 [T ] are isomorphic;
(0)
(ii) there exist a locally compact space Z, two continuous maps ρ : Z → G1
(0)
and σ : Z → G2 , a left action of G1 on Z with momentum map ρ and
a right action of G2 on Z with momentum map σ such that
(a) the actions commute and are free, the action of G2 is ρ-proper and
the action of G1 is σ-proper;
(0)
(0)
(b) the natural maps Z/G2 → G1 and G1 \Z → G2 induced from ρ
and σ are homeomorphisms.
Moreover, one may replace (b) by
(0)
(0)
(b)’ ρ and σ are open and induce bijections Z/G2 → G1 and G1 \Z → G2 .
Proof. It is clear that (i)’ implies (i). Conversely, if (i) is true, let (Vi ) be an open
(0)
cover of T by Hausdorff open subspaces, and T 0 = qVi . Let fi0 : T 0 → Gi be the
composition fi ◦ q, where q : T 0 → T is the obvious map. Then G1 [T 0 ] ' G2 [T 0 ].
The equivalence (i) ⇐⇒ (ii) follows from the proof of Proposition 2.30 if we
can show that (ii) is an equivalence relation. Let (Z1 , ρ1 , σ1 ) be an equivalence
between G1 and G2 , and (Z2 , ρ2 , σ2 ) be an equivalence between G2 and G3 . Let
(0)
(0)
Z = Z1 ×G2 Z2 , ρ = ρ1 × 1 : Z → G1 and σ = 1 × σ2 : Z → G3 . To prove
that (Z, ρ, σ) is an equivalence between G1 and G3 , it remains to show that Z
(0)
is locally compact. Let U3 be a Hausdorff open subspace of G3 . We show that
16
JEAN-LOUIS TU
3
σ −1 (U3 ) is locally compact. Replacing G3 by (G3 )U
U3 , we may assume that G2
acts freely and properly on Z2 . Let Γ be the groupoid (Z1 ×G(0) Z2 ) o G2 , and
2
R = (r, s)(Γ) ⊂ (Z1 ×G(0) Z2 )2 . Since the action of G2 on Z2 is free and proper,
2
there exists a continuous map ϕ : Z2 ×G(0) Z2 → G2 such that z2 = ϕ(z2 , z20 )z20 .
3
Then R = {(z1 , z2 , z10 , z20 ) ∈ (Z1 ×G(0) Z2 )2 ; z10 = z1 ϕ(z2 , z20 )} is locally closed.
2
By Proposition 2.12, Z = (Z1 ×G(0) Z2 )/G is locally compact.
2
2.6. The slice property. Roughly speaking, an action of G on a space Z is
said to have the slice property if Z is locally G-isomorphic to Y ×K G, where
K is a quasi-compact subgroupoid of G and Y is a Hausdorff K-space. Since
slice-proper actions may be easier to handle than proper actions, we examine
the relation between these two notions.
Lemma 2.37. Let G be a topological groupoid acting on a space Z. Suppose
that for every z ∈ Z, there exists a closed saturated neighborhood Z 0 of z such
that G acts properly on Z 0 . Then G acts properly on Z.
Proof. The only difficulty is to show that (r, s) : Z o G → Z × Z is closed.
Let F be a closed subspace of Z o G. Let (z, z 0 ) ∈ (r, s)(F ). Choose a closed
saturated neighborhood Z 0 of z such that G acts properly on Z 0 . Then (z, z 0 ) ∈
(r, s)(F ) ∩ (Z 0 × Z) = (r, s)(F ∩ (Z 0 × G)). Since Z 0 is closed and saturated,
(r, s)(Z 0 o G) ⊂ Z 0 × Z 0 , so (z, z 0 ) ∈ (Z 0 × Z 0 ) ∩ (r, s)(F ∩ (Z 0 × G)) = (r, s)(F ∩
(Z 0 ×G)) since the action of G on Z 0 is proper. Therefore, (z, z 0 ) ∈ (r, s)(F ). Definition 2.38. Let G be a topological groupoid acting on a topological space
Z. We say that the action of G on Z is slice-proper if for every z ∈ Z, there
exists a closed saturated neighborhood Z 0 of z, a quasi-compact subgroupoid
K of G, a Hausdorff K-space Y such that Z 0 is G-isomorphic to Y ×K G.
In the definition above, Y ×K G is the space Y ×G(0) G divided by the equivalence relation (y, g) ∼ (yk, k −1 g) for all k ∈ K.
Before we examine the relation with properness, we need a few lemmas.
Lemma 2.39. Let G be a quasi-compact topological groupoid such that G(0) is
Hausdorff. Let Z be a G-space. Then the canonical mapping π : Z → Z/G is
proper.
Proof. Let us show that π is closed. Let F ⊂ Z be a closed subspace, and
denote by ϕ : Z o G → Z the action. Since G(0) is Hausdorff, ϕ−1 (F ) is closed
in Z × G, and since G is quasi-compact, the first projection pr1 : Z × G → Z is
proper. It follows that π −1 (π(F )) = F G = pr1 (ϕ−1 (F )) is closed, thus π(F ) is
closed.
Applying the above to the G-space Z × T , where T is an arbitrary space, one
gets that π × 1T is closed, i.e. π is proper.
Proposition 2.40. Let G be a topological groupoid with G(0) Hausdorff acting
on a space Z. If the action is slice-proper, then it is proper.
NON-HAUSDORFF GROUPOIDS
17
Proof. By Lemma 2.37, we may assume that Z = Y ×K G, where K is a quasicompact subgroupoid of G. By Proposition 2.16, G acts properly on G. From
Proposition 2.20, the right action of G on Y ×G(0) G is proper.
By Lemma 2.39, the canonical mapping Y ×G(0) G → Y ×K G is proper. It
then follows from Proposition 2.19 that the action of G on Y ×K G is proper. It is known that the converse holds if G is a Hausdorff group and G, Z and
Z/G are locally compact metrizable [1, 6]. Now, we examine the case of étale
groupoids.
Lemma 2.41. Let G be an étale groupoid such that G(0) is Hausdorff. Let
x0 ∈ G(0) and let Γ ⊂ Gxx00 be a finite subgroup. Then there exists a neighborhood
A of x0 , an action of Γ on A and an étale morphism φ : A o Γ → GA
A such that
φ(x0 , γ) = γ for all γ ∈ Γ.
Proof. Let V0 be a neighborhood of x0 and let ρg : V0 → G be local sections of
r such that ρg (x0 ) = g for all g ∈ Γ. Let ϕg = s ◦ ρg . By continuity of the
product on G, and using the fact that G is étale, there exists a neighborhood
V1 ⊂ V0 of x0 such that for all g, h ∈ Γ and all y ∈ V1 ,
ρg (y)ρh (ϕg (y)) = ρgh (y).
Let A = ∩g∈Γ ρg (V1 ). The action of Γ on A is defined by y · γ = ϕγ (y) and the
morphism φ is φ(y, γ) = ϕγ (y).
Proposition 2.42. Let G be an étale groupoid such that G(0) is Hausdorff,
acting properly on a locally compact space Z with momentum map p : Z →
G(0) . Then the action is slice-proper. More precisely, if z0 ∈ Z, let Γ be
the stabilizer of z0 . Then there exists a compact neighborhood A of x = p(z), a
quasi-compact subgroupoid G1 of the form φ(AoΓ) where φ is as in Lemma 2.41,
a G1 -invariant compact neighborhood Y of z0 such that Y G is G-isomorphic to
Y ×G1 G.
Proof. We first show that there exists a neighborhood V of z0 such that V ∩V g 6=
∅ =⇒ g ∈ G1 .
Let ϕ : Z o G → Z × Z be the proper map ϕ(z, g) = (z, zg). Let C =
(Z o G) − (Z × G̊1 ). Since C is closed and G1 is a neighborhood of Γ, we have
(z0 , z0 ) ∈
/ ϕ(C) = ϕ(C). Let V be a neighborhood of z0 such that (V × V ) ∩
ϕ(C) = ∅. If V ∩ V g 6= ∅, then there exists z ∈ V such that ϕ(z, g) ∈ V × V ,
hence g ∈ G1 .
Now, let Y be a compact, G1 -invariant neighborhood of z0 such that Y ⊂
V . The map Y ×G1 G → Y G defined by (y, γ) 7→ yγ is clearly well-defined,
continuous, G-equivariant. It is bijective since Y ∩ Y g 6= ∅ =⇒ g ∈ G1 . We
show that it is a homeomorphism. The map ϕ : (Y G) ×G(0) G → (Y G) × (Y G)
defined by ϕ(z, g) = (z, zg) is closed (since by Corollary 2.23 the action of G on
Y G is proper), thus its restriction Y ×G(0) G = ϕ−1 (Y × (Y G)) → Y × (Y G) is
closed. Since Y is compact, Y → {∗} is proper, thus pr2 : Y × (Y G) → Y G is
closed. By composition, the map Y ×G(0) G → Y G, (y, g) 7→ yg is closed, hence
a homeomorphism.
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3. A topological construction
Let X be a locally compact space. Since X is not necessarily Hausdorff, a
filter1 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.
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 (Vs )s∈S of open sets such that s ∈ Vs , and Vs = X
except perhaps for finitely many s’s, one has ∩s∈S Vs 6= ∅;
(ii) for every finite family (Vi )i∈I of open sets such that S ∩ Vi 6= ∅ for all
i, one has ∩i∈I Vi 6= ∅.
Proof. (i) =⇒ (ii): let (Vi )i∈I as in (ii). For all i, choose s(i) ∈ S ∩ Vi . Put
Ws = ∩s=s(i) Vi , with the convention that an empty intersection is X. Then by
(i), ∅ =
6 ∩s∈S Ws = ∩i∈I Vi .
(ii) =⇒ (i): let (Vs )s∈S as in (i), and let I = {s ∈ S| Vs 6= X}. Then
∩s∈S Vs = ∩i∈I Vi 6= ∅.
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 6= b and {a, b} ⊂ V ∩ S. Then there exist Va , Vb 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 ΩV = {S ∈ HX| V ∩ S 6= ∅} and ΩQ = {S ∈
HX| Q ∩ S = ∅}. The topology on ĤX is generated by the ΩV ’s and ΩQ ’s (V
open and Q quasi-compact). More explicitly, a set is open if and only if it is
Q
a union of sets of the form ΩQ
(Vi )i∈I = Ω ∩ (∩i∈I ΩVi ) where (Vi )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 6⊂ S 0 and S, S 0 ∈ ĤX. Let s ∈ S − S 0 . Since S 0 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 0 = ∅. Then ΩV and ΩK are disjoint
neighborhoods of S and S 0 respectively.
For every filter F on ĤX, let
(1)
L(F) = {a ∈ X| ∀V 3 a open, ΩV ∈ F}.
1or a net; we will use indifferently the two equivalent approaches
NON-HAUSDORFF GROUPOIDS
19
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 6= ∅ =⇒ ΩV ∈ F;
(b) ∀Q quasi-compact, Q ∩ S = ∅ =⇒ ΩQ ∈ 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 ΩV ’s and ΩQ ’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 ΩK ∈ F, and condition (b)
implies ΩK ∈ F, thus ∅ = ΩK ∩ ΩK ∈ F, which is impossible: we have proved
the reverse inclusion L(F) ⊂ S.
Remark 3.5. This means that if Sλ → S, then a ∈ S if and only if ∀λ there
exists sλ ∈ Sλ such that sλ → a.
Example 3.6. Consider Example 2.3 with Γ = Z2 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. Therefore, HG is
homeomorphic to the disjoint union of [0, 1] and two points a, b, and G embeds
non continuously in HG as the subspace (0, 1] ∪ {a, b}.
Proposition 3.7. Let X be a locally compact space and K ⊂ X quasi-compact.
Then L = {S ∈ HX| S ∩ K 6= ∅} 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 S0 = L(F). Let us show that S0 ∩K 6= ∅:
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 ∈ S0 .
Let us show S0 ∈ HX: let (Vs ) (s ∈ S0 ) be a family of open subspaces of X
such that s ∈ Vs for all s ∈ S0 , and Vs = X for every s ∈
/ S1 (S1 ⊂ S0 finite).
By definition of S0 , Ω(Vs )s∈S1 = ∩s∈S1 ΩVs belongs to F, hence it is non-empty.
Choose S ∈ Ω(Vs )s∈S1 , then S ∩ Vs 6= ∅ for all s ∈ S1 . By Lemma 3.1(ii),
∩s∈S1 Vs 6= ∅. This shows that S0 ∈ HX.
Now, let us show that F converges to S0 .
• If V is open Hausdorff such that S0 ∈ ΩV , then by definition ΩV ∈ F.
• If Q is quasi-compact and S0 ∈ ΩQ , then ΩQ ∈ F, otherwise one would
have {S ∈ HX| S ∩ Q 6= ∅} ∈ F, which would imply as above that
S0 ∩ Q 6= ∅, a contradiction.
From Lemma 3.4, F converges to S0 .
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.
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Remark 3.9. Let us note that our construction is not functorial: if f : X → Y
is a continuous map between two locally compact spaces, then f does induce
a map HX → HY defined by S 7→ f (S), but that map is not continuous in
general. However, 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 ) = Ωf −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 ∈ Gx , write Sg = {sg| s ∈ S}.
Let us show that Sg ∈ HZ. Let Vs (s ∈ S) be open sets such that sg ∈ Vs .
By continuity, there exist open sets Ws 3 s and Wg 3 g such that for all
(z, h) ∈ Ws ×G(0) Wg , zh ∈ Vs . Let Vs0 = Ws ∩ p−1 (r(Wg )). Then Vs0 is an open
neighborhood of s, so there exists z ∈ ∩s∈S Vs0 . Since p(z) ∈ r(Wg ), there exists
h ∈ Wg such that p(z) = r(h). It follows that zh ∈ ∩s∈S Vs . 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 (Sλ , gλ ) → (S, g) and let
S 0 = L((Sλ , gλ )). Then for all a ∈ S there exists sλ ∈ Sλ such that sλ → a.
This implies sλ gλ → ag, thus ag ∈ S 0 . The converse may be proved in a similar
fashion, hence Sg = S 0 .
Applying this to any universal net (Sλ , gλ ) converging to (S, g) and knowing
from Proposition 3.8 that Φ(Sλ , gλ ) is convergent in ĤZ, we find that Φ(Sλ , gλ )
converges to Φ(S, g). This shows that Φ is continuous in (S, g).
3.2. The space H0 X. Let X be a locally compact space. Let Ω0V = {S ∈
HX| S ⊂ V }. Let H0 X be HX as a set, with the coarsest topology such that
the identity map H0 X → HX is continuous, and Ω0V is open for every relatively
quasi-compact open set V .
Lemma 3.11. Let X be a locally compact space. Then the map
H0 X → N∗ ∪ {∞}
S 7→ #S
is upper semi-continuous.
Proof. Let S ∈ H0 X such that #S < ∞. Let Vs (s ∈ S) be open relatively
compact Hausdorff sets such that s ∈ Vs , and let W = ∪s∈S Vs . Then S 0 ∈ H0 X
implies #(S 0 ∩ Vs ) ≤ 1, therefore S 0 ∈ Ω0W implies #S 0 ≤ #S.
Let us note that H0 X is Hausdorff (since HX is Hausdorff), but it is not
necessarily locally compact. For instance, let X = ∪t∈[0,1] {t} × Xt , where
Xt = {0, 1} if t = 1/n for some n ∈ N∗ , and Xt = {0} otherwise. The topology
NON-HAUSDORFF GROUPOIDS
21
on X is such that (t, n) 7→ t is étale from X to [0, 1]. Let V = [0, 1] × {0}. Then
Ω0V is open in H0 X. If K is a compact neighborhood of (0, 0) contained in Ω0V ,
then K contains W = [0, ε)×{0} for some ε > 0, hence K ⊃ W . But if n > 1/ε,
then {(1/n, 0), (1/n, 1)} ∈ W , which contradicts {(1/n, 0), (1/n, 1)} ∈
/ Ω0V .
3.3. Property (P). Let X be a topological space. For every S ∈ HX, let
K̃S = {a ∈ X| S ∪ {a} ∈ HX}. Let KS = K̃S − S.
Lemma 3.12. Let X be a locally compact space. Then for every S ∈ HX, KS
is closed, and is contained in any closed neighborhood of S.
Proof. Is easy to see that K̃S is the intersection of all sets of the form ∩i∈I Vi ,
where (Vi )i∈I is a finite family of open sets such that S ⊂ ∪i∈I Vi and S ∩ Vi 6= ∅
for all i. Thus, K̃S is closed, and contained in any closed neighborhood of S.
Every s ∈ S has a Hausdorff open neighborhood Ws . It is easy to see that
KS ∩ Ws = ∅, therefore S is open in S ∪ KS . The lemma follows.
Lemma 3.13. Let X be a locally compact space. Then for every quasi-compact
set K ⊂ X, F = ∪x∈K K̃{x} is closed.
Proof. Let y ∈
/ F . For every x ∈ K there exist Vx 3 x and Wx 3 y disjoint open.
Let A ⊂ K finite such that V = ∪a∈A Va contains K and let W = ∩a∈A Wa .
Then for every x ∈ K, X − W is a closed neighborhood of x, hence from
Lemma 3.12, K̃{x} ⊂ X − W . It follows that W ∩ F = ∅, whence y ∈
/ F.
Let us consider the property
(P) X is locally compact, and the closure of every quasi-compact subspace
of X is quasi-compact.
Proposition 3.14. Let X be a locally compact space. Let ∆ ⊂ X × X be the
diagonal. Consider the following properties:
(i) X satisfies property (P);
(ii) the closure of every compact subspace of X is quasi-compact;
(iii) every point in X has a closed quasi-compact neighborhood;
(iv) for every compact subspace K ⊂ X, ∪x∈K K̃{x} is quasi-compact;
(v) for every compact subspace K ⊂ X, there exists a quasi-compact set K 0
such that
∀S ∈ HX, S ∩ K 6= ∅ =⇒ K̃S ⊂ K 0 ;
(vi) the first projection pr1 : ∆ → X is proper;
(vii) the natural map H0 X → HX is a homeomorphism, and #S < ∞ for
every S ∈ HX;
(viii) for every compact subspace K ⊂ X, there exists CK > 0 such that
∀S ∈ HX, S ∩ K 6= ∅ =⇒ #S ≤ CK .
Then (i)–(vii) are equivalent, and imply (viii).
Proof. (i) =⇒ (ii) is obvious. (ii) =⇒ (iii) follows from the fact that every point
in X has a compact neighborhood K, and K is closed quasi-compact. To show
(iii) =⇒ (i), let K be a quasi-compact subspace of X. For all a ∈ K, let Ka
be a quasi-compact closed neighborhood of a. There exists A ⊂ K finite such
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that F = ∪a∈A Ka ⊃ K. Then F is closed, quasi compact, and contains K.
Therefore, K, being a closed subspace of F , is quasi-compact.
(iv) =⇒ (iii): from Lemma 3.13, F = ∪x∈K K̃{x} is a closed quasi-compact
neighborhood of x if K is a compact neighborhood of x.
(v) =⇒ (iv): suppose (v). Then ∪x∈K K̃{x} is a closed subspace of K 0 , hence
is quasi-compact.
(i) =⇒ (v): Let K1 be a quasi-compact neighborhood of K and let K 0 = K 1 .
From Lemma 3.12, we have S ⊂ K 0 .
(v) =⇒ (viii): Let K 0 as above and let (Vi )i∈I be a finite cover of K 0 by open
Hausdorff sets. For all b ∈ S, let Ib = {i ∈ I| b ∈ Vi }. By Lemma 3.2, the Ib ’s
(b ∈ S) are disjoint, whence one may take CK = #I.
(i) =⇒ (vi): let K ⊂ X compact. Let L ⊂ X quasi-compact such that
K ⊂ L̊. If (a, b) ∈ ∆ ∩ (K × X), then b ∈ L: otherwise, L × Lc would
be a neighborhood of (a, b) whose intersection with ∆ is empty. Therefore,
pr1−1 (K) = ∆ ∩ (K × L) is quasi-compact.
(vi) =⇒ (ii): let K ⊂ X be a compact set. Let L = pr2 (pr1−1 (K)). Then L
is quasi-compact and ∆ ∩ (K × X) = ∆ ∩ (K × L). If x ∈ K, there exists a
ultrafilter F on K convergent to x. Since K is compact, F also converges to an
element a ∈ K. It follows that (a, x) ∈ ∆ ∩ (K × X), hence x ∈ L. This shows
that K is a closed subspace of L, therefore K is quasi-compact.
(vi) =⇒ (vii): from (viii), #S < ∞ for all S ∈ HX. To show that H0 X →
HX is a homeomorphism, it remains to prove that Ω0V is open in HX for every
relatively quasi-compact open set V ⊂ X. Let S ∈ Ω0V , a ∈ S and K a compact
neighborhood of a. Let L = pr2 (∆ ∩ (K × X)). Then Q = L − V is quasicompact, and S ∈ ΩQ ⊂ Ω0V , therefore Ω0V is a neighborhood of each of its
K̊
points.
(vii) =⇒ (vi): let K be a compact subspace of X. By Proposition 3.7,
L = {S ∈ HX| S ∩ K 6= ∅} is compact. For every S ∈ L, since S is finite, there
exists a quasi-compact neighborhood QS of S. The open sets Ω0Q̊ cover L, so
S
there exists S ⊂ L finite such that L ⊂ ∪S∈S Ω0Q̊ . It follows that ∀S ∈ HX,
S
S ∩ K 6= ∅ =⇒ S ⊂ Q̊, where Q = ∪S∈S QS . In particular, for S = {(a, b)},
(a, b) ∈ ∆ and a ∈ K implies b ∈ Q, i.e. pr1−1 (K) = ∆ ∩ (K × Q).
Proposition 3.15. Let G be a locally compact proper groupoid. If G(0) satisfies
(P) (for instance if G(0) is Hausdorff ), then G satisfies (P).
Proof. Let K ⊂ G be a quasi-compact subspace. Then L = r(K) ∪ s(K) is
L
quasi-compact, thus GL
L is also quasi-compact. But K is closed and K ⊂ GL ,
therefore K is quasi-compact.
4. Haar systems
4.1. The space Cc (X). For every locally compact space X, Cc (X)0 will denote
the set of functions f ∈ Cc (V ) (V open Hausdorff), extended by 0 outside V .
Let Cc (X) be the linear span of Cc (X)0 . Note that functions in Cc (X) are not
necessarily continuous.
NON-HAUSDORFF GROUPOIDS
23
Proposition 4.1. Let X be a locally compact space, and let f : X → C. The
following are equivalent:
(i) f ∈ Cc (X);
(ii) f −1 (C∗ ) is relatively quasi-compact, and for every filter F on X, let
F̃ = i(F), where i : X → HX
P is the canonical inclusion; if F̃ converges
to S ∈ HX, then limF f = s∈S f (s).
Proof. Let us show (i) =⇒ (ii). By linearity, it is enough to consider the case
f ∈ Cc (V ), where V ⊂ X is open Hausdorff. Let K be the compact set
f −1 (C∗ ) ∩ V . Then f −1 (C∗ ) ⊂ K. Let F and S as in (ii). If S ∩ V = ∅, then
P
S ∈ ΩK , hence ΩK ∈ F̃, i.e. X − K ∈ F. Therefore, limF f = 0 = s∈S f (s).
P If S ∩ V = {a}, then a is a limit point of F, therefore limF f = f (a) =
s∈S f (s).
Let us show (ii) =⇒ (i) by induction on n ∈ N∗ such that there exist V1 , . . . Vn
open Hausdorff and K quasi-compact satisfying f −1 (C∗ ) ⊂ K ⊂ V1 ∪ · · · ∪ Vn .
For n = 1, for every x ∈ V1 , let F be a ultrafilter convergent
P to x. By
Proposition 3.8, F̃ is convergent; let S be its limit, then limF f = s∈S f (s) =
f (x), thus f|V1 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 V10 , . . . , Vn0 open sets, K1 . . . , Kn compact
such that K ⊂ V10 ∪ · · · ∪ Vn0 and Vi0 ⊂ Ki ⊂ Vi . Let F = (V10 ∪ · · · ∪ Vn0 ) − (V10 ∪
0 ). Then F is closed in V 0 and f
· · · ∪ Vn−1
|F is continuous. Moreover, f|F = 0
n
0 ) which is closed in K, hence quasi-compact,
outside K 0 = K − (V10 ∪ · · · ∪ Vn−1
and Hausdorff, since K 0 ⊂ Vn0 . Therefore, f|F ∈ Cc (F ). It follows that there
exists an extension h ∈ Cc (Vn0 ) of f|F . By considering f −h, we may assume that
f = 0 on F , so f = 0 outside K 0 = K1 ∪ · · · ∪ Kn−1 . But K 0 ⊂ V1 ∪ · · · ∪ Vn−1 ,
hence by induction hypothesis, f ∈ Cc (X).
Corollary 4.2. Let X be a locally compact space, f : X → C, fn ∈ Cc (X).
Suppose that there exists fixed quasi-compact set Q ⊂ X such that fn−1 (C∗ ) ⊂ Q
for all n, and fn converges uniformly to f . Then f ∈ Cc (X).
Lemma 4.3. Let X be a locally compact space. Let (Ui )i∈I be an open P
cover
of X by Hausdorff subspaces. Then every f ∈ Cc (X) is a finite sum f =
fi ,
where fi ∈ Cc (Ui ).
Proof. See [14, Lemma 1.3].
Lemma 4.4. Let X and Y be locally compact spaces. Let f ∈ Cc (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 fn ∈ Cc (V ) ⊗ Cc (W )
such that limn→∞ kf − fn k∞ = 0.
Proof. We may assume that X = V and Y = W . Let (Ui ) (resp. (Vj )) be
an open cover of X (resp. Y ) by Hausdorff subspaces. Then every element of
Cc (X × Y ) is a linear combination of elements of Cc (Ui × Vj ) (Lemma 4.3). The
conclusion follows from the fact that the image of Cc (Ui )⊗Cc (Vj ) → Cc (Ui ×Vj )
is dense.
Lemma 4.5. Let X be a locally compact space and Y ⊂ X a closed subspace.
Then the restriction map Cc (X) → Cc (Y ) is well-defined and surjective.
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Proof. Let (Ui )i∈I be a cover of X by Hausdorff open subspaces. The map
Cc (Ui ) → Cc (Ui ∩ Y ) is surjective (since Y is closed), and ⊕i∈I Cc (Ui ∩ Y ) →
Cc (Y ) is surjective (Lemma 4.3). Therefore, the map ⊕i∈I Cc (Ui ) → Cc (Y ) is
surjective. Since it is also the composition of the surjective map ⊕i∈I Cc (Ui ) →
Cc (X) and of the restriction map Cc (X) → Cc (Y ), the conclusion follows. Remark 4.6. Let G be a locally compact groupoid with G(0) Hausdorff. The
following are equivalent:
(i) G is Hausdorff;
(ii) G(0) is closed in G;
(iii) any net on G(0) has at most one limit in G;
(iv) the restriction of an element of Cc (G) to G(0) belongs to Cc (G(0) ).
Proof. (i) =⇒ (ii) =⇒ (iii) are obvious, and (ii) =⇒ (iv) is a consequence of
Lemma 4.5.
To show (iii) =⇒ (i), let gλ ∈ G and suppose that (gλ ) has two limits g and
h ∈ G. By continuity of the operations in a groupoid, gλ−1 gλ converges to g −1 g
and to g −1 h, thus g −1 g = g −1 h, which implies g = h.
To prove (iv) =⇒ (iii), let xλ ∈ G(0) and suppose that (xλ ) has two distinct
limits g and h ∈ G. Replacing h by h−1 h if necessary, we may assume that h is
an element x ∈ G(0) . Let V be a Hausdorff neighborhood of g. Let f ∈ Cc (V )
such that f (g) = 1, and suppose that its restriction to G(0) is continuous. Since
xλ → g, f (xλ ) converges to 1, thus by continuity f (x) = 1. It follows that
x ∈ V . Since x and g are limits of (xλ ) and V is Hausdorff, we get that g = x,
a contradiction.
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 Cc (G(0) ) is endowed with the Cr∗ (G)-valued scalar product hξ, ηi(g) =
ξ(r(g))η(s(g)). Its completion is a Cr∗ (G)-Hilbert module. However, if G is
not Hausdorff, the function g 7→ ξ(r(g))η(s(g)) does not necessarily belong to
Cc (G), therefore we need a different construction in order to obtain a Cr∗ (G)module.
Definition 4.7. [24, pp. 16-17] Let G be a locally compact groupoid such
that Gx is Hausdorff for every x ∈ G(0) . A Haar system is a family of positive
measures λ = {λx | x ∈ G(0) } such that
(i) ∀x ∈ G(0) , supp(λx ) = Gx ;
(ii) ∀x ∈ G(0) , ∀ϕ ∈ Cc (G),
Z
λ(ϕ) : x 7→
ϕ(g) λx (dg)
∈ Cc (G(0) );
g∈Gx
(iii) ∀x, y ∈ G(0) , ∀g ∈ Gyx , ∀ϕ ∈ Cc (G),
Z
Z
x
ϕ(gh) λ (dh) =
h∈Gx
h∈Gy
ϕ(h) λy (dh).
NON-HAUSDORFF GROUPOIDS
25
Note that the condition that Gx is Hausdorff is automatically satisfied if G(0)
is Hausdorff (Proposition 2.8).
Lemma 4.8. [24, p. 17] Let G be a locally compact groupoid with Haar system.
Then the range map r : G → G(0) is open.
Proof. Let g ∈ G. Let U0 be a Hausdorff open neighborhood of g. Let V ⊂ G(0)
be a Hausdorff open neighborhood of r(g), and Rlet U = U0 ∩ r−1 (V ). Let
f ∈ Cc (U )+ such that f (g) > 0, and let ϕ(x) = Gx f dλx . Then ϕ ∈ Cc (V ),
therefore ϕ−1 (C∗ ) is an open neighborhood of r(g) which is included in r(U ).
Consequently, r(U0 ) is a neighborhood of r(g).
Lemma 4.9. Let G be a locally compact groupoid with Haar system. Then for
every quasi-compact subspace K of G, supx∈G(0) λx (K ∩ Gx ) < ∞.
Proof. It is easy to show that there exists f ∈ Cc (G) such that 1K ≤ f . Since
supx∈G(0) λ(f )(x) < ∞, the conclusion follows.
Lemma 4.10. 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
(0) is continuous. Then for every f ∈ C (Z ×
p
c
p,r G), λ(f ) : z 7→
R: Z → G
p(z) (dg) belongs to C (Z).
f
(z,
g)
λ
c
g∈Gp(z)
Proof. By Lemma 4.5, f is the restriction
of Cc (Z × G).
R of an element
x
If f (z, g) = f1 (z)f2 (g), then ψ(x) = g∈Gx f2 (g) λ (dg) belongs to Cc (G(0) ),
therefore ψ ◦ p ∈ Cb (Z). It follows that λ(f ) = f1 (ψ ◦ p) belongs to Cc (Z).
By linearity, if f ∈ Cc (Z) ⊗ Cc (G), then λ(f ) ∈ Cc (Z).
Now, for every f ∈ Cc (Z × G), there exist relatively quasi-compact open
subspaces V and W of Z and G and a sequence fn ∈ Cc (V ) ⊗ Cc (W ) such that
fn converges uniformly to f . From Lemma 4.9, λ(fn ) converges uniformly to
λ(f ), and λ(fn ) ∈ Cc (Z). From Corollary 4.2, λ(f ) ∈ Cc (Z).
Proposition 4.11. 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 (λp(z) )z∈Z is a Haar system on Z o G.
Proof. Results immediately from Lemma 4.10.
5. The Hilbert module of a proper groupoid
5.1. The space X 0 . 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 0 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 0 , 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 0 , i.e. ∃x0 ∈ G(0) , S ⊂ Gxx00 .
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 Gxx00 .
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JEAN-LOUIS TU
q(S)
Denote by q : X 0 → G(0) the map such that S ⊂ Gq(S) . The map q is
continuous since it is the restriction to X 0 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 0 , convergent to S. Suppose that q(F) converges to S0 ∈ X 0 . Then S0 is a normal subgroup of S, and there exists
Ω ∈ F such that ∀S 0 ∈ Ω, S 0 is group-isomorphic to S/S0 . In particular,
{S 0 ∈ X 0 | #S = #S0 #S 0 } ∈ F.
Proof. Recall (Proposition 3.15) that G has property (P). In particular, S is
finite.
We shall use the notation Ω̃(Vi )i∈I = Ω(Vi )i∈I ∩ Ω0∪i∈I Vi . Let Vs0 ⊂ Vs (s ∈ S)
be Hausdorff, open neighborhoods of s, chosen small enough so that for some
Ω ∈ F,
(a) Ω ⊂ Ω̃(Vs0 )s∈S ;
(b) Vs01 Vs02 ⊂ Vs1 s2 , ∀s1 , s2 ∈ S.
(c) ∀s ∈ S − S0 , ∀S 0 ∈ Ω, q(S 0 ) ∈
/ Vs ;
(d) q(Ω) ⊂ Ω̃(Vs )s∈S0 ;
Let S 0 ∈ Ω. Let ϕ : S → S 0 such that {ϕ(s)} = S 0 ∩ Vs0 . Then ϕ is well-defined
since S 0 ∩ Vs0 6= ∅ (see (a)) and Vs0 is Hausdorff.
If s1 , s2 ∈ S then ϕ(si ) ∈ S 0 ∩ Vs0i . By (b), ϕ(s1 )ϕ(s2 ) ∈ S 0 ∩ Vs1 s2 . Since
Vs1 s2 is Hausdorff and also contains ϕ(s1 s2 ) ∈ S 0 , we have ϕ(s1 s2 ) = ϕ(s1 )ϕ(s2 ).
This shows that ϕ is a group homomorphism.
The map ϕ is surjective, since S 0 ⊂ ∪s∈S Vs0 (see (a)).
By (c), ker(ϕ) ⊂ S0 and by (d), S0 ⊂ ker(ϕ).
0 = {S ∈ X 0 | #S ≥ k}. By Lemma 3.11, X 0
Let X≥k
≥k is closed.
Example 5.3. Let N ⊂ H be normal subgroups of a discrete group Γ and let
G be the bundle of groups over [0, 1]2 with fiber G/N over [0, 1] × (0, 1], G/H
over (0, 1] × {0} and G over (0, 0). The topology is defined like in Example 2.3.
Let Sn = {(1/n, 0)} × (N/H), then Sn converges to S := {(0, 0)} × N and
q(Sn ) = (1/n, 0) converges to S0 := {(0, 0)} × H. We thus have checked by
hand that Sn is isomorphic to S/S0 for all n.
Suppose now that the range map r : G → G(0) is open. Then X 0 is endowed
with an action of G (Proposition 3.10) defined by S · g = g −1 Sg = {g −1 sg| s ∈
S}. To understand the groupoid better, let us introduce the notation (HG)yx =
{S ∈ HG| S ⊂ Gyx }. As above, we can see that HG is the union of all (HG)yx .
Moreover, by Proposition 3.10, g(HG)yx h = (HG)tz for all g ∈ Gty and h ∈ Gxz .
Denote by G0 the closure of G in HG. The space G0 admits continous left
and right actions of G by Proposition 3.10.
Let ρ(S) = SS −1 = {st−1 | s, t ∈ S} and σ(S) = S −1 S. We note that
ρ(S) ∈ X 0 . Indeed, if S is the set of limits of the net (gλ ), then SS −1 is the set
of limits of the net (xλ ), where xλ = gλ gλ−1 ∈ G(0) . Moreover, q(ρ(S)) = y if
S ∈ (HG)yx , and a similar assertion holds for σ.
Note that for all S ∈ G0 one has S = SS −1 S, since S = lim gλ implies
S = lim gλ gλ−1 gλ .
NON-HAUSDORFF GROUPOIDS
27
For all S, T ∈ G0 such that σ(S) = ρ(T ), we have ST ∈ G0 . Indeed, choose
g ∈ S and h ∈ T , then ST = gS −1 ST T −1 h = gS −1 SS −1 Sh = gS −1 Sh (since
SS −1 is a group), thus ST = Sh ∈ G0 . We thus have checked that G0 is endowed
with a groupoid law.
Proposition 5.4. The map (S, g) →
7 (Sg, g) is an isomorphism of topological
groupoids from X 0 o G → {(S, g) ∈ G0 × G| g ∈ S} with inverse (S, g) 7→
(Sg −1 , g).
Proof. Easy, left to the reader.
5.2. Construction of the Hilbert module. Now, let G be a locally compact,
proper groupoid, with G(0) Hausdorff. Let
p
E 0 = {f ∈ Cc (X 0 )| f (S) = #Sf (q(S)) ∀S ∈ X 0 }.
(q(S) ∈ G(0) is identified to {q(S)} ∈ X 0 .)
Define, for all ξ, η ∈ E 0 :
hξ, ηi(g) = ξ(r(g))η(s(g)).
Lemma 5.5. For every ξ, η ∈ E 0 , hξ, ηi belongs to Cc (G).
Proof. Let F be a ultrafilter on G convergent to A ∈ HG. There exist x,
y ∈ G(0) such that A ⊂ Gyx . Choose a ∈ A. Then r(F) and s(F) converge
to S = Aa−1 and S 0 = a−1 A respectively. Let k = #A = #S = #S 0 . Let
f (g) = ξ(r(g))η(s(g)). Then
X
√
√
f (a) = kξ(y)η(x) = kξ(y) kη(x)
a∈A
= ξ(S)η(S 0 ) = lim(ξ ◦ r)(η ◦ s) = lim f.
F
F
X0
Let K ⊂
be a compact set such that ξ = 0 and η = 0 outside K. Let
1
K1 = q(K) and K2 = GK
K1 . Then K2 is quasi-compact, and f = 0 outside K2 .
Therefore, from Proposition 4.1, f ∈ Cc (G).
Lemma 5.6. Let G be a locally compact, proper groupoid with G(0) Hausdorff.
Suppose that G is endowed with a Haar system. For all ξ ∈ E 0 and all f ∈
Cc (G), put
Z
ξ(g −1 Sg)f (g −1 ) λx (dg).
(ξf )(S) =
g∈Gq(S)
Then ξf ∈ E 0 .
√
Proof. It is clear that (ξf )(S) = #S(ξf )(q(S)). Let us show that ξf ∈
Cc (X 0 ).
Since G acts continuously on the left and on the right on itself, and since r
and s are open (see Lemma 4.8), it follows from Proposition 3.10 that G acts
continuously on the left and on the right on HG, so G acts on X 0 by conjugation.
The groupoid X 0 o G is endowed with the Haar system (λq(S) )S∈X 0 . Since the
map φ(S, g) = (g −1 Sg, g −1 ) is a homeomorphism from X 0 o G onto X 0 o G,
we have (ξ ⊗ f ) ◦ φ ∈ Cc (X 0 o G) i.e. (S, g) 7→ ξ(g −1 Sg)f (g −1 ) belongs to
Cc (X 0 o G). Therefore, ξf ∈ Cc (X 0 ).
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JEAN-LOUIS TU
Let us check the positivity of the scalar product. Recall that for all x ∈ G(0)
there is a representation πG,x : C ∗ (G) → L(L2 (Gx )) such that for all a ∈ Cc (G)
and all η ∈ Cc (Gx ),
Z
(πG,x (a)η)(g) =
a(h)η(gh) λs(g) (dh).
h∈Gs(g)
By definition, kakCr∗ (G) = supx∈G(0) kπG,x (a)k.
Z
η(g)a(h)η(gh) λs(g) (dh)λx (dg)
hη, πG,x (a)ηi =
g∈Gx , h∈Gs(g)
Z
(2)
η(g)a(g −1 h)η(h) λx (dg)λx (dh).
=
g∈Gx , h∈Gs(g)
Therefore,
Z
hη, πG,x (hξ, ξi)ηi = g∈Gx
2
ξ(s(g))η(g) λ (dg) ≥ 0.
x
By density of Cc (Gx ) in L2 (Gx ), πG,x (hξ, ξi) is a positive element of L(L2 (Gx ))
for all x ∈ G(0) , i.e. hξ, ξi is a positive element of Cr∗ (G).
Let us check that hξ, ηf i = hξ, ηi ∗ f for all ξ, η ∈ E 0 and all f ∈ Cc (G):
Z
hξ, ηf i(g) = ξ(r(g))
η(s(h))f (h−1 ) λs(g) (dh)
h∈Gs(g)
Z
=
ξ(r(g))η(s(k))f (k −1 g) λr(g) (dk)
k∈Gr(g)
= (hξ, ηi ∗ f )(g).
One easily checks that (ξf )f1 = ξ(f ∗ f1 ) for all ξ ∈ E 0 and all f , f1 ∈ Cc (G).
Therefore, we have established:
Proposition 5.7. Let G be a locally compact, proper groupoid. Assume that G
is endowed with a Haar system, and that G(0) is Hausdorff. Then the completion
E(G) of E 0 with respect to the norm kξk = khξ, ξik1/2 is a Cr∗ (G)-Hilbert module.
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 “cutR
off” function c ∈ Cc (G(0) )+ such that for every x ∈ G(0) , g∈Gx c(s(g)) λx (dg) =
p
1, and the function g 7→ c(r(g))c(s(g)) defines projection in Cr∗ (G). However,
if G is not Hausdorff, then the above function does not belong to Cc (G) is
general, thus we need another definition of a cutoff function.
Lemma 6.1. Let G be a locally compact, proper groupoid with G(0) Hausdorff.
0 ). Then X
(0)
Let X≥k = q(X≥k
≥k is closed in G .
Proof. It suffices to show that for every compact subspace K of G(0) , X≥k ∩ K
0
is closed. Let K 0 = GK
K . Then K is quasi-compact, and from Proposition 3.7,
NON-HAUSDORFF GROUPOIDS
29
0 = K 00 ∩ X 0
K 00 = {S ∈ HG| S ∩ K 0 6= ∅} is compact. The set q −1 (K) ∩ X≥k
≥k
is closed in K 00 , hence compact; its image by q is X≥k ∩ K.
Lemma 6.2. Let G be a locally compact, proper groupoid, with G(0) Haus0 → R∗
dorff. Let α ∈ R. For every compact set K ⊂ G(0) , there exists f : XK
+
0
−1
0
continuous, where XK = q (K) ⊂ X , such that
0
∀S ∈ XK
,
f (S) = f (q(S))(#S)α .
Proof. Let K 0 = GK
K . It is closed and quasi-compact. From Proposition 3.7,
0 is quasi-compact. For every S ∈ X 0 , we have S ⊂ K 0 . Since G has
XK
K
0
0 = ∅. We can thus proproperty (P), there exists n ∈ N∗ such that X≥n+1
∩ XK
0 ∩ q −1 (X
∗
ceed by reverse induction: suppose constructed fk+1 : XK
≥k+1 ) → R+
α
0
−1
continuous such that fk+1 (S) = fk+1 (q(S))(#S) for all S ∈ XK ∩q (X≥k+1 ).
0 ∩ q −1 (X
0
−1
Since XK
≥k+1 ) is closed in the compact set XK ∩ q (X≥k ), there
0 ∩ q −1 (X ) → R of f
exists a continuous extension h : XK
≥k
k+1 . Replacing h(x)
0 ∩ q −1 (X )) ⊂ R∗ . Put
by sup(h(x), inf fk+1 ), we may assume that h(XK
≥k
+
fk (S) = h(q(S))(#S)α . Let us show that fk is continuous.
0 ∩ q −1 (X ), and let S be its limit. Since q(F)
Let F be a ultrafilter on XK
≥k
0 .
is a ultrafilter on K, it has a limit S0 ∈ XK
−1
0
such that q(S1 ) = q(ψ(S1 )).
For every S1 ∈ q (X≥k ), choose ψ(S1 ) ∈ X≥k
0
0
0
Let S ∈ XK ∩ X≥k be the limit of ψ(F).
0 ∩q −1 (X )| #S = #S #S } is an element
From Lemma 5.2, Ω1 = {S1 ∈ XK
0
1
≥k
0 | #S 0 = #S #S } is an element of ψ(F).
of F, and Ω2 = {S2 ∈ X≥k
0
2
• If #S0 > 1, then S 0 ∈ X≥k+1 , so S and S0 belong to q −1 (X≥k+1 ).
Therefore, fk (S1 ) = (#S1 )α h(q(S1 )) converges with respect to F to
(#S)α
(#S)α
h(S0 ) =
fk+1 (S0 ) = fk+1 (S)
α
(#S0 )
(#S0 )α
= fk+1 (q(S))(#S)α = h(q(S))(#S)α = fk (S).
• If S0 = {q(S)}, then fk (S1 ) = (#S1 )α h(q(S1 )) converges with respect
to F to (#S)α h(q(S)) = fk (S).
Therefore, fk is a continuous extension of fk+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 0 → R+ continuous such that
(a) c(S) = c(q(S))#S for all S ∈ X 0 ;
(b) ∀α ∈ G(0) /G, ∃x ∈ π −1 (α), c(x) 6= 0;
(c) ∀K ⊂ G(0) compact, supp(c) ∩ q −1 (F ) is compact, where F = s(GK ).
If moreover G admits a Haar system, then there exists c : X 0 → R+ continuous
satisfying (a), (b), (c) and
Z
(0)
(d) ∀x ∈ G ,
c(s(g)) λx (dg) = 1.
g∈Gx
Proof. There exists a locally finite cover (Vi ) of G(0) /G by relatively compact
open subspaces. Since π is open and G(0) is locally compact, there exists Ki ⊂
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G(0) compact such that π(Ki ) ⊃ Vi . Let (ϕi ) be a partition of unity associated
0
to the cover (Vi ). For every i, from Lemma 6.2, there exists ci : XK
→ R∗+
i
0
continuous such that ci (S) = ci (q(S))#S for all S ∈ XKi . Let
X
c(S) =
ci (S)ϕi (π(q(S))).
i
It is clear that c is continuous from X 0 to R+ , and that c(S) = c(q(S))#S.
Let us prove (b): let x0 ∈ G(0) . There exists i such that ϕi (π(x0 )) 6= 0.
Choose x ∈ Ki such that π(x) = π(x0 ), then c(x) ≥ ci (x)ϕi (π(x0 )) > 0.
Let us show (c). Note that F = π −1 (π(K)) is closed, so q −1 (F ) is closed.
Let K1 be a compact neighborhood of K and F1 = π −1 (π(K1 )). Let J =
{i|
/ J, ci (ϕi ◦π◦q) = 0 on q −1 (F1 ), therefore c =
P Vi ∩π(K1 ) 6= ∅}. Then for all i ∈
−1
j∈J cj (ϕj ◦π◦q) in a neighborhood of q (F ). Since for all i, supp(ci (ϕi ◦π◦q))
is compact and since J is finite, supp(c) ∩ q −1 (F ) ⊂ ∪i∈J supp(ci (ϕi ◦ 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
X
X
lim ϕ = #Sc(s(a)) =
c(s(g)) =
ϕ(g).
F
g∈S
g∈S
For every compact set K ⊂ G(0) ,
{g ∈ G| r(g) ∈ K and ϕ(g) 6= 0}
⊂ {g ∈ G| r(g) ∈ K and s(g) ∈ supp(c)}
⊂ GK
q(supp(c)∩q −1 (F )) ,
so GK ∩ {g ∈ G| ϕ(g) 6= 0} is included in a quasi-compact set. Therefore,
(0)
for every
R l ∈ Cc (Gx ), g 7→ l(r(g))ϕ(g) belongs to Cc (G). It follows that
h(x) = g∈Gx ϕ(g) λ (dg) is a continuous function. Moreover, for every x ∈ G(0)
there exists g ∈ Gx such that ϕ(g) 6= 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 = Zn and H = {0}, the cutoff function
is the unique continuous extension to X 0 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 Hausdorffpand G(0) /G is compact. Let c be a cutoff function.
Then the function p(g) = c(r(g))c(s(g)) defines a selfadjoint projection p ∈
Cr∗ (G), and E(G) is isomorphic to pCr∗ (G).
p
Proof. Let ξ0 (x) = c(x). Then one easily checks that ξ0 ∈ E 0 , hξ0 , ξ0 i = p
and ξ0 hξ0 , ξ0 i = ξ0 , therefore p is a selfadjoint projection in Cr∗ (G). The maps
E(G) → pCr∗ (G)
ξ 7→ hξ0 , ξi = phξ0 , ξi
NON-HAUSDORFF GROUPOIDS
31
and
pCr∗ (G) → E(G)
a 7→ ξ0 a = ξ0 pa
are inverses from each other.
7. Generalized homomorphisms and C ∗ -algebra correspondences
P. Muhly, J. Renault and D. Williams introduced a notion of equivalence of
groupoids [20], and showed that two (second countable) locally compact, Hausdorff groupoids with Haar system are Morita-equivalent. M. Macho Stadler and
M. O’Uchi introduced a notion of correspondence of groupoids [19], and showed
that a correspondence from G1 to G2 , where G1 and G2 are locally compact,
Hausdorff, second countable groupoids with Haar system, induces a correspondence from Cr∗ (G2 ) to Cr∗ (G1 ), i.e. a Cr∗ (G1 )-module E and a ∗-homomorphism
Cr∗ (G2 ) → L(E). They deduce that a groupoid homomorphism f : G1 → G2
satisfying certain conditions induces a correspondence such that Cr∗ (G2 ) maps
to K(E), and therefore induces an element of KK(Cr∗ (G2 ), Cr∗ (G1 )).
In this section, we introduce a notion of generalized homomorphism for locally
compact groupoids which are not necessarily Hausdorff, and a notion of locally
proper generalized homomorphism. That notion is slightly weaker than the
notion of correspondence introduced by Macho Stadler and O’Uchi, but it was
chosen because a generalized homomorphism is, up to Morita-equivalence, a
homomorphism in the ordinary sense.
Then, we show that a locally proper generalized homomorphism from G1 to
G2 which satisfies an additional condition induces a Cr∗ (G1 )-module E and a
∗-homomorphism Cr∗ (G2 ) → K(E), hence an element of KK(Cr∗ (G2 ), Cr∗ (G1 )).
7.1. Generalized homomorphisms.
Definition 7.1. [12, 16] Let G1 and G2 be two locally compact groupoids
(0)
whose range maps Gi → Gi are open. A generalized homomorphism from G1
to G2 is a triple (Z, ρ, σ) where
(0) ρ
σ
(0)
G1 ←
−Z−
→ G2 ,
Z is endowed with a left action of G1 with momentum map ρ and a right action
of G2 with momentum map σ which commute, such that
(a) the action of G2 is free and ρ-proper,
(0)
(b) ρ induces a homeomorphism Z/G2 ' G1 .
In Definition 7.1, one may replace (b) by (b)’ or (b)” below:
(0)
(b)’ ρ is open and induces a bijection Z/G2 → G1 .
(b)” the map Z o G2 → Z ×G(0) Z defined by (z, γ) 7→ (z, zγ) is a homeo1
morphism.
Example 7.2. Let G1 and G2 be two groupoids whose range maps are open. If
(0)
f : G1 → G2 is a groupoid homomorphism, let Z = G1 ×f,r G2 , ρ(x, γ) = x and
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JEAN-LOUIS TU
σ(x, γ) = s(γ). Define the actions of G1 and G2 by g·(x, γ)·γ 0 = (r(g), f (g)γγ 0 ).
Then (Z, ρ, σ) is a generalized homomorphism from G1 to G2 .
(0)
That ρ is open follows from the fact that the range map G2 → G2 is open
and from Lemma 2.26. The other properties in Definition 7.1 are easy to check.
7.2. Locally proper generalized homomorphisms.
Definition 7.3. Let G1 and G2 be two locally compact groupoids such that
(0)
the range maps Gi → Gi are open. A generalized homomorphism from G1 to
G2 is said to be locally proper if the action of G1 on Z is σ-proper.
Our terminology is justified by the following proposition:
Proposition 7.4. Let G1 and G2 be two locally compact groupoids such that
(0)
(0)
the range maps Gi → Gi are open, and G2 is Hausdorff. Let f : G1 →
G2 be a groupoid homomorphism. Then the associated generalized groupoid
homomorphism is locally proper if and only if the map (f, r, s) : G1 → G2 ×
(0)
(0)
G1 × G1 is proper.
(0)
(0)
Proof. Let ϕ : G1 ×f ◦s,r G2 → (G2 ×s,s G2 ) ×r×r,f ×f (G1 × G1 ) defined by
ϕ(g1 , g2 ) = (f (g1 )g2 , g2 , r(g1 ), s(g1 )). By definition, the action of G1 on Z is
(2)
proper if and only if ϕ is a proper map. Consider θ : G2 ×s,s G2 → G2 given
by (γ, γ 0 ) = (γ(γ 0 )−1 , γ 0 ). Let ψ = (θ × 1) ◦ ϕ. Since θ is a homeomorphism,
the action of G1 on Z is proper if and only if ψ is proper.
(0)
Suppose that (f, r, s) is proper. Let f 0 = (f, r, s) × 1 : G1 × G2 → G2 × G1 ×
(0)
(0)
(0)
G1 × G2 . Then f 0 is proper. Let F = {(γ, x, x0 , γ 0 ) ∈ G2 × G1 × G1 ×
G2 | s(γ) = r(γ 0 ) = f (x0 ), r(γ) = f (x)}. Then f 0 : (f 0 )−1 (F ) → F is proper, i.e.
ψ is proper.
(0)
Conversely, suppose that ψ is proper. Let F 0 = {(γ, y, x, x0 ) ∈ G2 × G2 ×
(0)
(0)
G1 × G1 | s(γ) = y}. Then ψ : ψ −1 (F 0 ) → F 0 is proper, therefore (f, r, s) is
proper.
Our objective is now to show the
Proposition 7.5. Let G1 , G2 , G3 be locally compact groupoids such that the
(0)
range maps r : Gi → Gi are open. Let (Z1 , ρ1 , σ1 ) and (Z2 , ρ2 , σ2 ) be two
generalized groupoid homomorphisms from G1 to G2 and from G2 to G3 respectively. Then (Z, ρ, σ) = (Z1 ×G2 Z2 , ρ1 × 1, 1 × σ2 ) is a generalized groupoid
homomorphism. If (Z1 , ρ1 , σ1 ) and (Z2 , ρ2 , σ2 ) are locally proper, then (Z, ρ, σ)
is locally proper.
Proposition 7.5 shows that locally compact groupoids whose range maps are
open constitute a category whose arrows are generalized homomorphisms, 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 homomorphisms. In particular, local
properness of generalized homomorphisms is invariant under Morita-equivalence.
All the assertions of Proposition 7.5 will follow from Lemma 7.6 below
NON-HAUSDORFF GROUPOIDS
33
Lemma 7.6. Let G2 and G3 be locally compact groupoids whose range maps
are open and X a topological space. Let Z1 and Z2 be locally compact spaces.
Suppose there are maps
ρ1
σ
(0) ρ2
σ
(0)
1
2
X ←− Z1 −→
G2 ←− Z2 −→
G3 ,
a right action of G2 on Z1 with momentum map σ1 , such that ρ1 is G2 -invariant
and the action of G2 is ρ1 -proper, a left action of G2 on Z2 with momentum
map ρ2 and a right ρ2 -proper action of G3 on Z2 with momentum map σ2 which
commutes with the G2 -action.
Then the action of G3 on Z = Z1 ×G2 Z2 is ρ1 -proper.
To prove Lemma 7.6, we need a few preliminary lemmas.
Lemma 7.7. 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 0 : X × T → (X × T )/G be the canonical mappings. Since π is open (Lemma 2.34), f ◦ π 0 = π × 1 is open. Since π 0 is
continuous surjective, it follows that f is open.
Lemma 7.8. 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 f¯: Y /G → Z/G is proper.
Proof. We first show that f¯ is closed. Let π : Y → Y /G and π 0 : Z → Z/G be
the canonical mappings. Let A ⊂ Y /G be a closed subspace. Since f is closed
and π is continuous, (π 0 )−1 (f¯(A)) = f (π −1 (A)) is closed. Therefore, f¯(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 7.7, f¯ × 1T is closed.
Proof of Lemma 7.6. Let ϕ : Z2 o G3 → Z2 ×G(0) Z2 be the map (z2 , γ) 7→
2
(z2 , z2 γ). By assumption, ϕ is proper, therefore 1Z1 × ϕ is proper. Let F =
{(z1 , z2 , z20 ) ∈ Z1 × Z2 × Z2 | σ1 (z1 ) = ρ2 (z2 ) = ρ2 (z20 )}. Then 1Z1 × ϕ : (1 ×
ϕ)−1 (F ) → F is proper, i.e. Z1 ×G(0) (Z2 o G3 ) → Z1 ×G(0) (Z2 ×G(0) Z2 ) is
2
2
2
proper. By Lemma 7.8, taking the quotient by G2 , we get that the map
α : Z o G3 → Z1 ×G2 (Z2 ×G(0) Z2 )
2
defined by (z1 , z2 , γ) 7→ (z1 , z2 , z2 γ) is proper.
By assumption, the map Z1 o G2 → Z1 ×X Z1 given by (z1 , g) 7→ (z1 , z1 g)
is proper. Endow Z1 o G2 with the following right action of G2 × G2 : (z1 , g) ·
(g 0 , g 00 ) = (z1 g 0 , (g 0 )−1 gg 00 ). Using again Lemma 7.8, the map
β : Z1 ×G2 (Z2 ×G(0) Z2 ) = (Z1 o G2 ) ×G2 ×G2 (Z2 × Z2 )
2
→ (Z1 ×X Z1 ) ×G2 ×G2 (Z2 × Z2 ) ' Z ×X Z
is proper.
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JEAN-LOUIS TU
By composition, β ◦ α : Z o G3 → Z ×X Z is proper.
7.3. Proper generalized morphisms.
Definition 7.9. Let G1 and G2 be locally compact groupoids whose range
maps are open. A generalized homomorphism (Z, ρ, σ) from G1 to G2 is said
to be proper if it is locally proper, and if for every quasi-compact subspace K
(0)
of G2 , σ −1 (K) is G1 -compact.
Examples 7.10.
(a) Let X and Y be locally compact spaces and f : X →
Y a continuous map. Then the generalized homomorphism (X, Id, f ) is
proper if and only if f is proper.
(b) Let f : G1 → G2 be a continuous homomorphism between two locally
compact groups. Let p : G2 → {∗}. Then (G2 , p, p) is proper if and only
if f is proper and f (G1 ) is co-compact in G2 .
(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 homomorphism 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 7.11. Let G1 and G2 be locally compact groupoids with Haar system
(0)
(0)
such that G1 and G2 are Hausdorff, and (Z, ρ, σ) a locally proper generalized
homomorphism from G1 to G2 . Then one can construct a Cr∗ (G1 )-Hilbert module EZ and a map π : Cr∗ (G2 ) → L(EZ ). Moreover, if (Z, ρ, σ) is proper, then π
maps to K(EZ ). Therefore, it gives an element of KK(Cr∗ (G2 ), Cr∗ (G1 )).
Note by example (c) above, this generalizes the construction of Section 5:
if G is a locally compact proper groupoid with Haar system such that G(0) is
Hausdorff, and Z = G(0) then E(G) is isomorphic to EZ .
Corollary 7.12. (see [20]) Let G1 and G2 be locally compact groupoids with
(0)
(0)
Haar system such that G1 and G2 are Hausdorff. If G1 and G2 are Moritaequivalent, then Cr∗ (G1 ) and Cr∗ (G2 ) are Morita-equivalent.
Corollary 7.13. Let f : G1 → G2 be morphism between two locally compact
(0)
(0)
groupoids with Haar system such that G1 and G2 are Hausdorff. If the re(0) then f
striction of f to (G1 )K
K is proper for each compact set K ⊂ (G1 )
induces a correspondence Ef from Cr∗ (G2 ) to Cr∗ (G1 ). If in addition for every
(0)
(0)
compact set K ⊂ G2 the quotient of G1 ×f,r (G2 )K by the diagonal action of
G1 is compact, then Cr∗ (G2 ) maps to K(Ef ) and thus f defines a KK-element
[f ] ∈ KK(Cr∗ (G2 ), Cr∗ (G1 )).
Proof. See Proposition 7.4 and Definition 7.9 applied to the generalized mor(0)
phism Zf = G1 ×f,r G2 as in Example 7.2
Remark 7.14. WHen G is Hausdorff, the G(0) , G-correspondence given by the
space G endowed with the canonical left action of G(0) and the canonical right
NON-HAUSDORFF GROUPOIDS
35
action of G satisfies the assumptions of Theorem 7.11. The resulting C0 (G(0) )Hilbert module is actually L2 (G) endowed with the left regular representation
of Cr∗ (G). However this construction is not valid for G non-Hausdorff since G(0)
does not act properly on G (use Lemma 2.16)! The left regular representation
of G will be constructed differently in Section 8.
The rest of the section is devoted to proving Theorem 7.11.
Let us first recall the construction of the correspondence when the groupoids
are Hausdorff [19]. It is the closure of Cc (Z) with the Cr∗ (G1 )-valued scalar
product
Z
hξ, ηi(g) =
ξ(zγ)η(g −1 zγ) λσ(z) (dγ),
γ∈(G2 )σ(z)
where z is an arbitrary element of Z such that ρ(z) = r(g). The right Cr∗ (G1 )module structure is defined ∀ξ ∈ Cc (Z), ∀a ∈ Cc (G1 ) by
Z
(ξa)(z) =
ξ(g −1 z)a(g −1 ) λρ(z) (dg),
and the left action of
g∈(G1 )ρ(z)
∗
Cr (G2 ) is
Z
b(γ)ξ(zγ) λσ(z) (dγ)
(bξ)(z) =
γ∈(G2
)σ(z)
for all b ∈ Cc (G2 ).
We now come back to non-Hausdorff groupoids. For every open Hausdorff
set V ⊂ Z, denote by V 0 its closure in H((G1 n Z)VV ), where z ∈ V is identified
to (ρ(z), z) ∈ H((G1 n Z)VV ). Let EV0 be the set of ξ ∈ Cc (V 0 ) such that
ξ(S × {z})
√
for all S × {z} ∈ V 0 .
ξ(z) =
#S
P
Lemma 7.15. The space EZ0 = i∈I EV0i is independent of the choice of the
cover (Vi ) of Z by Hausdorff open subspaces.
Proof. It suffices
to show that for every open Hausdorff subspace V of Z, one
P
0
has EV ⊂ i∈I EV0i . Let ξ ∈ EV0 . Denote by qV : V 0 → V the canonical map
defined by qV (S × {z}) = z. Let K ⊂ V compact such that supp(ξ) ⊂ qV−1 (K).
There exists J ⊂ I finite such that K ⊂ ∪j∈J Vj . Let (ϕj )j∈J be a partition
of unity associated to P
that cover, and ξj = ξ.(ϕj ◦ qV ). One easily checks that
ξj ∈ EV0j and that ξ = j∈J ξj .
We now define a Cr∗ (G1 )-valued scalar product on EZ0 by
Z
hξ, ηi(g) =
ξ(zγ)η(g −1 zγ) λσ(z) (dγ),
γ∈(G2 )σ(z)
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 0 is another element, there exists
γ 0 ∈ G2 such that z 0 = zγ 0 , and the Haar system on G2 is left-invariant.
Moreover, the integral is convergent for all g ∈ G1 because the action of G2
on Z is proper.
Let us show that hξ, ηi ∈ Cc (G1 ) for all ξ, η ∈ EZ0 . We need a preliminary
lemma:
36
JEAN-LOUIS TU
Lemma 7.16. 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
Vx of x such that Vxc ∈ F. Extracting a finite cover (V1 , . . . , Vn ) 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 hξ, ηi ∈ Cc (G1 ), we can suppose that ξ ∈ EU0 and η ∈ EV0 ,
where U and V are open Hausdorff. Let F (g, z) = ξ(z)η(g −1 z), defined on
Γ = G1 ×r,ρ Z. Since the action of G1 on Z is proper, F is quasi-compactly
supported. Let us show that F ∈ Cc (Γ).
(0)
Let F be a ultrafilter on Γ, convergent in HΓ. Since G1 is Hausdorff, its
r(g )
limit has the form S = S 0 g0 × S 00 where S 0 ⊂ (G1 )r(g00 ) , S 00 ⊂ ρ−1 (r(g0 )).
r(g)
Moreover, S 0 is a subgroup of (G1 )r(g) by the proof of Lemma 5.1.
Suppose that there exist z0 , z1 ∈ S 00 and g1 ∈ S 0 g0 such that z0 ∈ U and
−1
g1 z1 ∈ V . By Lemma 7.16 applied to the proper map G1 o Z → Z × Z, there
exists
s0 ∈ S 0P
such that z0 = s0 z1 . We may assume that g0 = s0 g1 . Then
P
−1 0 −1
0 / stab(z ), then g −1 (s0 )−1 z ∈
F
(s)
=
0 /
0
0
s0 ∈S 0 ξ(z0 )η(g0 (s ) z0 ). If s ∈
s∈S
−1 0 −1
−1
−1
V since g0 z0 and g0 (s ) z0 are distinct limits of (g, z) 7→ g z with respect
to F and V is Hausdorff. Therefore,
X
F (s) = #(stab(z0 ) ∩ S 0 )ξ(z0 )η(g0−1 z0 )
s∈S
=
q
#(stab(z0 ) ∩ S 0 )ξ(z0 ) #(stab(g0−1 z0 ) ∩ (g0−1 S 0 g0 ))η(z0 )
p
= lim ξ(z)η(g −1 z) = lim F (g, z).
F
F
P
If for all z0 , z1 ∈ S 00 and all g1 ∈ S 0 g0 , (z0 , g1−1 z1 ) ∈
/ U ×V , then s∈S F (g, z) =
0 = limF F (g, z).
By Proposition 4.1,
R F ∈ Cc (Γ).
Since hξ, ηi(g) = γ∈(G2 )σ(z) F (g, zγ) λσ(z) (dγ), to prove that hξ, ηi ∈ Cc (G1 )
it suffices to show:
Lemma 7.17. Let G1 and G2 be two locally compact groupoids with Haar sys(0)
tem such that Gi are Hausdorff. Let (Z, ρ, σ) be a generalized homomorphism
from G1 to G2 . Let Γ = G1 ×r,ρ Z. Then for every F ∈ Cc (Γ), the function
Z
g 7→
F (g, zγ) λσ(z) (dγ),
γ∈(G2 )σ(z)
where z ∈ Z is an arbitrary element such that ρ(z) = r(g), belongs to Cc (G1 ).
NON-HAUSDORFF GROUPOIDS
37
Proof. Suppose
= f (g)h(z), where f ∈ Cc (G1 ) and h ∈ Cc (Z).
R first that F (g, z)
σ(z)
Let H(z) = γ∈(G2 )σ(z) h(zγ) λ
(dγ). By Lemma 7.18 below (applied to the
groupoid Z o G2 ), H is continuous. It is obviously G2 -invariant, therefore
(0)
H ∈ Cc (Z/G2 ). Let H̃ ∈ Cc (G1 ) ' Cc (Z/G2 ) correspond to H. The map
Z
g 7→
F (g, zγ) λσ(z) (dγ) = f (g)H̃(s(g))
γ∈(G2 )σ(z)
thus belongs to Cc (G1 ).
By linearity, the lemma is true for F ∈ Cc (G1 ) ⊗ Cc (Z). By Lemma 4.4
and Lemma 4.5, F is the uniform limit of functions Fn ∈ Cc (G1 ) ⊗ Cc (Z)
which are supported in a fixed quasi-compact set Q = Q1 × Q2 ⊂ G1 × Z. Let
Q0 ⊂ Z quasi-compact such that ρ(Q0 ) ⊃ r(Q1 ). Since the action of G2 on Z
is proper, K = {γ ∈ G2 | Q0 γ ∩ Q2 6= ∅} is quasi-compact. Using the fact that
(0)
G1 ' Z/G2 , it is easy to see that
Z
sup
1Q (g, zγ) λσ(z) (dγ)
(g,z)∈Γ γ∈(G2 )σ(z)
Z
≤
sup
z∈Q0
σ(z)
γ∈G2
Z
≤
sup
(0)
x∈G2
γ∈Gx
2
1Q2 (zγ)λσ(z) (dγ)
1K (γ)λx (dγ) < ∞
by Lemma 4.9. Therefore,
Z
lim sup F (g, zγ) − Fn (g, zγ) λσ(z) (dγ) = 0.
n→∞ g∈G1 γ∈Gσ(z)
2
The conclusion follows from Corollary 4.2.
In the proof of Lemma 7.17 we used the
Lemma 7.18. Let G be a locally compact, proper groupoid with Haar system,
such that Gx is Hausdorff for all x ∈ G(0) , and Gxx = {x} for all x ∈ G(0) . We
do not assume G(0) to be Hausdorff. Then ∀f ∈ Cc (G(0) ),
ϕ : G(0) → C
Z
x 7→
f (s(g)) λx (dg)
g∈Gx
is continuous.
Proof. Let V be an open, Hausdorff subspace of G(0) . Let h ∈ Cc (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) 7→ h(x)f (y) belongs to Cc (G(0) × G(0) ), the map
g 7→ h(r(g))f
(s(g)) belongs to Cc (G), therefore by definition of a Haar system,
R
x 7→ g∈Gx h(r(g))f (s(g)) λx (dg) = h(x)ϕ(x) belongs to Cc (G(0) ).
Since h ∈ Cc (V ) is arbitrary, this shows that ϕ|V is continuous, hence ϕ is
continuous on G(0) .
38
JEAN-LOUIS TU
Now, let us show the positivity of the scalar product. Fix z ∈ Z such that
ρ(z) = x. For all g, h ∈ Gx1 ,
Z
hξ, ξi(g −1 h) =
ξ(g −1 zγ)ξ(h−1 zγ) λσ(z) (dγ).
σ(z)
γ∈G2
Therefore, using (2),
Z
hη, πG1 ,x (hξ, ξi)ηi =
(3)
σ(z)
λσ(z) (dγ)
γ∈G2
Z
η(g)ξ(g
−1
g∈Gx
2
zγ) λ (dg) .
x
(0)
It follows that πG1 ,x (hξ, ξi) ≥ 0 for all x ∈ G1 , so hξ, ξi ≥ 0 in Cr∗ (G1 ).
Now, let us define a Cr∗ (G1 )-module structure on EZ0 by
Z
ξ(g −1 z)a(g −1 ) λρ(z) (dg),
(ξa)(z) =
g∈(G1
)ρ(z)
EZ0
for all ξ ∈
and a ∈ Cc (G1 ).
Let us show that ξa ∈ EZ0 . We need a preliminary lemma:
Lemma 7.19. Let X and Y be quasi-compact spaces, (Ωk ) an open cover of
X × Y . Then there exist finite open covers (Xi ) and (Yj ) of X and Y such that
∀i, j ∃k, Xi × Yj ⊂ Ωk .
Proof. For all (x, y) ∈ X × Y choose open neighborhoods Ux,y and Vx,y of x and
y such that Ux,y × Vx,y ⊂ Ωk for some k. For y fixed, there exist x1 , . . . , xn such
that (Uxi ,y )1≤i≤n covers X. Let Vy = ∩ni=1 Uxi ,y . Then for all (x, y) ∈ X × Y ,
0
0 ×V ⊂Ω .
there exists an open neighborhood Ux,y
of x and k such that Ux,y
y
k
Let (V1 , . . . , Vm ) = (Vy1 , . . . , Vym ) such that ∪1≤j≤m Vj = Y . For all x ∈ X,
0
0
let Ux0 = ∩m
j=1 Ux,yj . Let (U1 , . . . , Up ) be a finite sub-cover of (Ux )x∈X . Then for
all i and for all j, there exists k such that Ui × Vj ⊂ Ωk .
Let Q1 and Q2 be quasi-compact subspaces of G1 of Z respectively such that
a−1 (C∗ ) ⊂ Q1 and ξ −1 (C∗ ) ⊂ Q2 . Let Q be a quasi-compact subspace of Z such
that ∀g ∈ Q1 , ∀z ∈ Q2 , g −1 z ∈ Q. Let (Uk ) be a finite cover of Q by Hausdorff
open subspaces of Z. Let Q0 = Q1 ×r,ρ Q2 . Then Q0 is a closed subspace of
Q1 × Q2 . Let Ω0k = {(g, z) ∈ Q0 | g −1 z ∈ Uk }. Then (Ω0k ) is a finite open cover
of Q0 . Let Ωk be an open subspace of Q1 × Q2 such that Ω0k = Ωk ∩ Q0 . Then
{Q1 × Q2 − Q0 } ∪ {Ωk } is an open cover of Q1 × Q2 . Using Lemma 7.19, there
exist finite families of Hausdorff open sets (Wi ) and (Vj ) which cover Q1 and
Q2 , such that for all i, j and for all (g, z) ∈ Wi ×G(0) Vj , there exists k such
1
that g −1 z ∈ Uk .
Thus, we can assume by linearity and by Lemmas 4.3 and 7.15 that ξ ∈ EV0 ,
a ∈ Cc (W ), U = W −1 V , and U , V and W are open and Hausdorff.
Let Ω = {(g, S) ∈ W −1 × U 0 | g −1 qU (S) ∈ V }. Then the map (g, S) 7→
−1
(g , g −1 S) is a homeomorphism from Ω onto W ×r,ρ◦qV V 0 . Therefore, the map
NON-HAUSDORFF GROUPOIDS
39
(g, z) 7→ ξ(g −1 z)a(g −1 ) belongs to Cc (Ω) ⊂ Cc (G1 ×r,ρ◦qV U 0 ). By Lemma 4.10,
Z
S 7→ (ξa)(S) =
ξ(g −1 S)a(g −1 ) λρ◦qV (S) (dg)
ρ◦q (S)
g∈G1
V
√
belongs to Cc (U 0 ). It is immediate that (ξa)(S) = #S(ξa)(q(S)) for all
S ∈ U 0 , therefore ξa ∈ EU0 . This completes the proof that ξa ∈ EZ0 .
Finally, it is not hard to check that hξ, ηai = hξ, ηi ∗ a. Therefore, the
completion EZ of EZ0 with respect to the norm kξk = khξ, ξik1/2 is a Cr∗ (G1 )Hilbert module.
Let us now construct a morphism π : Cr∗ (G2 ) → L(EZ ). For every ξ ∈ EZ0
and every b ∈ Cc (G2 ), let
Z
(bξ)(z) =
b(γ)ξ(zγ) λσ(z) (dγ).
γ∈(G2 )σ(z)
Let us check that bξ ∈ EZ0 . As above, by linearity we may assume that ξ ∈ EV0 ,
b ∈ Cc (W ) and V W −1 ⊂ U , where V ⊂ Z, U ⊂ Z and W ⊂ G2 are open and
Hausdorff.
Let Φ(S, γ) = (Sγ, γ). Then Φ is a homeomorphism from Ω = {(S, γ) ∈
0
U ×σ◦qU ,r W | qU (S)γ ∈ V } onto V 0 ×σ◦qV ,s W . Let F (z, γ) = b(γ)ξ(zγ). Since
F = (ξ ⊗ b) ◦ Φ, F is an element of Cc (Ω) ⊂ Cc (U 0 ×σ◦qU ,r W ). By Lemma 4.10,
bξ ∈ Cc (U 0 ).
√
It is immediate that (bξ)(S) = #S(bξ)(q(S)). Therefore, bξ ∈ EU0 ⊂ EZ0 .
Let us prove that kbξk ≤ kbk kξk. Let
Z
η(g)ξ(g −1 zγ) λx (dg),
ζ(γ) =
g∈Gx
1
where z ∈ Z such that ρ(z) = r(g) is arbitrary. From (3),
hη, πG1 ,x (hξ, ξi)ηi = kζk2 2
σ(z)
L (G2
)
.
A similar calculation shows that
Z
hη, πG1 ,x (hbξ, bξi)ηi =
σ(z)
γ∈G2
λσ(z) (dγ)
Z
2
η(g)ξ(g −1 zγγ 0 )b(γ 0 ) λs(γ) (dγ 0 )
g∈Gx1
= hbζ, bζi ≤ kbk2 kζk2 .
By density of Cc (Gx2 ) in L2 (Gx2 ), kπG1 ,x (hbξ, bξi)k ≤ kbk2 kπG1 ,x (hξ, ξi)k. Taking
(0)
the supremum over x ∈ G1 , we get kbξk ≤ kbk kξk. It follows that b 7→ (ξ 7→
bξ) extends to a ∗-homomorphism π : Cr∗ (G2 ) → L(EZ ).
Finally, suppose now that (Z, ρ, σ) is proper, and let us show that Cr∗ (G2 )
maps to K(EZ ).
For every η, ζ ∈ EZ0 , denote by Tη,ζ the operator Tη,ζ (ξ) = ηhζ, ξi. Compact
operators are elements of the closed linear span of Tη,ζ ’s. Let us write an explicit
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formula for Tη,ζ :
Z
Tη,ζ (ξ)(z) =
ρ(z)
η(g −1 z)hζ, ξi(g −1 ) λρ(z) (dg)
g∈G1
Z
=
ρ(z)
η(g
−1
Z
z)
σ(z)
ζ(g −1 zγ)ξ(zγ) λσ(z) (dγ)λρ(z) (dg).
γ∈G2
g∈G1
Let b ∈ Cc (G2 ), let us show that π(b) ∈ K(EZ ). Let K be a quasi-compact
subspace of G2 such that b−1 (C∗ ) ⊂ K. Since (Z, ρ, σ) is a proper generalized homomorphism, there exists a quasi-compact subspace Q of Z such that
σ −1 (r(K)) ⊂ G1 Q̊. Before we proceed, we need a lemma:
Lemma 7.20. Let G2 be a locally compact groupoid acting freely and properly
(0)
on a locally compact space Z with momentum map σ : Z → G2 . Then for
every (z0 , γ0 ) ∈ Z o G2 , there exists a Hausdorff open neighborhood Ωz0 ,γ0 of
(z0 , γ0 ) such that
• U = {z1 γ1 | (z1 , γ1 ) ∈ Ωz0 ,γ0 } is Hausdorff;
• there exists a Hausdorff open neighborhood W of γ0 such that ∀γ ∈ G2 ,
∀z ∈ pr1 (Ωz0 ,γ0 ), ∀z 0 ∈ U , z 0 = zγ =⇒ γ ∈ W .
Proof. Let R = {(z, z 0 ) ∈ Z × Z| ∃γ ∈ G2 , z 0 = zγ}. Since the G2 -action
is free and proper, there exists a continuous function φ : R → G2 such that
φ(z, zγ) = γ. Let W be an open Hausdorff neighborhood of γ0 . By continuity
of φ, there exist open Hausdorff neighborhoods V and U0 of z0 and z0 γ0 such
that for all (z, z 0 ) ∈ R ∩ (V × U0 ), φ(z, z 0 ) ∈ W . By continuity of the action,
there exists an open neighborhood Ωz0 ,γ0 of (z0 , γ0 ) such that ∀(z1 , γ1 ) ∈ Ωz0 ,γ0 ,
z1 γ1 ∈ U0 and z1 ∈ V .
By Lemma 7.19, there exist finite covers (Vi ) of Q and (Wj ) of K such that
for every i, j, (Z ×G(0) G2 ) ∩ (Vi × Wj ) ⊂ Ωz0 ,γ0 for some (z0 , γ0 ).
2
By Lemma 6.2 applied to the groupoid (G1 n Z)VVii , for all i there exists
0
0
0
0
0
cP
i ∈ Cc (Vi )+ such that ci (S) = (#S)ci (qVi (S)) for all S ∈ Vi , and such that
0
i ci ≥ 1 on Q. Let
Z
fi (z) =
c0i (g −1 z) λρ(z) (dg)
ρ(z)
g∈G1
P
and let f = i fi . As in the proof of Theorem 6.3, one can show that for every
Hausdorff open subspace V of Z and every h ∈ Cc (V ), (g, z) 7→ h(z)c0i (g −1 z) belongs to Cc (G n Z), therefore hfi is continuous on V . Since h is arbitrary, it follows that fi is continuous, thus f is continuous. Moreover, f is G1 -equivariant,
nonnegative, and inf Q f > 0. Therefore, there exists f1 ∈ Cc (G1 \Z) such that
f1 (z) = 1/f (z) for all z ∈ Q. Let ci (z) = f1 (z)c0i (z). Let
Z
Z
Ti (ξ)(z) =
ci (g −1 z)b(γ)ξ(zγ) λρ(z) (dg)λσ(z) (dγ).
ρ(z)
g∈G1
Then π(b) =
for all i.
P
i Ti ,
σ(z)
γ∈G2
therefore it suffices to show that Ti is a compact operator
NON-HAUSDORFF GROUPOIDS
41
By linearity and by Lemma 4.3, one may assume that b ∈ Cc (Wj ) for some
j. Then, by construction of Vi (see Lemma 7.20), there exist open Hausdorff
sets U ⊂ Z and W ⊂ G2 such that {γ ∈ G2 | ∃(z, z 0 ) ∈ Vi × U, z 0 = zγ} ⊂ W ,
and {zγ| (z, γ) ∈ Vi ×σ,r W } ⊂ U .
The map (z, zγ) 7→ c(z)b(γ) defines an element of Cc (Vi0 × U ). Let L1 × L2 ⊂
Vi × U compact such that (z, zγ) 7→ c(z)b(γ) is supported on qV−1
(L1 ) × L2 .
i
Vi
U
By Lemma 6.2 applied to the groupoids (G1 n Z)Vi and (G1 n Z)U , there exist
d1 ∈ Cc (V√i0 )+ and d2 ∈ Cc (U 0 )+ such that d1 > 0 on L
√1 and d2 > 0 on L2 ,
0
d1 (S) = #Sd1 (qVi (S)) for all S ∈ Vi , and d2 (S) = #Sd2 (qU (S)) for all
S ∈ U 0 . Let
c(z)b(γ)
f (z, zγ) =
.
d1 (z)d2 (zγ)
Then f ∈ Cc (Vi ×G(0) U ). Therefore, f is the uniform limit of a sequence
1
P
fn =
αn,k ⊗ βn,k in Cc (Vi ) ⊗ Cc (U ) such that P
all the fn are supported in a
fixed compact set. Then Ti is the norm-limit of k Td1 αn,k ,d2 βn,k , therefore it
is compact.
Remark 7.21. The construction in Theorem 7.11 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 ann easy
exercise when G1 and G2 are Hausdorff.
8. The Hilbert module L2 (G)
In this section, G denotes a locally compact groupoid with Haar system
such that G(0) Hausdorff. Recall that in the Hausdorff situation, Cr∗ (G) is
canonically represented in a C0 (G(0) )-module denoted by L2 (G). For G nonHausdorff, Khoshkam and Skandalis [14] show that Cr∗ (G) is represented in
a C0 (Y )-Hilbert module (that we will denote by L2 (G)KS , where Y is the
spectrum of the C ∗ -algebra of Borel functions on G(0) generated by all f|G(0)
(f ∈ Cc (G)). In this section we give a definition of a C0 (X 0 )-Hilbert module
L2 (G), where X 0 is the closure of G(0) in HG (see Section 5). The reason why
we prefer L2 (G) is that
• G acts naturally on L2 (G), since G acts continuously in a quite obvious
way on X 0 (see Proposition 3.10).
• When G is proper, cutoff functions give an embedding of the C0 (G(0) )Hilbert module C0 (G(0) ) in the C0 (X 0 )-Hilbert module L2 (G), which
extends the fact that for compact groups, the trivial representation is
strongly contained in the regular representation.
Let us first compare the spaces X 0 and Y :
Proposition 8.1.
(a) The map q (see Section 5) factors through continuj0
j
ous proper surjections X 0 → Y → G(0) ;
(b) the canonical inclusion G(0) → X 0 factors through (not necessarily coni
i0
tinuous) maps G(0) → Y → X 0 ;
(c) we have i ◦ j = IdG(0) and i0 ◦ j 0 = IdY .
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P
Proof. (a) Let f ∈ Cc (G). By Proposition 4.1, if we define f 0 (S) = x∈S f (x)
for all S ∈ X 0 \G(0) , then f 0 is continuous on X 0 , therefore C0 (Y ) ⊂ Cb (X 0 ).
For every compact subset K of G(0) , q −1 (K) is closed in HG and included
in the compact set {S ∈ HG| S ∩ K 6= ∅} (see Proposition 3.7), therefore
q −1 (K) is compact. It follows that q is proper and thus C0 (G(0) ) ⊂ C0 (X 0 ).
Since C0 (X)C0 (Y ) = C0 (Y ) ([14]), we get C0 (G(0) ) ⊂ C0 (Y ) ⊂ C0 (X 0 ) and a
factorization q = j 0 ◦ j. Since q is proper and j is continuous surjective, j 0 is
proper. It is clear that j 0 is surjective, and the fact that j is proper surjective
is proven in [14].
(b) Recall that the Borel inclusion i : G(0) → Y is induced by the inclusion
C0 (Y ) ⊂ B(G(0) ), thus (b) and (c) are clear from the definitions.
We do not know whether X 0 and Y coincide or not. In Example 2.3, they both
coincide with the disjoint union of [0, 1] and an isolated point, and C0 (X 0 ) =
C0 (Y ) = C([0, 1]) + Cδ0 where δ0 is the Dirac function at 0.
Remark 8.2. If G is not Hausdorff then Y never coincides with G(0) by Remark 4.6.
Consider the convolution algebra Cc (X 0 oG). Again, by Remark 4.6, Cc (X 0 o
G) does not restrict to Cc (X 0 ), hence the scalar product
Z
∗
hξ, ηi◦ (S) = ξ ∗ η(S) =
ξ(S, g)η(S, g)λq(S) (dg)
g∈Gq(S)
(X 0 )-valued.
is not Cc
However, by definition of Y and the fact that C0 (Y ) ⊂
C0 (X 0 ), the restriction of S 7→ hξ, ηi◦ (S) to (i0 ◦i)(G(0) ) does extend in a unique
way to an element of Cc (X 0 ).
Definition 8.3. Endow Cc (X 0 o G) with the scalar product h·, ·i, where hξ, ηi
is the unique element of Cc (X 0 ) which coincides with hξ, ηi◦ on (i0 ◦ i)(G(0) ).
We define L2 (G) to be the completion of Cc (X 0 o G) with respect to that scalar
product.
Of course, the right C0 (X 0 )-module structure on L2 (G) is defined by (ξϕ)(S, g) =
ξ(S, g)ϕ(g −1 Sg) for all ξ ∈ Cc (X 0 o G) and ϕ ∈ Cc (X 0 ).
Example 8.4. In Example 2.3, L2 (G) = `2 (Γ) ⊕ ([0, 1] × `2 (Γ/H)) with the
scalar product hξ ⊕η, ξ ⊕ηi(x) = hη(x), η(x)i if x ∈ (0, 1], and hξ ⊕η, ξ ⊕ηi(0) =
hξ, ξi.
The next proposition shows that our definition of L2 (G) and the one in
[14] are related in a very simple way. The advantage of the construction of
Khoshkam and Skandalis is that C0 (Y ) is the “minimal” extension of C0 (G(0) )
necessary to represent Cr∗ (G) faithfully in a Hilbert module. On the other hand,
our space X 0 is defined more geometrically and moreover we don’t need to use
Kasparov’s generalized Stinespring theorem to prove the positivity of our scalar
product.
Proposition 8.5. L2 (G) = L2 (G)KS ⊗C0 (Y ) C0 (X 0 ). As a consequence, Cr∗ (G)
is faithfully represented in L2 (G).
NON-HAUSDORFF GROUPOIDS
43
Proof. Obvious, from the definitions of L2 (G) and L2 (G)KS , and Proposition 8.1.
In the next proposition, we consider L2 (G) as a continuous field of Hilbert
spaces over X 0 , with fiber at S ∈ X 0 equal to L2 (G)S := L2 (G) ⊗evS C, where
evS : X 0 → C is the evaluation map at S. We describe the fibers more explicitly:
for S = x ∈ G(0) , L2 (G)x = L2 (Gx , λx (dg −1 )). For S ∈
/ G(0) , L2 (G)S is more
complicated: let F be any ultrafilter on G(0) converging to S, then L2 (G)S is
the completion of Cc (G) with respect to the scalar product limF hξ, ηi(x).
Proposition 8.6. Let G be a locally compact groupoid with Haar system such
that G(0) is Hausdorff and G(0) /G is σ-compact. The following are equivalent:
(i) G is proper;
(ii) there exists a continuous bounded G-invariant section ξ of L2 (G) such
that hξ, ξi(S) = 1 for all S ∈ X 0 .
Proof. (i) =⇒ (ii): let c be a cutoff function (Theorem 6.3). In order to simplify
the proof, we will assume that G is quasi-compact (so that a section as in (ii) is
actually a vector in L2 (G)); the proof in the general case is thepsame but just
needs more notation. In the Hausdorff case, one defines ξ(g) = c(s(g)).
When G is not Hausdorff, we have to prove that this function extends to an
element of Cc (X 0 o G). To that end, define
s
c(s(g))
ϕ(S, g) =
#S
for all (S, g) ∈ X 0 o G. We must show that ϕ satisfies the properties of Proposition 4.1(ii). Suppose that (Sλ , gλ ) ∈ X 0 o G converges to (S, g) ∈ X 0 × G. We
may assume that the net is universal. Let xλ = s(gλ ), then, by Lemma 5.2,
xλ → S0 where S0 is a normal subgroup of g −1 Sg, and #Sλ converges to
#S/#S0 .
From the definition of a cutoff function
(Theorem 6.3.a), c(s(gλ )) converges
q
p
c(s(gλ ))
to #S0 c(s(g)), hence ϕ(Sλ , gλ ) =
#Sλ converges to #S0P c(s(g)).
On the other hand, (gλ ) converges in HG to gS0 , and
h∈S0 ϕ(S, gh) =
p
p
P
c(s(h)) = #S0 c(s(g)). We thus have proved that ϕ(Sλ , gλ ) conh∈S0
P
verges to g0 ∈gS0 ϕ(S, g 0 ), which is exactly condition (ii) in Proposition 4.1.
(ii) =⇒ (i): suppose that ξ is such an invariant section. Let f ∈ Cc (X 0 )
−1
arbitrary, and let η(S, g) = ξ(S,
element of L2 (G).
R g)f (g Sg). Then η is an
Let φ(γ) = hηr(γ) , γ · ηs(γ) i = g∈Gr(γ) η(s(g), g −1 )η(s(g), g −1 γ) λr(γ) (dg) (for all
γ ∈ G), and let us show that φ(γ) vanishes at infinity, in the sense that for all
ε > 0 there exists a quasi-compact set Q outside which |φ(γ)| < ε.
If η = 0 outside a quasi-compact set K ⊂ G, then φ = 0 outside the quasicompact set q({h−1 g| (h, g) ∈ K ×G(0) K}). The general case follows by a
continuity argument.
Now, Rfrom the definition of ξ, we have
φ(γ) = g∈Gr(γ) |ξ(s(g), g −1 )|2 f (r(γ))f (s(γ)) λr(γ) (dγ) = f (r(γ))f (s(γ)), thus
f (r(γ))f (s(γ)) vanishes at infinity.
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Take now K ⊂ G(0) an arbitrary compact subspace, then K 0 = q −1 (K)
is compact (Proposition 8.1.a), thus there exists f ∈ Cc (X 0 , [0, 1]) such that
f|K 0 = 1. Let Q be a quasi-compact set outside which f (r(γ))f (s(γ)) ≤ 1/2,
K
then GK
K ⊂ Q, thus GK is quasi-compact. This completes the proof that G is
proper.
9. The classifying space for proper actions
In [3], topological K-theory of a group is defined using the classifying space
for proper actions. It is therefore natural to define topological K-theory of a
groupoid using proper groupoid actions. At this point, one has two distinct
choices: either to consider proper actions of G on all locally compact spaces, or
to consider proper actions of G on locally compact, Hausdorff spaces. However,
the second approach leads to a disconnected classifying space and clearly doesn’t
give the right answer in K-theory (see Appendix B).
Let G be a locally compact groupoid acting properly on a locally Hausdorff
topological space Z.
We say that Z is “the” classifying space for proper actions of G if for every
locally compact space Y such that G acts properly on Y and such that Y /G
is compact, there exists a continuous, G-equivariant map f : Y → Z, unique
up to equivariant homotopy. If such a space exists, then it will be denoted by
EG, although it may not be uniquely determined up to homotopy, since it is
not assumed to be locally compact.
Note that when G is a locally compact Hausdorff groupoid such that G(0)
is σ-compact and the range map r : G → G(0) is open, then from Appendix A
there exists a locally compact Hausdorff model for EG. However, when G is
not Hausdorff this is not true in general.
The reason why we want the space Z to be locally Hausdorff is that whenever
Y is a proper, G-compact G-space and two G-maps f0 , f1 : Y → Z are homotopic, then they are homotopic inside a locally compact subspace (Lemma 10.3).
We do not know whether such a classifying space as defined above always
exists when G is not Hausdorff. The join construction ([3]) or the construction
of Kasparov and Skandalis ([13, Lemma 4.1]) yield spaces which are not locally
Hausdorff. In Example 2.3 with Γ = Z and H = {0}, the space Z = ([0, 1] ×
R)/((0, 1] × R), i.e. the space fibered over [0, 1] with fiber R at 0 and {0}
elsewhere, would seem at first sight to be a natural candidate for EG, but is
not locally Hausdorff.
However, there is a useful criterion to determine whether a given space is a
model for EG:
Theorem 9.1. Let G be a locally compact, étale groupoid acting properly on a
locally Hausdorff topological space T . Then T is the classifying space for proper
actions of G if and only if for every finite subgroup Γ of G there exist A ⊂ G(0)
and G1 = φ(A o Γ) as in Lemma 2.41, such that T is the classifying space for
proper actions of the groupoid G1 .
NON-HAUSDORFF GROUPOIDS
45
Proof. The condition is obviously necessary, since for every G1 -compact proper
G1 -space Y , the space Y ×G1 G is a G-compact proper G-space (Proposition 2.40). To show that it is sufficient, choose for every finite subgroup Γ
of G a compact subgroupoid GΓ as above. By Proposition 2.42, it is enough to
show by induction on n the assertion
(Pn ): “For every locally compact space Z such that there exist Z1 , . . . , Zn
closed saturated such that Z = ∪ni=1 Z̊i with Zi ' Yi ×GΓi G, Yi compact, there
exists a map Z → T which is unique up to G-homotopy.”
The case n = 1 follows from the assumption. Suppose that (Pn−1 ) is true
and let us show (Pn ). Let Z0 = Z2 ∪ · · · ∪ Zn . Let ϕ : Z → [0, 1] be a continuous
equivariant map such that ϕ−1 (0) ⊂ Z̊0 and ϕ−1 (1) ⊂ Z̊1 . Such a map exists
because Z/G is compact. By induction hypothesis, there exist G-maps f0 : Z0 →
T and f1 : Z1 → T . Since Z0 ∩ Z1 satisfies (Pn−1 ), there exists a homotopy
F : [0, 1] × (Z0 ∩ Z1 ) → T between f0 |Z0 ∩Z1 and f1 |Z0 ∩Z1 . Put
F (ϕ(z), z) if z ∈ Z0 ∩ Z1
f (z) =
fi (z) if ϕ(z) = i (i = 0, 1),
then f : Z → T is continuous G-equivariant.
To show uniqueness, let f0 , f1 : Z → T be G-maps. By induction hypothesis,
there exists a homotopy F0 : [0, 1] × Z0 → T from (f0 )|Z0 to (f1 )|Z0 and a
homotopy F1 : [0, 1] × Z1 → T from (f0 )|Z1 to (f1 )|Z1 . Consider the unit square
S = [0, 1]2 and let F : ∂S × (Z0 ∩ Z1 ) → T defined by F (i, t, z) = Fi (t, z) and
F (s, i, z) = fi (z) (i = 0, 1, s ∈ [0, 1], z ∈ Z0 ∩Z1 ). By induction hypothesis, F is
G-homotopic to a map which does not depend on the first variable (for instance,
(s, t, z) 7→ f0 (z)). Therefore, F extends to a map F : S × (Z0 × Z1 ) → T . Let
F (ϕ(z), t, z) if z ∈ Z0 ∩ Z1
H(t, z) =
Fi (t, z) if ϕ(z) = i (i = 0, 1).
Then H is a homotopy from f0 to f1 .
Let G be a locally compact groupoid. In many cases, one can construct
sequences Yn (n ∈ N) of locally compact, proper G-spaces, and an inductive
system fm,n : Ym → Yn , such that for every proper, G-compact G-space Y ,
• there exists n and a G-map f : Y → Yn ;
• if f and f 0 : Y → Yn are two G-maps, then there exists p > n such that
fn,p ◦ f and fn,p ◦ f 0 are G-homotopic.
We shall say that (Yn )n∈N is a classifying sequence of spaces. The following
proposition is an easy exercise left to the reader:
Proposition 9.2. Let G be a locally compact groupoid. Let Y1 → Y2 → . . .
be an inductive system of locally compact, proper G-spaces. Then (Yn )n∈N is
a classifying sequence of spaces if and only if the telescope of (Yn )n∈N is a
classifying space for proper actions of G.
(Recall that the telescope of (Yn )n∈N is ∪n∈N∗ [n, n + 1] × Yn divided by the
relation (n + 1, y) ∼ (n + 1, fn,n+1 (y)).)
One can formulate an analogue of Theorem 9.1 in the language of classifying
sequences.
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JEAN-LOUIS TU
Example 9.3. Let Γ be a discrete group, and let ` be a length function on Γ
such that every ball for the corresponding left-invariant distance is finite. Let
Yn be the Rips’ complex of Γ whose simplices consist of subspaces of diameter
≤ n. Then (Yn )n∈N is a classifying sequence for Γ.
Now, let us examine a non-Hausdorff example. Let Γ be a torsion-free discrete
group, H a normal subgroup. Fix two Hausdorff, locally compact models EΓ
and E(Γ/H) and a G/H-equivariant map h : EΓ/H → E(Γ/H) which is
compatible with homotopies between the two projections EΓ × EΓ → EΓ and
E(Γ/H) × E(Γ/H) → E(Γ/H) (for example, Γ = Z, H = {0}, EΓ = R,
EH = {0}). Let Yn = (([0, 1/n] × EΓ)/((0, 1/n] × H)) ∪ [1/n, 1] × E(Γ/H)
divided by the relation (1/n, z) ∼ (1/n, h(z)). We show that (Yn )n∈N is a
classifying sequence for the groupoid G = ([0, 1]×Γ)/((0, 1]×H) of Example 2.3.
Using Theorem 9.1, it is easy to reduce the problem to showing that for every
compact space Y and every continuous map p : Y → [0, 1],
• there exist n and a continuous [0, 1]-map Y → Yn ;
• if f , f 0 : Y → Yn are [0, 1]-maps, then there exists m > n such that
fn,m ◦ f and fn,m ◦ f 0 are [0, 1]-homotopic.
The first assertion is easy: choose a ∈ EΓ and let f : Y → Y1 defined by
f (y) = (p(y), a) if p(y) < 1 and f (y) = (1, h(a)) if p(y) = 1. Let us prove the
second assertion.
Since Γ is torsion-free, it acts freely on EΓ, thus there exists an open cover
(Ωi ) such that Ωi γ ∩ Ωi = ∅ for all γ 6= 1. Cover the compact set p−1 (0)
by finitely many compact subsets Kj satisfying f (Kj ) ⊂ {0} × Ωi(j) for some
i(j). By definition of Ωi there exists for all j an open neighborhood subset
Vj of Y , containing Kj , such that f|Vj lifts to a map f˜j : Vj → [0, 1] × EΓ.
We show that these liftings are compatible in some neighborhood of p−1 (0).
If not, then there exist j 6= j 0 and sequences yn ∈ p−1 ((0, 1]), hn ∈ H − {1}
such that f˜j (yn ) = f˜j 0 (yn )hn and p(yn ) → 0. By compactness, we may assume
that yn → y ∈ p−1 (0) and that hn = h is constant. By continuity, we get
f (y) = f (y)h, a contradiction since Γ acts freely on E Γ .
So, let m ∈ N∗ such that f˜j and f˜j 0 coincide on Vj ∩ Vj 0 ∩ p−1 ([0, 1/m]),
and such that p−1 ([0, 1/m]) is covered by the sets Vj . We thus get a lifting
f˜: p−1 [0, 1/m] → [0, 1] × EΓ, and we may assume that the same holds for f 0 .
The restrictions of fn,m ◦ f and fn,m ◦ f 0 to p−1 ([0, 1/m]) are [0, 1/m] × EΓhomotopic by definition of EΓ, and the restrictions of fn,m ◦ f and fn,m ◦ f 0 to
p−1 ([1/m, 1]) are homotopic in a consistent way with the previous homotopy
by assumption.
10. The assembly map
Since this section contains no theorems, we will sometimes be sketchy. However, we thought it useful to include it, since it disproves the belief that results
concerning the K-theory of Hausdorff groupoids can easily be extended to nonHausdorff ones.
NON-HAUSDORFF GROUPOIDS
47
10.1. A few lemmas.
Lemma 10.1. Let G be a topological groupoid such that G(0) is locally compact
and r : G → G(0) is open. Let Y , Z be two proper G-spaces such that Y is
locally compact, Y /G is compact and Z is locally Hausdorff. Let f : Y → Z be
a G-map. Then f is proper (and in particular f is closed).
Proof. Let us denote by p : Y → G(0) the momentum map, and by π : Y → Y /G
and π : Z → Z/G the canonical mappings. Since π is open surjective and Y
is locally compact, there exists a quasi-compact set K ⊂ Y such that π(K) =
Y /G. By quasi-compactness of K, there exist finitely many compact subspaces
Ki (i ∈ I) of Y such that K ⊂ ∪i∈I Ki , p(Ki ) is compact and f (Ki ) is compact
for all i. Then π(f (Ki )) is compact, hence closed. Therefore, π −1 (π(f (Ki )))
is closed. It thus suffices to consider the case where K, p(K) and f (K) are
compact.
Let Y 0 = K ×G(0) G = {(y, g) ∈ K × G| p(y) = r(g)} = K ×p(K) G. Since
the obvious map Y 0 → Y is continuous surjective, it suffices by [5, Proposition
I.10.1.5] to show that the composition Y 0 → Y → Z is proper. But consider
f ×1
pr2
K ×p(K) G −−→ f (K) ×p(K) G → f (K) × Z −−→ Z,
where the second map is (z, g) 7→ (z, zg). Let us show that the map K×p(K) G →
f (K) ×p(K) G is proper. The map f : K → f (K) is proper since K is compact,
therefore K ×G → f (K)×G is proper. Now, since p(K) is Hausdorff, K ×p(K) G
and f (K) ×p(K) G are closed in K × G and f (K) × G respectively.
The map f (K) ×p(K) G → f (K) × Z is proper because the action of G on Z
is proper, and f (K) × Z → Z is proper since f (K) is quasi-compact. It follows
that the composition Y 0 → Z of the three maps above is proper.
Lemma 10.2. Let Y and Z two topological spaces such that Y is locally compact
and Z is locally Hausdorff. Let f : Y → Z be a proper map. Then f (Y ) is locally
compact.
Proof. We can assume that Z = f (Y ). Let z ∈ Z, and let V be a Hausdorff
neighborhood of z. Since f is proper, f −1 (z) is quasi-compact. Since Y is locally
compact, there exists a quasi-compact neighborhood K of f −1 (z) contained in
f −1 (V ). Let L = f (K). We show that L is a neighborhood of z.
Let F = Y − K̊. Then F is closed. Since f is proper, f (F ) is closed. Since
z∈
/ f (F ) and Z = L ∪ f (F ), L is a neighborhood of z.
We have shown that every z ∈ f (Y ) has a compact neighborhood, therefore
f (Y ) is locally compact.
Lemma 10.3. Let G be a locally compact groupoid with G(0) Hausdorff, and
let Z be a locally Hausdorff proper G-space. Suppose Y is a locally compact,
proper, G-compact G-space and two G-maps f0 , f1 : Y → Z are related by a
G-homotopy F : [0, 1] × Y → Z, then the image Y 0 of F is a locally compact,
G-compact proper G-space and f0 , f1 are homotopic as maps from Y to Y 0 .
Proof. Follows from Lemmas 10.1 and 10.2.
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10.2. Topological K-theory. Let Y be a locally compact space. To every
open cover
` of Y by Hausdorff subsets U = (Ui ), we associate the groupoid
G[U] = i,j Ui ∩ Uj which is Morita-equivalent to Y . The C ∗ -algebra C ∗ (U) =
Cr∗ (G[U]) is independent of the choice of the cover up to Morita equivalence.
We will sometimes employ the abusive notation C ∗ (Y ).
Examples 10.4. (a) If Y is Hausdorff, then C ∗ ((Ui )i∈I ) is the closure in
C0 (Y, K(`2 (I)) of ⊕i,j Cc (Ui ∩ Uj ).
(b) If Y = ({0, 1} × [0, 1])/ ∼, with (0, t) ∼ (1, t) for all t ∈ (0, 1], and Ui =
{i} × [0, 1] ⊂ Y , then C ∗ ((Ui )i=0,1 ) = {f : [0, 1] → M2 (C)| f (0) is diagonal}.
Suppose that Y = ∪Ui and Z = ∪Vk are two locally compact spaces and
f : Y → Z is a proper map. Composing f with the Morita equivalences Y ∼
G[U] and Z ∼ G[V], one gets a proper generalized morphism G[U] → G[V],
hence (Theorem 7.11) an element of KK(C ∗ (V), C ∗ (U)).
Remark 10.5. More concretely, the generalized morphism is given by Zf =
{(i, y, k)| y ∈ Ui , f (y) ∈ Vk }.
Suppose now that G acts on Y . The problem when we want to define the
equivariant K-homology of Y by KKG (C ∗ (Y ), C0 (G(0) )) is that G does not act
on C ∗ (Y ). However, let Yx ⊂ Y be the fiber over x ∈ G(0) , and denote by Ux the
induced cover on Yx , i.e. the cover by the sets Yx ∩Ui (i ∈ I). Then every g ∈ Gyx
induces a Morita-equivalence between the fibers C ∗ (Y )x = C ∗ (Ux ) and C ∗ (Y )y ,
using the composition of Morita-equivalences G[Uy ] ∼
= G[g ∗ Uy ] ∼ G[Ux ], where
∗
g Uy denotes the cover (Ui g)i∈I of Yx . We thus see that, in some sense, G
“acts by Morita equivalences” on C ∗ (Y ). The appropriate notion is that of
Fell bundle over a groupoid [30, 9, 15, 21]. Since the definitions in the case of
Hausdorff groupoids are also recalled in the appendix of [28], we will be brief.
Definition 10.6. Let X be a locally Hausdorff topological space. A continuous
(resp. upper semicontinuous) field of Banach spaces E over X consists of a
`
family (Ex )x∈X of Banach spaces together with a topology on Ẽ = x∈X Ex
such that
(i) the topology on Ex induced from that on Ẽ is the norm-topology;
(ii) the projection π : Ẽ → X is continuous and open;
(iii) the operations (e, e0 ) ∈ Ẽ ×X Ẽ 7→ e + e0 ∈ Ẽ and (λ, e) ∈ C × Ẽ →
λe ∈ Ẽ are continuous;
(iv) the norm Ẽ → R+ is continuous (resp. u.s.c.);
(v) if kei k → 0 and π(ei ) → x then ei → 0x ;
(vi) for all e ∈ Ex there exists a neighborhood V of x and a continuous
section ξ : V → Ẽ such that ξ(x) = e.
If (Ui ) is an open cover of X by (say Hausdorff) open subsets, giving a
continuous (resp. u.s.c.) field of Banach spaces over X is equivalent to giving
a family (Ẽi ) of continuous (resp. u.s.c.) fields over Ui with isomorphisms
ϕij : Ẽj|Uij → Ẽi|Uij satisfying ϕij ◦ ϕjk = ϕik on Uijk . (We have used the
notation Uij = Ui ∩ Uj ).
NON-HAUSDORFF GROUPOIDS
49
Definition 10.7. Let G be a locally compact groupoid and denote by m : G(2) →
G the multiplication map. A continuous (resp. u.s.c.) Fell bundle over G is
a continuous (resp. u.s.c.) field of Banach spaces (Eg )g∈G over G together
with an associative bilinear product (ξ, η) ∈ Eg × Eh 7→ ξη ∈ Egh whenever
(g, h) ∈ G(2) , and an antilinear involution ξ ∈ Eg 7→ ξ ∗ ∈ Eg−1 such that for
any (g, h) ∈ G(2) , and (e1 , e2 ) ∈ Eg × Eh ,
(i) ke1 e2 k ≤ ke1 kke2 k;
(ii) (e1 e2 )∗ = e∗2 e∗1 ;
(iii) ke∗1 e1 k = ke1 k2 ;
(iv) e∗1 e1 is a positive element of the C ∗ -algebra Es(g) ;
(v) the product (e, e0 ) ∈ m∗ (Ẽ) 7→ ee0 ∈ Ẽ, and the involution e ∈ Ẽ 7→ e∗
are continuous;
(vi) for all (g, h) ∈ G(2) , the image of the product Eg × Eh → Egh spans a
dense subspace of Egh .
Remark 10.8. Note that (i)–(iii) imply that Ex , x ∈ G(0) , is a C ∗ -algebra, so
(iv) makes sense. Moreover, when G(0) is Hausdorff, the space A = C0 (G(0) , Ẽ)
of sections over G(0) that vanish at infinity is a C0 (G(0) )-algebra with fiber Ex
at x ∈ G(0) .
Suppose that A is a C ∗ -algebra and G is a locally compact groupoid with G(0)
Hausdorff. A Fell bundle over G given with an isomorphism A ∼
= C0 (G(0) , Ẽ)
will be called a generalized action of G on A. To explain the terminology, if G
acts on A and αg : As(g) → Ar(g) denotes the action, then there is a canonical
Fell bundle with fiber Eg = Ar(g) , product (e, e0 ) 7→ eαg (e0 ) (∀(g, h) ∈ G(2) ,
∀(e, e0 ) ∈ Eg × Eh ) and adjoint e 7→ αg−1 (e∗ ).
Suppose that A is a Fell bundle over G. Let us denote by  the pull-back of
A by the inverse map g 7→ g −1 , i.e. Âg = Ag−1 .
In what follows, G is a locally compact, σ-compact groupoid with Haar system, such that G(0) is Hausdorff.
Definition 10.9. Let A and B be C ∗ -algebras endowed with generalized actions
A and B of G. Let E be a C ∗ -correspondence from A to B. We say that
the correspondence E is equivariant if there is an isomorphism of s∗ A, r∗ B
correspondences
W : s∗ E ⊗s∗ B B̂ → Â ⊗t∗ A r∗ E
such that for every (g, h) ∈ G(2) ,
(IdAh−1 ⊗ Wg ) ◦ (Wh ⊗ IdBg−1 )
∈ L(Es(h) ⊗Bs(h) Bh−1 ⊗Br(h) Bg−1 , Ah−1 ⊗Ar(h) Ag−1 ⊗Ar(g) Er(g) )
is equal to
Wgh ∈ L(Es(h) ⊗Bs(h) Bh−1 g−1 , Ah−1 g−1 ⊗Ar(g) Er(gh) )
via the identifications A(gh)−1 ∼
= Ah−1 ⊗Ar(h) Ag−1 and B(gh)−1 ∼
= Bh−1 ⊗Br(h)
Bg−1 .
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JEAN-LOUIS TU
In the definition above, W is an isomorphism of fields of correspondences
over G, the fiber at g ∈ G being a As(g) , Br(g) -correspondences. Alternatively,
the restriction of these fields of correspondences over any Hausdorff open subset
U of G are correspondences between C0 (U )-algebras.
As in [16, 28], one can define an equivariant Kasparov bifunctor KKG .
Definition 10.10. Let A and B be two C ∗ -algebras endowed with generalized
actions of G. An equivariant Kasparov A, B-bimodule is a pair (E, F ) where E
is a Z/2Z-graded, equivariant A, B-correspondence and F ∈ L(E) is a degree 1
operator such that for all a ∈ A,
(i) a(F − F ∗ ) ∈ K(E);
(ii) a(F 2 − 1) ∈ K(E);
(iii) [a, F ] ∈ K(E);
(iv) W (s∗ F ⊗ Id)W ∗ ∈ L(Â ⊗r∗ A r∗ E) is a r∗ F -connection for Â.
Property (iv) above has to be interpreted in the following way when G is not
Hausdorff: for any open Hausdorff subset V of G, we denote by WV and ÂV , the
restrictions to V of W and Â. Then, ÂV is considered as a rV∗ A-Hilbert module,
and (iv) means that for all V , the operator WV (s∗V F ⊗Id)WV∗ ∈ L(AˆV ⊗rV∗ A rV∗ E)
is a rV∗ F -connection for ÂV .
As usual, KKG (A, B) is defined to be the set of equivariant Kasparov A,
B-bimodules divided by homotopy. Then KKG (A, B) is an abelian group,
and (A, B) is a bifunctor, covariant in B and contravariant in A with respect to G-equivariant correspondences. There is a bilinear associative product
KKG (A, D) × KKG (D, B) → KKG (A, B) (the proof is the same as in [28]).
Let B be a C ∗ -algebra endowed with a generalized action of G. We can
define K∗top (G; B), the topological K-theory of G with coefficients in B, to
be the abelian group generated by elements of KKG (C ∗ (Y ), B) where Y is a
locally compact proper, G-compact G-space, divided by following equivalence
relation: if u ∈ KKG (C ∗ (Y ), B) and u0 ∈ KKG (C ∗ (Y 0 ), B), then u ∼ u0 if and
only if there exists a proper G-compact G-space Y 00 , G-maps f : Y → Y 00 and
f 0 : Y 0 → Y 00 such that f∗ u = f∗0 u0 . Note that f∗ is well-defined since f is proper
(Lemma 10.1) and the correspondence of Theorem 7.11 is G-equivariant.
Alternatively, if there exists a classifying space for proper actions EG, then
K∗top (G; B) is the inductive limit of KKG (C ∗ (Y ), B) where Y runs over locally compact, G-compact subspaces of EG. The two definitions are equivalent
because of Lemma 10.3.
Now, the assembly map
µ : K∗top (G; B) → K∗ (Cr∗ (B))
is defined by the composition
j
[EY oG ]⊗·
G
KK∗G (C ∗ (Y ), B) −→
KK(C ∗ (Y ) or G, Cr∗ (B)) −−−−−−→ K∗ (Cr∗ (B)),
where Cr∗ (B) is the reduced C ∗ -algebra of the Fell bundle B (analogue to the
reduced crossed-product algebra), and the module EY oG of the proper groupoid
Y oG defines an element of K-theory [EY oG ] by Proposition 6.5. The definition
NON-HAUSDORFF GROUPOIDS
51
of the descent map jG is analogue to the one in [16]. Let us sketch it. Suppose
we are given a Kasparov A, B-bimodule (E, F ), then jG ([(E, F )]) is the class of
the Kasparov bimodule (E ⊗B Cr∗ (B), F ⊗ 1) endowed with the following left
action of Cr∗ (A):
Z
(π(a)ξ)(g) =
ah ⊗ ξh−1 g λr(g) (dh)
Gr(g)
for all a ∈ Cc (G; A), ξ ∈ Cc (r∗ E ⊗r∗ B B). The element ah ⊗ ξh−1 g belongs to
the module Ah ⊗As(h) Es(h) ⊗Bs(h) Bh−1 g which is identified to Er(g) ⊗Br(g) Bg via
the isomorphism W , thus the formula above indeed defines an element π(a)ξ of
Cc (r∗ E ⊗r∗ B B).
With that definition, one might wonder whether µ is an isomorphism in the
case of proper groupoids. However, since L2 (G) is a C0 (X 0 )-module and cutoff
functions are continuous on X 0 and not on X, the method of [13, Theorem 5.4]
(see also [25]) only allows to prove that µ is an isomorphism in the case when B
is endowed with a generalized action of X 0 o G. This is not in general the case
unless C0 (G(0) ) happens by accident to be a X 0 o G-algebra like in Example 2.3
for Γ finite.
One might also wonder whether the method of [10, 26] extends to nonHausdorff amenable groupoids. First one needs to define the notion of amenable
groupoid. Consider the assertions
(A1) ∃ξi ∈ L2 (G) such that g 7→ kg(ξi )s(g) − (ξi )r(g) k vanishes at infinity on
o G;
(A2) ∃ξi ∈ L2 (G) such that g →
7 kg(ξi )s(g) − (ξi )r(g) k vanishes at infinity on
X0
G;
(AT) There exists a X 0 o G-equivariant continuous field of real affine Hilbert
spaces on X 0 and an arbitrary section x 7→ ξx such that kgξs(g) − ξr(g) k tends
to +∞ when g → ∞ in G.
Then (A1) implies (A2) (using properness of the second projection X 0 o G →
G), and (A2) implies (AT) (like in [4, 26], see also [8]).
In Example 2.3 with H = {1}, G satisfies (A2) if and only if Γ is amenable,
and G satisfies (AT) if and only if Γ is a-T-menable; in Example 2.4 there
is no reason why G should satisfy either amenability condition (A1) or (A2)
even when Γ is amenable. The Baum–Connes conjecture with coefficients for
Hausdorff amenable, and even a-T-menable groupoids is true [26, 10] but the
groupoid of Example 2.3 with H = {1} and Γ = F2 is a counterexample to
the Baum–Connes conjecture [11], and yet it satisfies property (AT). It would
be interesting, either to find a counterexample to the Baum–Connes conjecture
for an amenable foliation groupoid or to prove it; this would probably require
other ideas than those in [10].
Appendix A. Existence of a Hausdorff model for EG
In this appendix we examine conditions underwhich there exists a locally
compact Hausdorff model for EG.
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JEAN-LOUIS TU
Let G be a locally compact groupoid. For every x ∈ G(0) , let Kx = {g ∈
G| {x, g} ∈ HG}. Let Hx ⊂ Gxx be the group generated by Kx , and HG =
∪x∈G(0) Hx . Then HG is a closed subgroupoid of G.
Lemma A.1. Let G be a locally compact groupoid such that r : G → G(0) is
open. Let Z be a Hausdorff space endowed with an action of G with momentum
map p : Z → G(0) . Let Y ⊂ Z and assume that p|Y is open. Then S(Y ) = {g ∈
G| ∀y ∈ Y, p(y) = r(g) =⇒ yg = y} is closed, and contains HG .
Proof. Let us show that S(Y ) is closed. Let g ∈ S(Y ) and y ∈ Y be such that
p(y) = r(g). Let V 3 y and V 0 3 yg be open subspaces of Z. There exist open
sets V1 and W such that y ∈ V1 ⊂ V , g ∈ W , V1 W ⊂ V 0 . Since p|Y is open,
p(V1 ∩ Y ) is an open neighborhood of p(y) = r(g), and since r is open, r(W ) is
an open neighborhood of r(g). Thus, there exist h ∈ W ∩ S(Y ) and y 0 ∈ V1 ∩ Y
such that p(y 0 ) = r(h). One has y 0 h = y 0 ∈ V 0 ∩ V1 , hence V ∩ V 0 6= ∅. Since Z
is Hausdorff, it follows that y = yg for every y ∈ p−1 (r(g)), i.e. g ∈ S(Y ). This
completes the proof that S(Y ) is closed.
Since S(Y ) is a subgroupoid of G, it remains to prove that Kx ⊂ S(Y ). Let
y ∈ Y , x = p(y) and g ∈ Kx . Let V 3 y and V 0 3 yg be open subspaces of
Z. There exist open sets V1 and W such that y ∈ V1 ⊂ V , g ∈ W , V1 W ⊂ V 0 .
Since p(V1 ∩ Y ) and r(W ) are open neighborhoods of p(y) = r(g), one may
assume that r(W ) ⊂ p(V1 ∩ Y ). Since gx−1 = g, there exist Wx 3 x and Wg 3 g
open such that Wg Wx−1 ⊂ W .
Since g ∈ Kx , Wx ∩ Wg 6= ∅. Let h ∈ Wx ∩ Wg . One has r(h) = hh−1 ∈ W .
Let y1 ∈ V1 ∩ Y such that p(y1 ) = r(h). Then y1 = y1 r(h) ∈ V 0 , whence
y1 ∈ V ∩ V 0 . Therefore, V ∩ V 0 6= ∅. Since Z is Hausdorff, one concludes that
y = yg, therefore Kx ⊂ S(Y ).
Proposition A.2. Let G be a locally compact groupoid such that r : G → G(0)
is open and G(0) is Hausdorff and σ-compact. Consider the following assertions:
(i) there exists a locally compact, Hausdorff model for EG;
(i)’ there exists a Hausdorff model for EG;
(ii) there exists a Hausdorff space Z, a map p : Z → G(0) such that G acts
properly on Z with momentum map p and a locally compact subspace
Y ⊂ Z such that p|Y is open surjective;
(0)
(iii) HG ∩ GK
K is quasi-compact for every compact subspace K of G ;
K
(iv) HG ∩ GK is quasi-compact for every quasi-compact subspace K of G(0) ;
(v) HG is a proper groupoid.
(v)’ HG is a proper groupoid, and r : HG → G(0) is open.
Then (v)’ =⇒ (i) =⇒ (i)’ =⇒ (ii) =⇒ (iii) ⇐⇒ (iv) ⇐⇒ (v).
Proof. (iv) =⇒ (iii): obvious.
(iii) =⇒ (v): if K and L are compact subspaces of G(0) , then (HG )L
K =
HG ∩ GK∩L
is
quasi-compact,
so
H
is
proper.
G
K∩L
K
(v) =⇒ (iv): follows from the fact that HG ∩ GK
K = (HG )K .
(i) =⇒ (i)’: obvious.
(i)’ =⇒ (ii): since G is a locally compact, proper G-space (see Proposition 2.16) and G/G = G(0) is σ-compact, by the universal property of EG
NON-HAUSDORFF GROUPOIDS
53
there exists a G-map f : G → EG. Let Y = f (G(0) ). Since p ◦ f|G(0) is the identity map, f induces a bijection from G(0) onto Y , therefore f|G(0) : G(0) → Y
and p|Y are homeomorphisms. In particular, Y is locally compact and p|Y is
open surjective.
(ii) =⇒ (iii): let K ⊂ G(0) be a compact set. Since HG is closed in G,
K
K
HG ∩ GK
K is closed in S(Y ) ∩ GK , hence it suffices to prove that S(Y ) ∩ GK is
quasi-compact. Let L ⊂ Y be a compact set such that p(L) = K. Such a set
exists because Y is locally compact Hausdorff and p|Y is open and surjective.
Then S(Y ) ∩ GK
K is closed in the quasi-compact set {g ∈ G| ∃z ∈ L, zg ∈ L},
thus it is quasi-compact.
(v)’ =⇒ (i): set Z = HG \G. Let π : G → Z be the canonical mapping. We
prove that Z is Hausdorff, i.e. that Z = HZ. Since G acts on HZ, it suffices
to prove that for all x ∈ G(0) and all g ∈ Gxx − HG , there exist disjoint open
neighborhoods of π(x) and of π(g) in Z.
Let K be an open neighborhood of x in G(0) . Let K 0 = HG ∩ GK
K . It is a
K
quasi-compact subspace of G. For every a ∈ HG ∩ GK , there exist Va 3 a and
Va,g 3 g such that Va and Va,g are disjoint open sets, otherwise one would have
a ∈ Gxx and ga−1 ∈ Kx , which would contradict g ∈
/ HG .
By quasi-compactness, there exist two disjoint open sets V and V 0 such
that K 0 ⊂ V and g ∈ V 0 . Since K 0 is quasi-compact, there exist an open
and K 0 W ⊂ V . Thus, we have
neighborhood W of x such that W ⊂ GK̊
K̊
0
0
HG W ∩ V = ∅, whence HG W ∩ HG V = ∅. It follows that Z is Hausdorff.
Let us show that the action of G on Z is proper. Since r : HG → G(0) is open,
by Example 2.33 Z o G is Morita-equivalent to HG n (G/G) = HG n G(0) = HG
which is proper. Moreover, by Lemma 2.34, Z is locally compact.
Let Z0 = G/HG . The above shows that Z0 is locally compact Hausdorff,
and that the left action of G on Z0 is proper. Let M (Z0 ) be the set of positive
measures on Z0 whose image by r : G/HG → G(0) is of the form λδx (1/2 <
λ ≤ 1, x ∈ G(0) ). It is not hard to show that M (Z0 ) is locally compact (see
for instance [25, Proposition 6.13]). For every f ∈ Cc (Z0 )+ , let Ωf = {µ ∈
M (Z0 )| µ(f ) > 12 kf k∞ }. Then {g ∈ G| Ωf1 g ∩ Ωf2 6= ∅} is relatively quasicompact, hence by Proposition 2.14, G acts properly on M (Z0 ).
If Z 0 is a locally compact σ-compact proper G space, let c be a cutoff function on Z 0 o G. For every z ∈ Z 0 , consider the measure c(zg) λp(z) (dg) on
Gp(z) . Denote by ϕ(z) its image by the obvious map Gp(z) → G/H. Then
ϕ : Z 0 → M (Z0 ) is continuous G-equivariant, and two G-maps Z 0 → M (Z0 ) are
connected by a linear homotopy.
Appendix B. The classifying space for proper actions on
Hausdorff spaces
Let G be a locally compact groupoid. Let Z be a Hausdorff, locally compact
space endowed with a proper action of G such that Z/G is σ-compact. We say
that Z is universal for proper actions on Hausdorff spaces if for every Hausdorff,
locally compact space Z 0 with a proper action of G such that Z 0 /G is σ-compact,
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JEAN-LOUIS TU
there exists a G-map f : Z 0 → Z, and if f is unique up to G-homotopy. Then
the space Z is unique up to G-homotopy, and will be denoted by E h G.
Let us first examine an example of E h G.
Lemma B.1. Let G be a locally compact groupoid with G(0) Hausdorff. Suppose
that G = G1 ∪G2 , where G1 is an open subgroupoid and G2 a closed subgroupoid,
(0)
such that G1 ∩ G2 = ∅. If every S ∈ HG which is in the closure of G1 and
such that S ⊂ G2 , is infinite, then G has a classifying space for proper actions
if and only if it is the case for G1 and G2 ; then E h G is the disjoint union of
E h G1 and E h G2 .
Proof. Let Z be a locally compact, Hausdorff space, and p : Z → G(0) such
(0)
that G acts properly on Z with momentum map p. Let Z1 = p−1 (G1 ) and
(0)
Z2 = p−1 (G2 ). Then Z1 is an open saturated subspace of G. Let us show that
it is closed. Suppose that z ∈ Z2 is the limit of a ultrafilter F on Z1 . Then
p(F) converges to p(z) in G(0) , thus it converges to an element S ∈ HG such
p(z)
that S ⊂ Gp(z) , so every s ∈ S is also a limit point of p(F). By assumption,
S is infinite. As z 0 p(z 0 ) = z 0 for every z 0 ∈ Z, by continuity zs = z for every
s ∈ S. This contradicts the fact that G acts properly on Z.
Therefore, (Z1 , Z2 ) is a partition of Z by open and closed subspaces. The
conclusion follows easily.
Example B.2. Consider Example 2.3 with Γ = Z and H = {0}. By Lemma B.1,
E h G = (0, 1] q ({0} × R).
We thus see in this elementary example (which is a simplified version of the
holonomy groupoid of Reeb’s foliation) that if one defines topological K-theory
using the classifying space E h G, then it does not coincide with the K-theory
of Cr∗ (G). Therefore, we won’t discuss the existence of E h G any further.
References
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Jean-Louis Tu
University Pierre et Marie Curie
Institut de Mathématiques
175, rue du Chevaleret
75013 Paris
France
[email protected]