AMENABLE EQUIVARIANT MAPS DEFINED ON A GROUPOID

17 Pages·2005·0.15 MB·English

AMENABLEEQUIVARIANTMAPSDEFINEDONAGROUPOID
MA˘DA˘LINAROXANABUNECI
ABSTRACT. Weprovetheequivalencebetweentheamenabilityofanequivariantmap
definedonagroupoidandtheamenabilityofasemidirectproduct. Also,wegivea
generalizationofthefollowingwellknownresult(J.P.Pier, 21.2of[12]): ifG isan
amenablelocallycompactnon-compactgroup,andmisaninvariantmeanonL∞(G),
thenm(f)=0foreveryf ∈C0(G).Weestablishasimilarpropertyofanapproximate
stronglyinvariantmean[1,Definition3.1.24]forthemapx7→(r(x),d(x)),whererand
daretherange,respectivelythedomainmap,ofalocallycompactgroupoidGwith
non-compactisotropygroups.
1. INTRODUCTION
Thenotionofamenabilityforgroupoidswasintroducedin[13]andwasextensively
studiedin[1]. In[1]amoregeneralnotionofamenabilityforequivariantmapswas
introduced. WeshallestablishthatanequivariantmapπfromaBorelgroupoidG to
aBorelG-spaceSisamenableifandonlyifthesemidirectproductS∗G isamenable.
Weshallalsoextendfromgroupstogroupoidsthefollowingwellknownresult: ifG
isanamenablenon-compactlocallycompactHausdorfftopologicalgroupendowed
withaHaarmeasureν, and m isaninvariantmeanonL∞(G), then m(f)=0forall
f ∈C (G),thespaceofcomplex-valuedcontinuousfunctionsonGvanishingatinfinity
0
[12,Section21.1]. InthispaperweshallconsidertheprincipalgroupoidRassociated
withG,andadecomposition
λu= βs dαu(s,t)
Z t
oftheHaarsystemλonGovertheprincipalgroupoidR.Weshallassumethatthereis
anapproximatestronglyinvariantmean(g ) forthemapθ:G→R,θ(x)=(r(x),d(x))
n n
withrespectto(λ,β,µ◦α),whereµisaquasi-invariantmeasurefortheHaarsystemλ.
Thisconditionisequivalenttotheamenabilityofθandisweakerthantheamenability
ofr:G→G(0).Weshallprovethatthereisasubsequenceof(g ) ,stilldenoted(g ) ,
n n n n
suchthat
lim ϕ(x)g (x)dβu(x)=0
n→∞Z n v
foranycompactlysupportedBorelboundedfunctionϕonGandµ◦α-a.e.(u,v)∈R.
2000MathematicsSubjectClassification. 43A07,22A22.
Keywordsandphrases. Amenablemap,amenablerepresentation,locallycompactgroupoid,invariant
mean.
ThisresearchwaspartiallysupportedbytheMECgrantAt127/2004.
1
2 MA˘DA˘LINAROXANABUNECI
In order to establish notation and to give the reader some essential information
about groupoids and amenable maps, we include here a list of definitions that can
befoundin[9,10,13,8,1].
1.1. Groupoidsandsemi-directproducts. Weshallusethedefinitionofatopological
groupoidgivenbyRenaultin[13,DefinitionI.2.1]. ForagroupoidG,G(0) willdenote
itsunitspaceandG(2) thesetofthecomposablepairs. Usually,elementsofG willbe
denotedbylettersasx, y,orz,andtheelementsofG(0) byletterssuchasu,v,orw.
Theinversemapiswrittenx7→x−1[:G→G] andtheproductmapiswritten(x,y)7→xy
[:G(2)→G].TherangeandthesourcemapsfromGtoG(0)willbedenotedrespectively
byr andd. ThefibersoftherangeandthesourcemapsaredenotedGu=r−1({u})and
G =d−1({v}), respectively. More generally, given the subsets A, B ⊂G(0), we define
v
GA=r−1(A),G =d−1(B),andGA=r−1(A)∩d−1(B).ThereductionofGisG|A=GA.
B B A
ABorelgroupoidisagroupoidGendowedwithaBorelstructure,suchthatG(2)isa
BorelsetintheproductstructureG×G,andtheproductmapandtheinversemapare
Borelfunctions.WeareexclusivelyconcernedwithanalyticBorelstructure.
AtopologicalgroupoidconsistsofagroupoidGandatopologycompatiblewiththe
groupoidstructure. Thismeansthattheproductmapandtheinversemaparecon-
tinuousfunctions,whereG(2)hastheinducedtopologyfromG×G.Weareconcerned
withtopologicalgroupoidswhicharesecondcountable, locallycompact, andHaus-
dorff.Itwasshownin[10]thatmeasuredgroupoids(inthesenseof[6,Definition2.3])
maybeassumed,withnolossofgenerality,tohavelocallycompacttopologies.
IfX isalocallycompactspace,C (X)denotesthespaceofcomplex-valuedcontin-
c
uousfunctionswithcompactsupport. TheBorelsetsofatopologicalspacearetaken
tobetheelementsoftheσ-algebrageneratedbytheopensets.
AHaarsystemonalocallycompactgroupoidG isafamilyofpositiveRadonmea-
suresonG,{λu:u∈G(0)},havingthefollowingproperties[13,DefinitionI.2.2]:
(1) Forallu∈G(0), supp(λu)=Gu.
(2) Forall f ∈C (G)
c
u7→ f(x)dλu(x) [:G(0)→C]
Z
iscontinuous.
(3) Forall f ∈C (G)andallx∈G,
c
f(y)dλr(x)(y)= f(xy)dλd(x)(y).
Z Z
IntheBorelsetting,asystemofmeasures{λu:u∈G(0)}willbecalledBorelHaarsys-
tem(inshortBHS)ontheBorelgroupoidGifeachλuhasitssupportinGu,isinvariant
inthesenseof(3),isproperinthesensethatthereisapositiveBorelfunction f onG
0
suchthatλu(f )=1forallu,andforall f ≥0BorelonG,
0
u7→ f(x)dλu(x) [:G(0)→R]
Z
isareal-extendedBorelmap.
Theimageofλubytheinversemapx7→x−1isdenotedλ .
u
AMENABLEEQUIVARIANTMAPSDEFINEDONAGROUPOID 3
IfµisameasureonG(0),thenthemeasureν= λudµ(u),definedby
R
f(y)dν(y)= f(y)dλu(y) dµ(u), f ≥0 Borel,
Z ZµZ ¶
iscalledthemeasureonGinducedbyµ.Theimageofνbytheinversemapx7→x−1is
denotedν−1. Themeasureµissaidquasi-invariantifitsinducedmeasureνisequiv-
alenttoitsinverseν−1. Ifµisaquasi-invariantmeasureonG(0)andλisthemeasure
inducedonG,thentheRadon-Nikodymderivative∆=dν/dν−1iscalledthemodular
functionofµ.
A(left)BorelG-spaceisaBorelspaceSendowedwithaBorelsurjectionr:S→G(0)
andaBorelmap(x,s)7→xsfromthespace
G∗S={(x,s):d(x)=r(s)}
toS,satisfyingthefollowingconditions:
(1) r(xs)=r(x)forall(x,s)∈G∗S,andr(s)s=sforalls∈S.
(2) If(x ,x )∈G(2)and(x ,s)∈G∗S,then(x x )s=x (x s).
1 2 2 1 2 1 2
Thespace
S∗G={(s,x):r(s)=r(x)}
hasagroupoidstructure,calledsemi-directproduct,withthefollowingoperations
(s,x)(x−1s,y)=(s,xy), (s,x)−1=(x−1s,x−1).
LetusfixaBHSλ={λu:u∈G(0)}onG. LetSbeaBorelG-space. Then{ε ×λr(s):
s
s∈S}isaBHSforS∗G,whereε istheunitpointmassats.ApositivemeasureµonS
s
iscalledquasi-invariantwithrespecttoνifthemeasureµ◦λonS∗Gdefinedby
fdµ◦λ= f(s,x)dλr(s)(x)dµ(s)
Z Z
forallBorelnonnegativefunctiononS∗G,isequivalenttoitsimageundertheinverse
map(s,x)→(x−1s,x−1).
LetusdenotebyB (G,ν)thespaceofBorelfunctions f onGwith
b
u7→ f dλu
Z
bounded.B (G,ν)actsbyconvolutiononL∞(S)accordingtotheformula
b
f ∗ϕ(s)= f(x)ϕ(x−1s)dλr(s)(x), f ∈B (G,ν), ϕ∈L∞(S).
b
Z
1.2. AmenableBorelequivariantmaps. Concerningamenability,weshallusedefini-
tionsandnotationfrom[1],whichwerecallbelow.
LetT,SbetwoBorelspacesandπ:T →SaBorelmap.ABorelπ-system(oraBorel
systemofmeasuresforπ)isafamilyβ={βs:s∈S}ofmeasuresonπ−1({s})suchthat
(1) ForeverynonnegativeBorelfunction f onT,
sβ7→(f) f(t)dβs(t)
Z
definesaBorelmap.
(2) ThereisapositiveBorelfunctiong onT suchthatβ(g)=1.
4 MA˘DA˘LINAROXANABUNECI
Condition(2)isequivalentto:
(∗): Y is the union of an increasing sequence (A ) of Borel subsets such that
n n
s7→βs(A )isboundedforalln,andβs6=0foralls∈S[5,Lemme3,p.37].
n
Letβbesuchaπ-systemandc beaclassofmeasuresonS. Weshallfixapositive
measureµbelongingtoc,andthespaceL∞(S,c)willbeidentifiedtoL∞(S,µ),which
willbedenotedbyL∞(S).Weintroducethemeasureµ◦βonT definedby
f dµ◦β= β(f)dµ
Z Z
foreverynonnegativeBorelfunction f :T →R.
WedefinetheBanachspaceL∞(S,L1(T,β))ofµ◦β-measurablefunctionsf :T →C
suchthats7→ |f|dβsisµ-essentiallybounded,normedby
R
kfkπ,1=kβ(|f|)k∞,
andthesetL∞(S,L1(T,β))+ofnonnegativeelementswithnorm≤1inL∞(S,L1(T,β)).
1
WedenoteL∞(T)=L∞(T,µ◦β).InthiscontextameanisapositiveunitalL∞(S)-linear
map[1,Definition1.3.4]
m:L∞(T)→L∞(S).
Now,letusfixtwo(left)BorelG-spacesT andS. ABorelmapπfromT toSissaid
tobeequivariantifr(π(t))=r(t)forallt ∈T,andπ(xt)=xπ(t)forall(x,t)∈G∗T. If
π:T →SisanequivariantBorelmap,thenaBorelπ-systemβ={βs:s∈S}ofmeasures
iscalledinvariant(orG-invariant)ifx−1βs=βx−1sforevery(s,x)∈S∗G,wherex−1βs
isdefinedby
f(t)d(x−1βs)(t)= f(x−1t)dβs(t).
Z Z
TheBorelπ-systemβ={βs:s∈S}ofmeasuresisG-quasi-invariantifthereisaBorel
cocycle(i.e.agroupoidhomomorphism)q:T∗G→R+suchthat
∗
f(x−1t)q(t,x)dβs(t)= f(t)dβx−1s(t)
Z Z
forall(s,x)∈S∗G,and f ≥0BorelonT. Itisnothardtoseethatifaπ-systemβ=
{βs:s∈S}isG-quasi-invariantandifµisaquasi-invariantmeasureonS,thenµ◦βis
aquasi-invariantmeasureonT.Conversely,anyquasi-invariantmeasureonT canbe
obtainedinthatway.
Aninvariantmeanforπisameanm:L∞(T)→L∞(S)whichsatisfiestheinvariance
property
m(f ∗ϕ)=f ∗m(ϕ)
forallf ∈B (G,ν)andϕ∈L∞(T)[1,Definition3.1.4].TheG-equivariantmapπ:T →S
b
iscalledamenablewithrespectto(λ,µ,β)ifthereisaninvariantmean m:L∞(T)→
L∞(S)[1,Definition3.2.1].
Anapproximatestronglyinvariantmeanforπisanet(g ) satisfyingthefollowing
i i
[1,Definition3.1.24]:
(1) β(g )=1foralli.
i
AMENABLEEQUIVARIANTMAPSDEFINEDONAGROUPOID 5
(2) Forall f ∈L1(S∗G),
lim f(s,x)|g (x−1t)−g (t)|dλr(s)(x)dβs(t)dµ(s)=0.
i i
i Z
Theexistenceofinvariantmeanm:L∞(T)→L∞(S)fortheG-equivariantmapπ:
T →Sisequivalentwiththeexistenceofanapproximatestronglyinvariantmean(g )
i i
forπ(Propositions3.1.7,3.1.21,3.1.22of[1]).
AlsotheamenabilityoftheG-equivariantmapπ:T →S (withrespectto(λ,µ,β))
isequivalenttotheexistenceofaninvariantmeasurableπ-systemofmeans(Propo-
sitions 3.1.27, 3.2.5 of [1]). An invariant measurable π-system of means is a family
m={ms:s∈S}ofstates(ormeans)msonL∞(T,βs)suchthat,foreveryφ∈L∞(T),
(1) s7→ms(φ)isµ-measurable.
(2) xmx−1s(φ)=ms(φ)forµ◦λ-a.e.(s,x)∈S∗G,where
xmx−1s(φ):=mx−1s(t7→φ(xt)).
In[1]theamenabilityofameasuredgroupoid(G,λ,µ)(agroupoidGendowedwith
aHaarsystemλandaquasi-invariantmeasureµ)wasdefinedastheamenabilityof
therangemapwithrespectto(λ,µ)[1,Definition3.2.8].
1.3. Groupoidrepresentations. LetH ={H(s)}s∈S beafamilyofHilbertspacesin-
dexedbyasetS.Letusformthedisjointunion
S∗H ={(s,ξ):ξ∈H(s)},
andletp:S∗H →S bethenaturalprojection,p(s,ξ)=s. Apair(S∗H,p)iscalled
aHilbertbundleoverS. Foreachs∈S,thespaceH(s),whichcanbeidentifiedwith
p−1({s})={s}×H(s), is called the fibre over s. A section of the bundle is a function
f :S→S∗H suchthatp(f(s))=sforalls∈S.Givenasection f,wemaywrite f(s)=
(s,fˆ(s))forauniquelydeterminedelement
fˆ∈ H(s)= φ:S→ H(s) : φ(s)∈H(s)foralls ;
½ ¾
sY∈S s[∈S
andgivenanelement fˆ∈ s∈SH(s)wemaydefineasection f(s)=(s,fˆ(s)). Because
ofthislinkbetweensectionQsofS∗H andelementsof s∈SH(s),weshalloftenabuse
notationandwrite f(s)insteadof fˆ(s). AnanalyticBQorelHilbertbundleisaHilbert
bundle(S∗H,p),whereS∗H isendowedwithananalyticBorelstructurewhichsat-
isfiestheaxioms:
(1) AsubsetEisBorelifandonlyifp−1(E)isBorel.
(2) Thereisasequence{f } ofsections,calledafundamentalsequence,suchthat
n n
(a) eachfunction f˜n:S∗H →C,definedby f˜n(s,ξ)=(fn(s),ξ)H(s), isBorel.
(b) foreachpair(fn,fm)offundamentalsections, s 7→(fn(s),fm(s))H(s) de-
finesaBorelfunction.
(c) thefunctions{f˜} andpseparatethepointsofS∗H.
n n
LetG beagroupoidand{λu :u∈G(0)}beaHaarsystemonG. LetG(0)∗H bea
Hilbertbundle.WewriteIso(G(0)∗H)for
{(u,L,v) : L:H(v)→H(u)Hilbertspaceisomorphism}
6 MA˘DA˘LINAROXANABUNECI
endowedwiththeweakestBorelstructuresothatthemaps
(u,L,v)7→(Lf (v),f (u))
n m
areBorelforeverynandm,where(f ) isafundamentalsequenceforG(0)∗H.
n n
Iso(G(0)∗H)isagroupoidintheoperations:
(u,L ,v)(v,L ,w)=(u,L L ,w), (u,L,v)−1=(v,L−1,u).
1 2 1 2
AunitaryrepresentationofGconsistsofaquasi-invariantmeasureµ,aHilbertbun-
dleG(0)∗H,aconullsubsetUofG(0),andaBorelmap
L:G| →Iso G(0)∗H| ,
U U
¡ ¢
whereG(0)∗H| istherestrictionofG(0)∗H toU,suchthat
U
(1) L(x)=(d(x),Lˆ(x),r(x)) andLˆ(x):H(d(x))→H(r(x))isaHilbertspaceiso-
morphismforµ◦λ-a.e.x∈G|
U.
(2) Lˆ(u)=I ,theidentityoperatoronH(u),forµ-a.e.u∈U.
u
(3) Lˆ(x)Lˆ(y)=Lˆ(xy)for (λu×λ )dµ(u)-a.e.(x,y)∈G(2).
u
(4) Lˆ(x−1)=Lˆ(x)−1forµR◦λ-a.e.x.
If(G(0)∗H,L,µ)isarepresentationofthegroupoidG,weabusenotationandwrite
L(x)insteadofLˆ(x).ForanyrepresentationL,thereisaBorelhomomorphismL :G→
0
Iso(G(0)∗H)thatpreservestheunitspaceG(0),inthesensethat
L (x)=(d(x),Lˆ (x),r(x)),
0 0
whereLˆ (x):H(d(x))→H(r(x))isaHilbertspaceisomorphism,suchthatL agrees
0 0
µ◦λ-a.e.withL.
2. THEEQUIVALENCEBETWEENTHEAMENABILITYOFANEQUIVARIANTMAPDEFINED
ONAGROUPOIDANDTHEAMENABILITYOFASEMIDIRECTPRODUCT
Wefirstsetupsomenotation.DenotebyB(H)thespaceoftheboundedoperators
ontheHilbertspaceH. LetS∗H beananalyticBorelHilbertbundleoverS,andlet
(f ) be a fundamental sequence for this bundle. We write S∗B(H) for the set of
n n
A:S→ s∈SB(H(s))withtheproperties:
(1)SA(s)∈B(H(s))foralls∈S.
(2) s7→〈A(s)f (s),f (s)〉isBorelforallm,n∈N.
n m
(3) sup kA(s)k<∞.
s
LetµbeameasureonS. WewriteL∞(S,H,µ)forthesetof A:S→ s∈SB(H(s))
forwhich S
(1) A(s)∈B(H(s))forµ-a.e.s∈S.
(2) s7→〈A(s)f (s),f (s)〉isµ-measurableforallm,n∈N.
n m
(3) ks7→kA(s)kk <∞.
∞
Inordertoprovetheequivalencebetweentheamenabilityofanequivariantmap
definedonagroupoidandtheamenabilityofasemidirectproduct,weshallusethe
notionofamenablerepresentationofagroupoidintroducedin[4,Definition2].
AMENABLEEQUIVARIANTMAPSDEFINEDONAGROUPOID 7
Definition1. LetG beaBorelgroupoidendowedwithaBHSλ={λu :u∈G(0)}. Let
µbeaquasi-invariantmeasurefortheHaarsystemλ.Aunitaryrepresentation(G(0)∗
H,L,µ)ofthegroupoidG issaidtobeamenableifthereexistsaninvariantfamilyof
states{Mu:u∈G(0)}onL∞(G(0),H,µ),i.e.
(1) Mu ∈B(H(u))∗,Mu ≥0,Mu(I)=1(thismeansthatMu isastateonH(u))
forµ-a.e.u∈G(0).
(2) u7→Mu(A(u))isµ-measurableforallA∈L∞(G(0),H,µ).
(3) Mr(x)(L(x)AL(x−1))=Md(x)(A)forµ◦λ-a.e.x∈GandallA∈B(H(d(x))).
Thisdefinitionextendsfromgroupstogroupoidsthenotionofamenabilityforan
arbitraryunitaryrepresentation,introducedbyE.B.Bekkain[2].
In[4, Theorem6]wecharacterizedtheamenablegroupoidsbyamenabilityofall
theirunitaryrepresentationsasfollows.
Theorem2. LetG beaBorelgroupoidendowedwithaBHSλ={λu :u∈G(0)}. Letµ
beaquasi-invariantmeasurefortheHaarsystemλ. Thenthefollowingconditionsare
equivalent:
(1) (G,λ,µ)isamenable.
(2) Everyunitaryrepresentation(G(0)∗H,L,µ)ofGisamenable.
(3) TheleftregularrepresentationReg isamenable.
µ
Weshallusethistheoremforasemi-directproductS∗G. Ifλ={λu :u∈G(0)}isa
BHSonG,thenweendowS∗GwiththeBHS{ε ×λr(s):s∈S}.Aunitaryrepresentation
s
(S∗H,L,µ)ofS∗Gisamenableifthereexistsaninvariantfamilyofstates{Ms:s∈S}
onL∞(S,H,µ).Thismeansthat
(1) Ms∈B(H(s))∗,Ms≥0,Ms(I)=1forµ-a.e.s∈S.
(2) s7→Ms(A(s))isµ-measurableforallA∈L∞(S,H,µ).
(3) Ms(L(s,x)AL(s,x)−1) = Mx−1s(A) for µ◦λ-a.e. (s,x) ∈ S ∗G and all A ∈
B(H(x−1s)).
LetusdefineananalogofReg foranequivariantmap.ConsideraG-equivariant
µ
mapπ:G→S.HereSisa(left)BorelG-spaceandalsoGisaBorelG-spacebyleftmul-
tiplication.Letβ={βs:s∈S}beaG-invariantπ-systemofmeasuresandµbeaquasi-
invariantmeasurefortheHaarsystemonS∗G.Letg beapositiveBorelfunctionsuch
0
thatβ(g )=1. Let H(s)=L2(βs),toobtainaHilbertbundleS∗H onS. TheBorel
0
structureonS∗H isgivenbyasequenceofsections(f ) definedasfollows:let(h )
n n n n
be a sequence of bounded, non-negative Borel functions onG that separate points,
andletusdefine f :S→S∗H,by f (s)=(s,fˆ(s)),where fˆ(s)(x)=h (x)g−1/2(x),
n n n n n 0
x∈π−1({s}). Letµbeaquasi-invariantmeasurefortheHaarsystemonS∗G. Letus
definearepresentation(µ,S∗H,R )where
π,µ
Rˆ (s,x):L2(βx−1s)→L2(βs)
π,µ
isgivenbytheformula
Rˆ (s,x)f (y)=f(x−1y).
π,µ
Letuscallthisrepresentationo¡fS∗Gtheu¢nitaryrepresentationassociatedtoπinduced
bytheG-action,anddenoteitR .
π,µ
8 MA˘DA˘LINAROXANABUNECI
Theorem3. LetG beaBorelgroupoidendowedwithaBHSλ={λu :u∈G(0)}. LetS
bea(left)BorelG-spaceandletπ:G→S beaG-equivariantmap. Letβ={βs : s∈S}
beaG-invariantπ-systemofmeasureandµbeaquasi-invariantmeasurefortheHaar
systemonS∗G.Thenthefollowingconditionsareequivalent:
(1) π:G→Sisanamenablemapwithrespectto(λ,µ,β).
(2) Everyunitaryrepresentation(S∗H,L,µ)ofS∗Gisamenable.
(3) TherepresentationR isamenable.
π,µ
Proof. (1⇒2) Let {ms : s ∈S} be an invariant measurable π-system of means with
respect to (λ,µ,β) and let (S∗H,L,µ) be a unitary representation of S∗G. Since
(S∗H,L,µ)isarepresentationofS∗G,thereisaBorelhomomorphismL :S∗G→
0
Iso(S∗H)thatagreesµ◦λ -a.e.withL,whereλ ={ε ×λr(s):s∈S}. WereplaceLby
S S s
L .LetuschooseC∈S∗B(H)withC(s)≥0andkC(s)k =1foralls∈S.Foreachs∈S
0 1
andeachA∈B(H(s)),defineφs :Gs→Cby
A
φs(x)=Tr AL(π(x),x)C(x−1π(x))L(π(x),x)−1
A
=Tr¡AL(s,x)C(x−1s)L(s,x)−1 , x∈G¢s.
¡ ¢
Thenwehave
|φs(x)|≤kAL(s,x)C(x−1s)L(s,x)−1k ≤kL(s,x)−1AL(s,x)kkC(x−1s)k ≤kAk.
A 1 1
Thusφs ∈L∞(G,βs).Foreachs∈S,defineMs:B(H(s))→Cby
A
Ms(A)=ms(φs), A∈B(H(s)).
A
Letverifytheinvarianceofthefamilyofstates{Ms:s∈S}.Bydefinitionwehave
Mx−1s(A)=mx−1s y7→Tr(AL(π(y),y)C(y−1π(y))L(π(y),y)−1)
=ms y→T¡r AL(x−1π(y),x−1y)C(y−1π(y))L(x−1π(y),¢x−1y)−1
=ms¡y7→Tr¡AL(x−1s,x−1)L(s,y)C(y−1s)L(s,y)−1L(x−1s,x−1)−¢1¢
=ms¡y7→Tr¡L(x−1s,x−1)−1AL(x−1s,x−1)L(s,yt)C(y−1s)L(s,y)−¢1¢
=ms¡φs ¡ ¢¢
L(x−1s,x−1)−1AL(x−1s,x−1)
=Ms¡L(x−1s,x−1)−1AL(x−1s,¢x−1)
=Ms¡L(s,x)AL(s,x)−1 ¢
forµ◦λ-a.e.x∈G¡andallA∈B(H¢(d(x))).
Theimplication(2⇒3)istrivial.
(3⇒1) For each s ∈S, let Ts be the operator of multiplication by φ∈L∞(G,βs),
φ
actingonL2(G,βs).ThenwehaveTs=1,Ts≥0ifφ≥0,and
1 φ
R (s,x)Tx−1sR (s,x)−1=Ts forallφ∈L∞(G,βs).
π φ π xφ
Foreachs∈Sdefinems(φ)=Ms(Ts),φ∈L∞(G,βs).Wehave
φ
ms( φ)=Ms(Ts )=Ms R (s,x)Tx−1sR (s,x)−1 =Mx−1s(Tx−1s)=mx−1s(φ)
x xφ π φ π φ
¡ ¢
for µ◦λ-a.e. (s,x)∈S∗G and all φ ∈ L∞(G,βx−1s). Thus {ms :s ∈S} is an invariant
(cid:3)
measurableπ-systemofmeanswithrespectto(β,λ,µ).
AMENABLEEQUIVARIANTMAPSDEFINEDONAGROUPOID 9
Corollary4. LetG beaBorelgroupoidendowedwithaBHSλ={λu :u∈G(0)}. LetS
bea(left)BorelG-spaceandletπ:G→S beaG-equivariantmap. Letβ={βs : s∈S}
beaG-invariantπ-systemofmeasureandµbeaquasi-invariantmeasurefortheHaar
systemλ ={ε ×λr(s):s∈S}onS∗G.Thenπisamenablewithrespectto(λ,µ,β)ifand
S s
onlyif(S∗G,λ ,µ)isamenable.
S
Proof. AccordingtoTheorem2,(S∗G,λ ,µ)isamenableifandonlyifanyunitaryrep-
S
resentation (S∗H,L,µ) of S∗G is amenable. But applying Theorem 3, any unitary
representation(S∗H,L,µ)ofS∗Gisamenableifandonlyifπ:G→Sisanamenable
(cid:3)
mapwithrespectto(λ,µ,β).
3. THEG-EQUIVARIANTMAPθ
LetGbealocallycompactsecondcountablegroupoidandRbeitsassociatedprin-
cipalgroupoid. R istheimageofG underthehomomorphismθ:G →R definedby
θ(x)=(r(x),d(x)).WeendowRwiththequotienttopologyinducedfromGbythemap
θ. ThistopologyconsistsofthesetswhoseinverseimagesbyθinG areopen. Inpar-
ticular,G andR areBorelgroupoids. Moreover,R istheimageofG underθ,soitisa
σ-compactgroupoid.Gcanbeviewedasa(left)G-spaceactingonitselfbytranslation.
AlsoRcanbeviewedasa(left)G-spacetakingr:R→G(0),r(u,v)=u,andtheaction
x(d(x),v)=(r(x),v),
definedfromG∗R={(x,(u,v)):d(x)=u}intoR.ThemapθisG-equivariantbecause
θ(xy)=(r(x),d(y))=x(r(y),d(y))=xθ(y)forall(x,y)∈G(2)=G∗G.
Letλ={λu:u∈G(0)}beaBHSonG. InotherwordsλisaG-invariantr-systemof
measures(r:G→G(0)).
Let us recall some results on the structure of the Haar systems, as developed by
Renault in [14, Section 1], andalso by RamsayandWalter in [11, Section 2]. In [14,
Section1]aBHSfortheisotropygroupbundleG0={x∈G:r(x)=d(x)}isconstructed.
OnewaytodothisistochooseacontinuousfunctionF withconditionallysupport,
0
whichisnonnegativeandequalto1ateachu∈G(0).Foreachu∈G(0),choosealeft
Haarmeasureβu onGu sothattheintegralofF withrespecttoβu is1,thendefine
u u 0 u
βu =xβv if x ∈Gu, where xβv(f)= f(xy)dβv(y)asusual. If z isanotherelement
v v v v v
inGu,thenx−1z∈Gv,andsinceβv iRsaleftHaarmeasureonGv,itfollowsthatβu is
v v v v v
independentofthechoiceofx.A1-cocycleδonGsuchthat,foreveryu∈G(0),δ|Guu is
themodularfunctionforβu,isalsodefinedin[14]. δandδ−1=1/δareboundedon
u
compactsetsinG. Thesystemofmeasuresβ={βv :(u,v)∈R}satisfiesthefollowing
u
conditions:
(1) supp(βu)=Guforall(u,v)∈R.
v v
(2) (u,v)7→ f dβv isBorelforeverynonnegativeBorelfunction f onG.
u
(3) f(y)dβRr(x)(y)= f(xy)dβd(x)(y)forallx∈Gandvwith(v,r(x))∈R.
v v
(4) Rsup βu(K)<∞fRorallcompactK⊂G.
u,v v
(5) f(y)dβu(y)= f(y−1)δ(y−1)dβv(y).
v u
R R
Conditions(1−4)aboveimplythatβisaninvariantθ-systemofmeasures.
10 MA˘DA˘LINAROXANABUNECI
Withthisapparatusinplace, RenaultdescribesadecompositionoftheBHS{λu :
u∈G(0)}forGoverR,provingthatthereisauniqueBHSαforRwiththepropertythat
λu= βs dαu(s,t) forallu∈G(0).
Z t
In[11,Section2]RamsayandWalterprovethat
supαu(θ(K)t)<∞ forallcompactK⊂G.
u
Weshallcallthepairofsystemofmeasures
{βuv}(u,v)∈R,{αu}u∈G(0)
¡ ¢
describedaboveadecompositionoftheHaarsystem{λu :u∈G(0)}overtheprincipal
groupoidassociatedtoG.Alsoweshallcallδthe1-cocycleassociatedtothedecomposi-
tion.
Foreachu∈G(0)themeasureαu isconcentratedon{u}×[u]. Thereforethereisa
measureµu concentratedon[u]suchthatαu=ε ×µu. Since{αu:u∈G(0)}isaHaar
u
system,wehaveµu=µv forall(u,v)∈R,andthefunction
u7→ f(s)dµu(s)
Z
isBorelforallf ≥0BorelonG(0).Foreachuthemeasureµuisquasi-invariant,cf.[11,
Section2].Thereforeµuisequivalenttod∗(vu)([9,Lemma4.5]).
Letussummarize:λ={λu:u∈G(0)}isaBHSonG,θ:G→RisaG-equivariantmap,
β={βv :(u,v)∈R}isaninvariantθ-systemofmeasures,andα={αu:u∈G(0)}isaan
u
invariantr-systemofmeasures(r:R→G(0),r(u,v)=u).
Letµbeaquasi-invariantmeasure(withrespecttoλ)onG(0).Letusnotethatµ◦α
isaquasi-invariantmeasureonR.Ifθ:G→Risamapwhichisamenablewithrespect
to (λ,β,µ◦α), then there is an invariant mean m :L∞(G)→L∞(R). The invariance
propertyofmcanbewrittenas
m(f ∗ϕ)(u,v)= f(x)m(ϕ)(d(x),v)dλu(x), (u,v)∈R,
Z
forall f ∈B (G,λ)andϕ∈L∞(G). Theexistenceoftheinvariantmeanm:L∞(G)→
b
L∞(R)isequivalenttotheexistenceofanapproximatestronglyinvariantmean. This
isanet(g ) inL∞(R,L1(G,β))+suchthat
i i 1
(1) β(g )=1.
i
(2) Forall f ∈L1(R∗G),
lim f((r(y),d(y)),x)|g (x−1y)−g (y)|dλu(x)dλu(y)dµ(u)=0.
i i
i Z
Wemayequivalentlyworkwithasequence(g ) insteadofanet(g ) .
n n i i
4. TOPOLOGICALLYNULLFUNCTIONSONGROUPOIDS
In the following considerations we use the notation from the preceding section.
In [3], we considered a locally compact second countable groupoidG having non-
compactisotropygroups,endowedwithaHaarsystemλ={λu:u∈G(0)}andaquasi-
invariantmeasureµ.Weassumedthatthereisanapproximatestronglyinvariantmean