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