Amenability of Groupoids Arising from Partial Semigroup Actions and Topological Higher Rank Graphs

0.35 MB·English

by  Jean N. Renault

AMENABILITY OF GROUPOIDS ARISING FROM PARTIAL
SEMIGROUP ACTIONS AND TOPOLOGICAL HIGHER RANK
5
GRAPHS
1
0
2
JEANN.RENAULTANDDANAP.WILLIAMS
r
a
M
Abstract. WeconsidertheamenabilityofgroupoidsGequippedwithagroup
valued cocycle c : G → Q with amenable kernel c−1(e). We prove a general

7 resultwhichimplies,inparticular,thatGisamenablewheneverQisamenable
1 andifthereiscountablesetD⊂Gsuchthatc(Gu)D=Qforallu∈G(0).
We show that our result is applicable to groupoids arising from partial

] semigroup actions. We explore these actions in detail and show that these
A
groupoids includethosearisingfromdirectedgraphs, higherrankgraphs and
O even topological higher rank graphs. We believe our methods yield a nice
alternativegroupoidapproachtotheseimportantconstructions.
.
h
t
a
m
1. Introduction
[

It is often important to establish the amenability of groupoids that arise in
2
v applications. For example, amenability implies the equality of the reduced and
7 universal norms on the associated groupoid algebras. This is important in the
2 study ofthe Baum-ConnesconjectureforgroupoidC∗-algebrasasitis the reduced
0
algebrathatplaysthekeyrole,whiletheuniversalalgebrahasthebetterfunctorial
3
properties. Forexample,TuhasshownthattheC∗-algebraofanamenablegroupoid
0
. withaHaarsystemsatisfiestheBaum-ConnesconjectureandtheUCT[24]. Inthe
1
classification program, amenability implies nuclearity which is typically a crucial
0
hypothesis.
5
1 Thesortofgroupoidswewishtofocusonarethosearisingfromthemuchstudied
: C∗-algebras associated to higher-rank graphs, and more recently, to topological
v
higher-rank graphs. As a specific example, we recently considered the C∗-algebras
i
X
of groupoids associated to k-graphs (see [20]). Such groupoids have a canonical
r Zk-valued cocycle c, and it is not hard to see that c−1(0) is amenable. In some
a
cases, c is not only surjective, but strongly surjective in that c(Gu) = Zk for all
u ∈ G(0). Then the amenability of G is a consequence of [1, Theorem 5.3.14].
However,there are interestingexamples inwhich c neednotbe stronglysurjective,
and examples show that the problem of the amenability of G turns out to be very
subtle. A result of Spielberg [22, Proposition 9.3] asserts that if G is ´etale and
if c : G → Q is a continuous cocycle into a countable amenable group, then G
is amenable whenever c−1(e ) is. Although this result is satisfactory for most k-
Q
graphs,theproofisunsatisfyinginthatitcircumventsgroupoidtheorybyinvoking
Date:January15,2015; RevisedMarch16,2015.
1991 Mathematics Subject Classification. Primary22A22,46L55, 46L05.
Key words and phrases. Groupoids, amenable groupoids, cocycle, semigroup actions, higher
rankgraphs,topological higherrankgraphs.
Thesecondauthor wassupportedbyagrantfromtheSimonsFoundation.
1
2 JEANN.RENAULTANDDANAP.WILLIAMS
the nontrivial result that for ´etale groupoid, C∗(G) is nuclear if and only if G
is amenable [15, Corollary 2.17]. In particular, this result is valid only for ´etale
groupoids.
Herewewanttoproveageneralresultthatsubsumesbothcocycleresultsfrom[1]
and from [22]. That such a result will have delicate hypotheses is foreshadowedby
theobservationthatonecanhaveasurjectivecontinuouscocyclecfromagroupoid
into an amenable group Q such that c−1(e ) amenable and still have G fail to be
G
amenable. Forexample,letQbeanamenablegroupwithanonamenablesubgroup
N (which obviously is not closed in Q). Let G be the group bundle G = Q N
and let c be the identity map: c(γ)=γ. `
Nevertheless, we obtain a quite general result: Theorem 4.2. It implies in par-
ticular, that if c : G → Q is a continuous cocycle into an amenable group Q with
amenable kernel such that there is a countable set D so that c(Gu)D = Q for all
u ∈ G(0), then G is amenable. Even in this form, we recover both results above
and remove the hypothesis that G be ´etale from Spielberg’s result.
Although our results apply to topological groupoids with Haar systems, it is
interestingthatourproofreliesonthenotionsofBorelgroupoids,Borelamenability
andBorelequivalence. Atacrucialjuncture,weareabletoshowthatourgroupoid
is Borel equivalent to a Borel amenable groupoid. Since we also show that Borel
equivalencepreservesBorelamenability,wecanappealtotheresultfrom[19]which
demonstrates that topologicalamenability is equivalent to Borelamenabiltiy — at
least for locally compact groupoids with Haar systems.
Having proved our cocycle result, it is crucial to show how it can be applied
to groupoids which are currently being studied in the literature. To do this, we
show that our results can be applied to groupoids arising from (partial) semigroup
actions. Akeyrˆoleis playedbythe notionofdirected action(Definition 5.2) which
givesa partition ofthe spaceinto orbits. We explorethese actions in detail. Then,
making significant use of hard work due to Nica and Yeend, we are able to show
thatsuchgroupoids include those arising fromdirectedgraphs,higherrank graphs
and even topological higher rank graphs. We think our methods using semigroup
actionsyieldanicealternativegroupoidapproachtotheseimportantconstructions.
To prove our cocycle result, we work with locally Hausdorff, locally compact
groupoidswhich are alwaysassumed to be secondcountable and to possess a Haar
system. Note that a secondcountable, locally Hausdorff,locally compact groupoid
isthecountableunionofcompactHausdorffsets. HenceitsunderlyingBorelstruc-
ture is standard. In Section 2 we review the notion of Borel amenability and some
of the basic properties of Borel groupoids we need in the sequel. In Section 3 we
recallthenotionofBorelequivalenceandprovethatitpreservesBorelamenability.
InSection4,weprovethemainamenabilityresult. InSection5weintroducesemi-
groupactions. Whentheactionisdirected,thereisasemi-directproductgroupoid.
Thisconditionputsintolighttwoclassesofsemigroups: Oresemigroupsandquasi-
lattice ordered semigroups. Under reasonable hypotheses, our main result applies
and the semi-direct product groupoid is amenable. In Section 6, we show how the
groupoid corresponding to a topological higher rank graph (and therefore to the
manysubcasesoftopologicalhigherrankgraphs)canberealizedasthesemi-direct
productgroupoidofadirectedsemigroupactionofP =Nd. Iffact,weconsiderP-
graphs,whereP isanarbitrarysemigroupratherthansimplyNd asintheoriginal
definition. The natural assumption is that the semigroup be quasi-lattice ordered.
AMENABILITY OF GROUPOIDS 3
If moreover the semigroup is a subsemigroup of an amenable group and the graph
satisfiesapropernesscondition,thenthisgroupoidisamenableaswell. Besidesthe
amenabilityproblem,thesectionrevealsatightconnectionbetweenhigherrankC*-
algebras and Wiener-Hopf C*-algebras. In fact, we use the techniques introduced
by Nica in [14]; in particular the Wiener-Hopf groupoid of a quasi-lattice ordered
semigroupdefined by Nica appears as a particular case of our generalconstruction
for topological higher rank graphs.
Acknowledgments. We are very grateful to an anonymous referee for a thorough
reading of our paper and for pointing out the relevance of the work of Exel[8] and
Brownlowe-Sims-Vittadello [4]. In addition the referee pointed out a gap in the
originalproof of Theorem 5.13, and suggesteda line of attack which resulted in an
improved version of the result.
2. Borel Amenability
For the details on Borel groupoids, proper Borel amenability and proper Borel
G-spaces, we refer to [1] and to [1, §2.1a] in particular. Recall that in order for
a groupoid to act on (the left) a space X, we require a map r : X → G(0). If
X
there is no ambiguity, we write simply r in place of r and call it the moment
X
map.1 As in [1], to avoid pathologies we will always assume that our Borel spaces
are analytic Borel spaces. We recall some of the basics here. If X and Y are Borel
spaces and π :X →Y is Borel surjection, then a π-system is a family of measures
m={my} such that each my is supported in π−1(y) and such that
y∈Y
y 7→ f(x)dmy(x)
Z
X
is Borel for any nonnegative Borel function f on X. If G is a Borel groupoid, X
and Y are Borel G-spaces and π is G-invariant, then we have that m is invariant
if γ·my =mγ·y whenever s(γ)=r(y). (By definition, γ·my(E)=my(γ−1·E).)
Definition 2.1 ([1, Definition 2.1.1(b)]). Suppose that G is a Borel groupoid and
that X and Y are Borel G-spaces. A surjective Borel G-map π : X → Y is G-
properlyamenable ifthereisainvariantBorelπ-systemm={my} ofprobability
y∈Y
measures on X. Then we say that m is an invariant mean for π.
Remark 2.2. Notice that if π : X → Y is a Borel G-map, then the induced map
π˙ : G\X → G\Y is Borel with respect to the quotient Borel structures. Suppose
that π is G-properly amenable and that {λy} is an invariant mean for π. Let
p : X → G\X be the quotient map. Then the push forward p λy is a probability
∗
measure supported on π˙−1(y˙). By invariance, it depends only on y˙ and we can
denote this measure by λ˙y˙. If b is a bounded Borel function on G\Y, then
b(z˙)dλ˙y˙(z˙)= b(p(x))dλy(x).
Z Z
G\Y X
Hence {λ˙y˙} is a Borel π˙-system of probability measures on G\X.
y˙∈G\Y
Definition 2.3 ([1, Defintion 2.1.2]). A Borel groupoid G is proper if the range
map r : G → G(0) is G-properly amenable. A Borel G-space X is proper if the
projection p:X ∗G→X is G-properly amenable.
1Thetermprojectionmap,anchormapandstructuremaphavealsobeenusedintheliterature.
4 JEANN.RENAULTANDDANAP.WILLIAMS
Remark 2.4. Thus X is a proper G-space if and only if there is a family {mx}
x∈X
of probability measuresmx on G supported in Gr(x) such that x7→mx(f) is Borel
for all non-negative Borel functions on G and such that γ·mx =mγ·x.
IfX andY are(left)G-spaces,thenthe fiberedproductX∗Y isaG-spacewith
respectto the diagonalaction. If X and Y are Borel G-spaces, then we give X∗Y
the Borel structure as a subset of X ×Y with Borel structure generated by the
Borel rectangles. In particular, if X is a G-space, then X∗G={(x,γ)∈X ×G:
r(x)=r(γ)} is a G-space with respect to the diagonal action.2
Definition 2.5. Suppose that G is a Borel groupoid and that X and Y are Borel
G-spaces. AsurjectiveBorelG-mapπ :X →Y isG-amenable (orsimplyamenable
if there is no ambiguity about G) if there is a sequence {mn}n∈N of Borel systems
ofprobabilitymeasuresy 7→my onX whichisapproximatelyinvariantinthesense
n
thatforallγ ∈G, kγ·my−mγ·yk convergesto0,wherek·k isthetotalvariation
n n 1 1
norm. We say that {mn}n∈N is an approximate invariant mean for π.
ThenotionofBorelamenabilityforgroupoidswasfirstformalizedin[19,Defini-
tion 2.1] — however, here we use the formulation in which conditions (ii) and (iii)
of [19, Definition 2.1] have been replaced by (ii′).
Definition 2.6. ABorelgroupoidGis(Borel)amenable ifthe rangemapr :G→
G(0) isG-amenable. ABorelG-spaceX isamenable iftheprojectionp:X∗G→X
is G-amenable.
A key result about Borel amenable maps we need is the following.
Lemma 2.7. Let G be a Borel groupoid. If there is a proper Borel G-space X such
that the moment map r :X →G(0) is Borel amenable, then G is Borel amenable.
Proof. Since,byassumption,the mapπ :X∗G→X isproperlyamenable,there
X
is an invariantsystemm={mx} of probabilitymeasures for π . We can view
x∈X X
each mx as a measure on GrX(x) such that γ ·mx = mγ·x. Since rX is amenable,
thereisanapproximateinvariantmean{µ }. Thenµu is aprobabilitymeasureon
n n
r−1(u), and for all γ ∈G, lim kγ·µs(γ)−µr(γ)k =0. Define
X n n n 1
λu = mxdµu(x).
n Z n
X
Then λu is a probability measure supported in Gu. The system λ = {λu}
n n n u∈G(0)
is certainly Borel, and
kγ·λs(γ)−λr(γ)k = γ·mydµs(γ)(y)− mxµr(γ)(x)
n n 1 (cid:13)ZX n ZX n (cid:13)
(cid:13) (cid:13)
(cid:13) (cid:13)
= mγ·xdµs(γ)(y)− mxdµr(γ)(x)
(cid:13)ZX n ZX n (cid:13)
(cid:13) (cid:13)
(cid:13) (cid:13)
= mxd(γ·µs(γ)−µr(γ))(x)
(cid:13)ZX n n (cid:13)
(cid:13) (cid:13)
≤k(cid:13)γ·µs(γ)−µr(γ)k . (cid:13)
n n 1
2Infact,X∗Gisagroupoid—sometimescalledthetransformationgroupoidforGactingon
X. Thisgroupoidisdenoted byX⋊G in[1]. After identifyingits unitspacewithX,the range
map (x,γ)7→x is clearly G-equivariant. Hence, in Defintion 2.3, we defined a G-space X to be
properlyamenableexactlywhenthegroupoidX⋊Gisproperlyamenable.
AMENABILITY OF GROUPOIDS 5
It follows that λ is an approximate invariant mean for r : G → G(0). Thus G is
Borel amenable. (cid:3)
Of course, a proper Borel groupoid is Borel amenable. Our next lemma should
be compared with [1, Proposition 5.3.37].
Lemma 2.8. Suppose that G is a Borel groupoid which is the increasing union of
a sequence of proper Borel subgroupoids G . Then G is Borel amenable.
n
Proof. Byassumption,wecanfindansystemm =(mu) ofinvariantproba-
m n u∈G(n0)
bilitymeasuresonG ;thatis, γ·ms(γ) =mr(γ) forallγ ∈G . Usingthestandard
n n n n
theory of disintegration of measures for example (see [25, Theorem I.5]), we can
extend each m to a Borel system of probability measures on all of G(0). Then
m
that m=(m ) is an approximate invariant mean for r :G→G(0). (cid:3)
n
3. Borel Equivalence
The definition of equivalence for Borelgroupoids is given in the appendix of [1].
In light of [19], it turns out to be a much more significant notion than originally
thought.
Definition 3.1 ([1, Definition A.1.11]). Let G and H be Borel groupoids. A
(G,H)-Borel equivalence is a Borel space Z such that
(a) Z is a free and proper left Borel G-space,
(b) Z is a free and proper right Borel H-space,
(c) The G- and H-actions commute,
(d) r :Z →G(0) induces a Borel isomorphism between Z/H and G(0), and
(e) s:Z →H(0) induces a Borel isomorphism between G\Z and H(0).
In this case, we say that G and H are Borel equivalent.
Itisprovedin[1,Theorem2.2.17]thatequivalenceoflocallycompactgroupoids
preserves(topological)amenability. HereweshowasimilarresultholdsintheBorel
case using virtually the same argument.
Theorem3.2. SupposethatGandH areequivalentBorelgroupoids. IfH isBorel
amenable (resp., properly amenable), then so is G.
Proof. Let Z be a (G,H)-equivalence. Consider the commutative diagram
G r // G(0)
OO OO
pG r
Z∗G pZ //Z
p q
(cid:15)(cid:15) q (cid:15)(cid:15)
Z // G\Z,
where p(z,γ):=γ−1·z.
First, assume that G is properly amenable. To see that H is properly amenable
it will suffice, by [1, Corollary 2.1.7], to see that that H-map s : Z → H(0) is
properly amenable.
6 JEANN.RENAULTANDDANAP.WILLIAMS
Let λ = {λu} be an invariant mean for r : G → G(0). We first build an
u∈G(0)
invariant mean λ for p :Z∗G→Z: let λz be given by
Z Z Z
λz(f)= f(z,λ)dλrZ(z)(γ).
Z Z
G
Consider the measure on Z obtained by the push forward q λz. Since invariance
∗ Z
implies η−1·λr(η) =λs(η) for all η ∈G, it follows that
q λη·z(b)= b(p(η·z,γ))dλη·z(γ)
∗ Z Z Z
G
= b(γ−1η·z)dλrZ(η·z)(γ)= b(γ−1·z)d(η−1·λr(η))(γ)
Z Z
G G
=q λz(b).
∗ Z
Hence the push forward q λz depends only on z˙ in G\Z. Since Z is a Borel
∗ Z
equivalence, we can identify the quotient map q : Z → G\Z with the moment
map s : Z → H(0). Thus we get an s-system of measures ν = {νv} where
v∈H(0)
νv =q λz for any z with s(z)=v. It will suffice to see that ν is invariant. But if
∗ Z
s(z)=r(h), then
b(z)d(νr(h)·h)(z)= b((γ−1·z)·h)dλr(z)(γ)= b(γ−1·(z·h))dλr(z·h(γ)
Z Z Z
Z G G
= b(z)dνs(h)(z).
Z
Z
This completes the proof for proper amenability.
NowweassumethatGisBorelamenableandthatλ={λ }isanapproximately
n
invariantmeanforr :G→G(0)withλ ={λu} . ByLemma2.7,itwillsuffice
n n u∈G(0)
to show that the H-map s : Z → H(0) is Borel amenable. The argument parallels
that above argument for proper amenability.
We begin by lifting each λ to a system λ = {λz } of probability mea-
n n,Z n,Z z∈Z
sures for p :Z∗G→Z:
Z
λz (f)= f(z,γ)dλr(z)(γ).
n,Z Z n
G
We let µz = q λz . Since λ is not necessarily invariant, we can’t assert that
n ∗ n,Z n,Z
µz depends only on z˙. However
n
b(w)dµz(w)= b(γ−1·z)dλr(z)(γ),
Z n Z n
Z G
and µz is supported on q−1(z)=G·z.
n
Since Z is a properG-space,by definition there is a G-invariantsystemofprob-
ability measures ρ = {ρz} for the Borel G-map p : Z ∗ G → Z which we
z∈Z Z
view as a family of measures on G such that suppρz ⊂ Gr(z). As in Remark 2.2,
ρ drops to a system ρ˙ = {ρ˙z˙} for the quotient map q : Z → G\Z. Since
z˙∈G\Z
(z,γ)7→γ−1·z identifies G\(Z ∗G) with Z, we can view these as measures on Z.
Explicitly, ρ˙z˙ =p ρz and
∗
ρ˙z˙(b)= b(γ−1·z)dρz(γ).
Z
G
AMENABILITY OF GROUPOIDS 7
Usingtheinvarianceofρwegetmeasuresdependingonlyonz˙ supportedonq−1(z˙)
by averagingthe µz with respect to ρ˙z˙:
n
(3.1) νz˙ = µz dρ˙z˙(z)= µγ−1·zdρz(γ).
n Z n Z n
Z G
Asinthefirstpartoftheproof,weusethefactthatZ implementsanequivalence
to identify the quotient map q : Z → G\Z with the moment map s : Z → H(0).
Thus we get an s-system {νv} where νv = νz˙ for any z such that s(z) = v.
v∈H(0)
We will complete the proof by showing that ν is an approximately invariant mean
for s. Therefore we need to see that for all h∈H, kνr(h)·h−νs(h)k tends to zero
n n 1
with n. Notice that if (z,γ)∈Z∗G, then
kµz −µγ−1·zk ≤kλr(γ)−λ·λs(γ)k .
n n 1 n n 1
Next we claim that for fixed z ∈ Z, lim kνz˙ −µzk = 0. To see this we employ
n n 1
(3.1), view ρz as a measure on Gr(z), and deduce that
(3.2) kνz˙ −µzk ≤ kµγ−1·z−µzkdρz(γ)≤ kλr(γ)−γ·λs(γ)k dρz(γ).
n n 1 Z n n Z n n 1
G G
Since for each γ, lim kλr(γ)−γ·λs(γ)k =0, we see that (3.2) goes to zero by the
n n n 1
Lebesgue Dominated Convergence Theorem.
Now fix h∈H and ǫ>0. Let z ∈Z be such that s(z)=r(h) and let z′ =z·h
and observe that µz ·h=µz′. Let M be such that n≥M implies that
n n
kνz˙′ −µz′k< ǫ.
n n 2
Then for n≥M, we have
kνr(h)·h−νs(h)k ≤kνr(h)·h−µz ·hk +kµz ·h−µz˙k+kµz˙ −νs(h)k
n n 1 n n 1 n n n n
=kνz˙′ −µz′k +0+kµz′ −νz˙′k<ǫ.
n n 1 n n
Thus s:Z →H(0) is Borel amenable. This completes the proof. (cid:3)
We close this section with two technical results which will be of use in the next
section. Recallfrom [19, Definition 2.3]that a Borel approximate invariant density
on G is a sequence (g ) of non-negative Borel functions on G such that
n
g (γ)dλu(γ)≤1 for all n, g (γ)dλu(γ)→1 for all u∈G(0) and
Z n Z n
G G
g (γ−1γ′)−g (γ′) dλr(γ)(γ′)→0 for all γ ∈G.
Z n n
G(cid:12) (cid:12)
(cid:12) (cid:12)
Lemma 3.3. Let G be a Borel groupoid with a Borel Haar system {λu} . Let
u∈G(0)
{Y }∞ be a countable cover of G(0) by invariant Borel subsets such that each G
i i=1 |Yi
is Borel amenable. Then G is Borel amenable.
Proof. Since each Y is invariant, u ∈ Y implies that (G )u = Gu. Hence λ
i i |Yi
restricts to a BorelHaar system on G . Hence by [19, Proposition2.4], the Borel
|Yi
amenability of G implies that there is a Borel approximate invariant density),
|Yi
{gi}∞ , on G .
n n=1 |Yi
Let B = Y and if i ≥ 2, let B = Y \ i−1Y . Then the {B } are a pairwise
1 1 i i j=1 j i
disjoint cover of G(0) by invariant Borel setSs (some of which might be empty). Let
bi be the characteristic function of B . Then each bi is a Borel function on G(0)
i
8 JEANN.RENAULTANDDANAP.WILLIAMS
(takingthe values0 and1)suchthatb vanishesoffY andforeachu∈G(0), there
i i
is one and only one i such that bi(u)=1. In particular, we trivially have
∞
bi(u)=1 for all u∈G(0).
X
i=1
For each n, define g on G by
n
g (γ)= gi ·bi,
n n
X
i
where gi ·bi(γ)=gi(γ)bi(s(γ)).
n n
By invariance,
g (γ)dλu(γ)= gi(γ)dλu(γ),
Z n Z n
G G
where i is such that bi(u) = 1. Now it is clear that {g } is a Borel approximate
n
invariant density for G. Hence G is Borel amenable as claimed. (cid:3)
Recall that by assumption, all of our Borel spaces, and our Borel groupoids in
particular, are analytic Borel spaces.
Lemma3.4. LetGbeaBorel groupoid actingfreelyon (theright of) aBorel space
X such that the quotient map q : X →X/G has a Borel cross section. Then X is
a proper Borel G-space.
Proof. LetX⋊GbethetransformationgroupoidfortherightG-action.3 Themap
(x,γ)7→(x,x·γ) is a Borelas well as analgebraic isomorphismof X⋊G onto the
equivalence relation groupoid X ∗ X = {(x,y) ∈ X ×X : q(x) = q(y)}. Since
q
X∗ X andX⋊GarebothBorelsubsetsofthecorrespondingproductspaces,they
q
arethemselvesanalyticBorelspaces. HenceX⋊GandX∗ X areBorelisomorphic
q
by Corollary2 of[2, Theorem3.3.4]. Hence itsuffices to see that X∗ X is proper.
q
WecanidentifyX withtheunitspaceofX∗ X. Foreachx∈X,letmx =δ ×
q x
δ wherec:X/G→X is our Borelcrosssectionfor q. Since c(q(x))=c(q(y))
c(q(x))
if(x,y)∈X∗ X,wehave(y,x)·mx =my. HenceX isaproperBorelG-space. (cid:3)
q
4. Cocycles
In this paper, by a cocycle on a groupoid G we mean a homomorphism c:G→
Q into a group Q. As in [17, Definition 1.1.9],4 the corresponding skew-product
groupoid is
G(c)={(a,γ,b)∈Q×G×Q:b=ac(γ)}.
Then multiplication is given by (a,η,b)(b,γ,d) = (a,ηγ,d) and inversion by
(a,γ,b)−1 =(b,γ−1,a). We can identify the unit space of G(c) with G(0)×Q, and
then the range and source maps are given as expected: r(b,γ,a) = (r(γ),b) while
s(b,γ,a) = (s(γ),a). If G is a locally Hausdorff, locally compact groupoid with
a Haar system {λu} and if Q is a locally compact group, then provided c
u∈G(0)
3HereX⋊G={(x,γ)∈X×G:s(x)=r(γ)}hasmultiplicationdefinedby(x,γ)(x·γ,η)=
(x,γη).
4In[17],G(c)isdescribedaspairs(γ,a)∈G×Q. Thegroupoidsareisomorphicviathemap
(a,γ,b)7→(γ,a).
AMENABILITY OF GROUPOIDS 9
is continuous, G(c) is a locally Hausdorff, locally compact groupoid with Haar
system β ={β(u,a)} where
(u,a)∈G(0)×Q
β(u,a)(g)= g(a,γ,ac(γ))dλu(γ).
Z
G
Later it will be useful to note that Q acts on the left of G(c) by (groupoid)
automorphisms:
q·(b,γ,a)=(qb,γ,qa).
The action of Q on G(0) ×Q, identified with the unit space of G(c), is given by
q·(u,a)=(u,qa).
Part of our interest in G(c) is due to the following result.
Proposition 4.1 ([17, Proposition II.3.8]). Suppose that G is a second countable
locally Hausdorff, locally compact groupoid with a Haar system and that c is a
continuous homomorphism from G into a locally compact group Q. If G(c) and Q
are both amenable, then so is G.
Proof. By [19, Definition 2.7], we need to construct a topological approximate in-
variantmean{g }inC(G). Itwillsufficetoshowthatforeachpairofcompactsets
n
K ⊂ G and L ⊂ G(0), and each ǫ > 0, there is a nonnegative function f ∈ C(G)
such that
(A) f(γ)dλu(γ)≤1 for all u∈G(0),
Z
G
(B) f(γ)dλu(γ)≥1−ǫ for all u∈L and
Z
G
(C) |f(γ−1γ′)−f(γ′)|dλr(γ)(γ′)≤ǫ for all γ ∈K.
Z
G
Since Q is amenable and c(K) is compact, there is a nonnegative function k ∈
C (Q) such that with respect to a fixed right Haar measure α on Q we have
c
ǫ
k(a)dα(a)=1 and |k(ab)−k(a)|dα(a)≤ for all b∈c(K).
Z Z 2
Q Q
On the other hand, the (topological) amenability of G(c) allows us to find a non-
negative function g ∈C(G(c)) such that
(a) g(a,γ,ac(γ))dλu ≤1 for all (u,a)∈G(0)×Q,
Z
G
(b) g(a,γ,ac(γ))dλu ≥1−ǫ for all (u,a)∈L×suppk and
Z
G
(c) |g(ac(γ),γ−1γ′,ac(γ′))−g(a,γ′,ac(γ′))|dλr(γ) ≤ ǫ for all (a,γ) ∈
Z
G
suppk·c(K)−1×K.
Now we define a nonnegative function on G via
f(γ):= k(a)g(a,γ,ac(γ))dα(a).
Z
G
Ifg iscontinuousandsupportedinacompactHausdorffsubset,thenf istoo. Since
in general, g a finite sum of such functions, we have f ∈C(G). Then it is easy to
check f satisfies (A) and (B) above. On the other hand,
|f(γ−1γ′)−f(γ′)|dλr(γ)(γ′)
Z
G
10 JEANN.RENAULTANDDANAP.WILLIAMS
= k(a) g(a,γ−1γ′,ac(γ−1γ′))−g(a,γ′,ac(γ′)) dλr(γ)(γ′)
Z Z
G Q (cid:0) (cid:1)
= k(ac(γ)) g(ac(γ),γ−1γ′,a(c(γ′))−g(a,γ′,ac(γ′)) dλr(γ)(γ′)dα(a)
Z Z
G Q (cid:0) (cid:1)
+ k(ac(γ))−k(a) g(a,γ′,ac(γ′))dλr(γ)(γ′)dα(a)
Z Z
G Q(cid:0) (cid:1)
which, in view of our assumptions on k and g, is
ǫ ǫ
≤ + =ǫ. (cid:3)
2 2
Theorem4.2. SupposethatGisalocallyHausdorff, locally compactsecondcount-
able groupoid with a Haar system and that c : G → Q is a continuous homomor-
phism into a locally compact group Q such that the kernel c−1(e) is amenable. Let
s˜: G → G(0) ×Q be the map given by s˜(γ) = (s(γ),c(γ)). Let Y be the image of
s˜ in G(0) ×Q. Then Y is G(c)-invariant. Suppose that there are countably many
q ∈Q such that
n
G(0)×Q= q ·Y,
n
[
n
then G(c) is amenable. In particular, if Y is open in G(0) × Q, then G(c) is
amenable. If in addition, Q is amenable, then so is G.
It is not hard to check that Y is invariant: if s(a,γ,ac(γ)) = (s(γ),ac(γ)) ∈ Y,
then there is an η ∈ G such that s˜(η) = (s(γ),ac(γ)). Hence s(η) = s(γ) and
c(η)=ac(γ). But then s˜(ηγ−1)=(r(γ),c(η)c(γ)−1)=(r(γ),a)=r(a,γ,ac(γ)).
Our key tool is the following observation.
Proposition4.3. LetG,Q,candY =s˜(G)beasinthestatementofTheorem4.2.
Then the Borel groupoids G(c) and c−1(e) are Borel equivalent.
|Y
Remark 4.4. The proof is complicated by the fact that we are not assuming G is
Hausdorff,noris itclearwhether c−1(e)hasa Haarsystem. Thus wecannotapply
[1, Corollary 2.1.17] to establish that the actions are proper. Instead, we resort to
the definition.
Proof. We want to show that G is a (c−1(e),G(c) )-Borel equivalence. We let
|Y
c−1(e) act on the left by multiplication in G. (Hence the moment map for the left
action is just r.) For the right action of G(c) we use the moment map s˜. Thus
|Y
η·(a,γ,ac(γ)) makes sense only when (η,γ)∈G(2) and c(η)=a. Then we define
(4.1) η·(c(η),γ,c(ηγ))=ηγ.
To see that G is a proper right G(c) -space, we need to see that the transfor-
|Y
mation groupoid G⋊G(c) for the right G(c) -action is proper. But the map
|Y |Y
(η,(c(η),γ,c(ηγ))7→(η,γ)
isclearlyabijectionofG⋊G(c) ontoG⋊G. Itiseasytoseethatitisagroupoid
|Y
homomorphism as well. Since the right action of any Borel groupoid on itself is
proper (by a left to right modification of [1, Example 2.1.4(1)]), it follows that G
is a proper G(c) -space. The action is clearly free.
|Y
To see that the left action is proper in the Borel sense, we first note that it
is certainly free and proper topologically. Hence the orbit space is itself a locally