amenability and coamenability of algebraic quantum groups

26 Pages·2014·1.96 MB·English

IJMMS31:10(2002)577–601
PII.S016117120210603X
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AMENABILITY AND COAMENABILITY OF ALGEBRAIC
QUANTUM GROUPS
ERIKBÉDOS,GERARDJ.MURPHY,andLARSTUSET
Received5June2001andinrevisedform25January2002
Wedefineconceptsofamenabilityandcoamenabilityforalgebraicquantumgroupsinthe
sense of Van Daele (1998). We show that coamenability of an algebraic quantum group
alwaysimpliesamenabilityofitsdual.Variousnecessaryand/orsufficientconditionsfor
amenability or coamenability are obtained. Coamenability is shown to have interesting
consequencesforthemodulartheoryinthecasethatthealgebraicquantumgroupisof
compacttype.
2000MathematicsSubjectClassification: 46L05,46L65,16W30,22D25,58B32.
1. Introduction. Theconceptofamenabilitywasfirstintroducedintotherealmof
quantumgroupsbyVoiculescuin[25]forKacalgebras,andstudiedfurtherbyEnock
and Schwartz in [7] and by Ruan in [20] (which also deals with Hopf von Neumann
algebras).Ontheotherhand,amenabilityandcoamenabilityof(regular)multiplicative
unitaries have been defined by Baaj and Skandalis in [1], and studied in [2, 3, 6].
AmenabilityandcoamenabilityforHopfC∗-algebrashavebeenconsideredbyNgin
[17,18].
Thepresentpaperisacontinuationoftheearlierpaper[5]oftheauthors,inwhich
westudiedtheconceptofcoamenabilityforcompactquantumgroupsasdefinedby
Woronowicz[15,28].WeshowedtherethatthequantumgroupSU (2)iscoamenable
q
andthatacoamenablecompactquantumgrouphasafaithfulHaarintegral.Combin-
ingtheseresultsgivesanewproofofNagy’stheoremthattheHaarintegralofSU (2)
q
isfaithful[16].
Inthispaper,weextendtheclassofquantumgroupsforwhichwestudytheconcept
ofcoamenability,andwealsoinitiateastudyofthedual notionofamenability.The
quantum groups we consider are the algebraic quantum groups introduced by Van
Daelein[24].Thisclassissufficientlylargetoincludecompactquantumgroupsand
discrete quantum groups (for a more precise statement of what is meant here, see
Proposition3.2).Analgebraicquantumgroupadmitsadualthatisalsoanalgebraic
quantumgroup;moreover,thereisaPontryagin-typedualitytheoremtotheeffectthat
thedoubledualiscanonicallyisomorphictotheoriginalalgebraicquantumgroup(see
Theorem3.1).
If Γ is a discrete group, there are associated to it in a natural way two algebraic
quantum groups, namely (A,∆) = (C(Γ),∆), and its dual (Aˆ,∆ˆ), where C(Γ) is the
groupalgebraofΓ and∆isthecomultiplicationonC(Γ)givenby∆(x)=x⊗x,forall
x∈Γ.Then(A,∆)iscoamenableifandonlyif(Aˆ,∆ˆ)isamenable;moreover,eachof
theseconditionsisinturnequivalenttoamenabilityofΓ (seeExamples3.4and4.6).
578 ERIK BÉDOS ET AL.
Wefirstrelatecoamenabilityofanalgebraicquantumgroup(A,∆)toapropertyof
the multiplicative unitary W naturally associated to it. This provides a link between
our theory and that of Baaj and Skandalis [1], although we make no use of their re-
sults.Thenwealsoobtainseveralotherequivalentformulationsofcoamenability(in
Theorem4.2)thatgeneralizewell-knownresultsinthegroupalgebracase.
Onebasicquestioninthetheoryiswhethercoamenabilityofanalgebraicquantum
group (A,∆) is always equivalent to amenability of its dual (Aˆ,∆ˆ). In fact, we show
thatcoamenabilityof(A,∆)alwaysimpliesamenabilityof(Aˆ,∆ˆ),butitisconceivable
thattheconverseisnotalwaystrue.When(A,∆)isofcompacttype(i.e.,thealgebra
A is unital) and its Haar state is tracial, the converse is known to hold, as may be
deducedfromadeepresultofRuan[20,Theorem4.5].
In another direction, we prove below that if M is the von Neumann algebra asso-
ciated to an algebraic quantum group (A,∆), then coamenability of (A,∆) implies
injectivity of M (Theorem4.8). It may also be deduced from [20, Theorem 4.5] that
theconverseistrueinthecompacttracialcase.Relatedtothis,wegiveadirectproof
in Theorem4.9, that if (A,∆) is of compact type and its Haar state is tracial, then
injectivityofM impliesamenabilityofthedual(Aˆ,∆ˆ).
In the final section of this paper, we investigate the modular properties of a coa-
menablealgebraicquantumgroupofcompacttype.TheunitalHaarfunctionalϕ of
suchanalgebraicquantumgroup(A,∆)isaKMS-statewhenextendedtotheanalytic
extensionA ofA.Weshow,inthecasethat(A,∆)iscoamenable,themodulargroup
r
canbegivenadescriptionintermsofthemultiplicativeunitaryof(A,∆).
Wewillcontinueourinvestigationsoftheconceptsstudiedinthispaperinasub-
sequentpaper[4].Extensionsoftheseresultstothegeneralcaseoflocallycompact
quantumgroupswillalsobeconsidered.
Wenowgiveabriefsummaryofhowthepaperisorganized.Section2establishes
some preliminaries on multiplier algebras, C∗-algebra and von Neumann algebras.
We give a careful treatment of slice maps in connection with multiplier C∗-algebras
(Theorem2.1). This is a material that is often assumed in the literature, but does
not appear to have been anywhere explicitly formulated and established in terms
of known results. Section3 sets out the background material on algebraic quantum
groupsthatwillbeneededinthesequel.Themostimportantresultsofthepaperare
tobefoundinSection4,wheremostoftheresultsmentionedintheearlierpartofthis
introductionareproved.Thefinalsection,Section5,discussessomeconsequencesof
coamenabilityforthemodularpropertiesofanalgebraicquantumgroupofcompact
type.
Weendthisintroductorysectionbynotingsomeconventionsofterminologythat
willbeusedthroughoutthepaper.
Everyalgebrawillbean(notnecessarilyunital)associativealgebraoverthecomplex
fieldC.TheidentitymaponasetVwillbedenotedbyι ,orsimplybyι,ifnoambiguity
V
isinvolved.
IfV andW arelinearspaces,V(cid:4)denotesthelinearspaceoflinearfunctionalsonV
andV⊗W denotesthelinearspacetensorproductofV andW.Theflipmap χ from
V⊗W toW⊗V isthelinearmapsendingv⊗w ontow⊗v,forallv∈V andw∈W.
IfV andW areHilbertspaces,V⊗W denotestheirHilbertspacetensorproduct;we
AMENABILITY AND COAMENABILITY OF ALGEBRAIC QUANTUM GROUPS 579
denotebyB(V)andB (V)theC∗-algebrasofboundedlinearoperatorsandcompact
0
operatorsonV,respectively.Ifv∈Vandw∈W,ω denotestheweaklycontinuous
v,w
bounded linear functional on B(V) that maps x onto (x(v),w). We set ω =ω .
v v,v
Wewilloftenalsousethenotationω todenotearestrictiontoaC∗-subalgebraof
v
B(V)(thedomainofω willbedeterminedbythecontext).
v
IfV andW arealgebras,V⊗W denotestheiralgebratensorproduct.IfV andW are
C∗-algebras,thenV⊗W denotestheirC∗-tensorproductwithrespecttotheminimal
(spatial)C∗-norm.
2. C∗-algebrapreliminaries. Inthissectionwereviewsomeresultsrelatedtomul-
tiplieralgebras, especiallymultiplieralgebras ofC∗-algebras, andalsowereviewel-
ementsofthetheoryof(multiplier)slicemaps.ThesetopicsarefundamentaltoC∗-
algebraicquantumgrouptheorybutthosepartsoftheirtheorythataremostrelevant
arescatteredthroughouttheliteratureandareoftenpresentedonlyinaverysketchy
form.Therefore,fortheconvenienceofthereaderandinordertoestablishnotation
andterminology,wepresentabrief,butsufficientlydetailed,backgroundaccount.
First,weintroducethemultiplieralgebraofanondegenerate∗-algebra.Thisgen-
eralizestheusualideaofthemultiplieralgebraofaC∗-algebra.Recallthatanonzero
algebraAisnondegenerateif,foreverya∈A,a=0ifab=0,forallb∈Aorba=0,
forallb∈A.Obviously,allunitalalgebrasarenondegenerate.IfAandB arenonde-
generatealgebras,soisA⊗B.
DenotebyEnd(A)theunitalalgebraoflinearmapsfromanondegenerate∗-algebra
Atoitself.LetM(A)denotethesetofelementsx∈End(A)suchthatthereexistsan
element y ∈End(A) satisfying x(a)∗b =a∗y(b), for all a,b ∈A. Then M(A) is a
unital subalgebra of End(A). The linear map y associated to a given x ∈ M(A) is
uniquely determined by nondegeneracy and we denote it by x∗. The unital algebra
M(A)becomesa∗-algebrawhenendowedwiththeinvolutionx(cid:1)x∗.
When A is a C∗-algebra, the closed graph theorem implies that M(A) consists of
bounded operators. If we endow M(A) with the operator norm, it becomes a C∗-
algebra.ItisthentheusualmultiplieralgebrainthesenseofC∗-algebratheory.
SupposenowthatAissimplyanondegenerate∗-algebraandthatAisaselfadjoint
idealina∗-algebraB.Forb∈B,defineL ∈M(A)byL (a)=ba,foralla∈A.Then
b b
the map L:B →M(A), b (cid:1)L , is a ∗-homomorphism. If A is an essential ideal in
b
B in the sense that an element b of B is necessarily equal to zero if ba=0, for all
a∈A, or ab =0, for all a∈A, then L is injective. In particular, A is an essential
selfadjoint ideal in itself (by nondegeneracy) and therefore we have an injective ∗-
homomorphismL:A→M(A).WeidentifytheimageofAunderLwithA.ThenAis
anessentialselfadjointidealofM(A).Obviously,M(A)=AifandonlyifAisunital.
If T is an arbitrary nonempty set, denote by F(T) and K(T) the nondegenerate
∗-algebras of all complex-valued functions on T and of all finitely supported such
functions,respectively,wheretheoperationsarethepointwise-definedones.Clearly,
K(T) is an essential ideal in F(T), and therefore we have a canonical injective ∗-
homomorphismfromF(T)toM(K(T));amoment’sreflectionshowsthatthishomo-
morphismissurjectiveandwethereforecan,anddohenceforth,usethistoidentify
M(K(T))withF(T).
580 ERIK BÉDOS ET AL.
IfA and B arenondegenerate ∗-algebras, thenitiseasilyverifiedthatA⊗B isan
essential selfadjoint ideal in M(A)⊗M(B). Hence, by the preceding remarks, there
exists a canonical injective ∗-homomorphism from M(A)⊗M(B) into M(A⊗B). We
use this to identify M(A)⊗M(B) as a unital ∗-subalgebra of M(A⊗B). In general,
thesealgebrasarenotequal.
If π :A→B is a homomorphism, it is said to be nondegenerate if the linear span
of the set π(A)B={π(a)b|a∈A, b∈B} and the linear span of the set Bπ(A) are
both equal to B. In this case, there exists a unique extension to a homomorphism
π:M(A)→M(B)(see[23]),whichisdeterminedbyπ(x)(π(a)b)=π(xa)b,forevery
x∈M(A), a∈A and b∈B. Note that π is a ∗-homomorphism whenever π is a ∗-
homomorphism. We will henceforth use the same symbol π to denote the original
mapanditsextensionπ.
Ifπ:A→B isa∗-homomorphismbetweenC∗-algebras,wewillusethetermnon-
degenerateonlyinitsusualsenseinC∗-theory.Thus,inthiscase,π isnondegenerate
iftheclosedlinearspanofthesetπ(A)B={π(a)b|a∈A, b∈B}isequaltoB.
IfωisalinearfunctionalonAandx∈M(A),wedefinethelinearfunctionalsxω
andωx onAbysetting(xω)(a)=ω(ax)and(ωx)(a)=ω(xa),foralla∈A.
Wesayωispositiveifω(a∗a)≥0,foralla∈A;ifωispositive,wesayitisfaithful
if,foralla∈A,ω(a∗a)=0⇒a=0.
SupposegivenC∗-algebrasAandB.Ifω∈A∗,thelinearmapdefinedbytheassign-
menta⊗b(cid:1)ω(a)bextendstoanorm-boundedlinearmapω⊗ιfromA⊗BtoB.We
callω⊗ιaslicemap.Obviously,ifτ∈B∗,wecandefinetheslicemapι⊗τ:A⊗B→A
inasimilarmanner.ThenextresultshowshowwecanextendthesemapstoM(A⊗B).
Thisresultisfrequentlyusedintheliterature,usuallywithoutexplicitexplanationof
howω⊗ιistobeunderstoodorhowitisconstructed.Similarremarksapplytothe
corollary.
Theorem 2.1. Let A and B be C∗-algebras and let ω ∈ A∗. Then the slice map
ω⊗ι:A⊗B→B admits a unique extension to a norm-bounded linear map ω⊗ι :
M(A⊗ B)→M(B)thatisstrictlycontinuousontheunitballofM(A⊗B).
Ifx∈M(A⊗B)andb∈B,then
(cid:1) (cid:2) (cid:1) (cid:2)
b(ω⊗ι)(x)=(ω⊗ι) (1⊗b)x , (ω⊗ι)(x)b=(ω⊗ι) x(1⊗b) . (2.1)
Proof. To prove this, we may assume that ω is positive, since the set of posi-
tiveelementsofA∗ linearlyspansA∗.Inthiscase,theslicemapω⊗ιiscompletely
positive [26, page 4] and is easily seen to be strict in the sense defined by Lance
in [14, page 49]. Hence, by [14, Corollary 5.7], ω⊗ι admits an extension to a norm-
bounded linear map ω⊗ι:M(A⊗B)→M(B) that is strictly continuous on the unit
ball of M(A⊗B). Uniqueness of ω⊗ι and the properties in the second paragraph
of the statement of the theorem now follow immediately from the strict continuity
condition.
Of course, an analogous result holds in the preceding theorem for an element
τ∈ B∗.
Recall that a norm-bounded linear functional on a C∗-algebra A has a unique ex-
tension to a norm-bounded strictly continuous functional on M(A). We will usually
AMENABILITY AND COAMENABILITY OF ALGEBRAIC QUANTUM GROUPS 581
denotetheoriginalfunctionalanditsextensionbythesamesymbol.Thisresult,which
ispointedoutintheappendixof[11],followseasilyfromaresultofTaylor[22]thatas-
sertsthat,foreachω∈A∗,thereexistanelementa∈Aandafunctionalθ∈A∗such
thatω(b)=θ(ab),forallb∈A.WethendefineωonM(A)bysettingω(x)=θ(ax),
for all x ∈ M(A). If ω ∈ A∗ and τ ∈ B∗, it follows that the norm-bounded linear
functionalω⊗τ:A⊗B→Cadmitsauniqueextensiontoastrictlycontinuousnorm-
boundedlinearfunctionalonM(A⊗B).Inagreementwithourstandingconvention,
wewilldenotetheextensionbythesamesymbolω⊗τ.Usingtheseobservations,we
havethefollowingcorollary.
Corollary 2.2. LetAandB beC∗-algebrasandletω∈A∗ andτ∈B∗.Letx∈
M(A⊗B).Then
(cid:1) (cid:2) (cid:1) (cid:2)
(ω⊗τ)(x)=ω (ι⊗τ)(x) =τ (ω⊗ι)(x) . (2.2)
WewillalsoneedtoconsiderslicemapsinthecontextofvonNeumannalgebras.
LetM,NbevonNeumannalgebrasonHilbertspacesHandK,respectively.Wedenote
thevonNeumannalgebratensorproductbyM⊗¯N(thisistheweakclosureoftheC∗-
tensorproductM⊗N inB(H⊗K)).WedenotebyM∗ thepredualofM consistingof
thenormalelementsofM∗.Recallthatforanyω∈M∗ andτ∈N∗,wecandefinea
uniquefunctionalω⊗¯τ∈(M⊗¯N)∗suchthat(cid:8)ω⊗¯τ(cid:8)=(cid:8)ω(cid:8)(cid:8)τ(cid:8)and(ω⊗¯τ)(x⊗y)=
ω(x)τ(y), for all x∈M and y ∈N. If ω∈M∗, we show now how we can define a
slice map ω⊗¯ι from M⊗¯N to N. For any x∈M⊗¯N, the assignment τ (cid:1)(ω⊗¯τ)(x)
defines a bounded functional on N∗. Since N∗∗ = N, there exists a unique element
z∈N such that (ω⊗¯τ)(x)=τ(z), for all τ ∈N∗. We define (ω⊗¯ι)(x) to be equal
to z. Thus, (ω⊗¯τ)(x)=τ((ω⊗¯ι)(x)), as one would expect of a slice map. Clearly,
(cid:8)(ω⊗¯ι)(x)(cid:8)≤(cid:8)ω(cid:8)(cid:8)x(cid:8). The map ω⊗¯ι which sends x ∈M⊗¯N to (ω⊗¯ι)(x)∈N is
obviouslylinearandnorm-bounded.Finally,itisevidentthatω⊗¯ιisanextensionof
theusualslicemapω⊗ι:M⊗N→N.Inasimilarfashion,foreachτ∈N∗,onecan
defineaslicemapι⊗¯τ:M⊗¯N→M.
WefinishthissectiononC∗-algebrapreliminariesbyrecallingbrieflyausefulfact
concerningcompletelypositivemapsthatwillbeneededinthesequel.Supposethat
π:A→BisacompletelypositiveunitallinearmapbetweenunitalC∗-algebrasAand
B. If a∈A and π(a∗)π(a)=π(a∗a), then π(xa)=π(x)π(a), for all x ∈A [26,
page5].Inparticular,ifuisaunitaryinAforwhichπ(u)=1,itfollowseasilythat
π(u∗xu)=π(x),forallx∈A.
3. Algebraicquantumgroups. WebeginthissectionbydefiningamultiplierHopf
∗-algebra.Referencesforthissectionare[13,23,24].
Let A be a nondegenerate ∗-algebra and let ∆ be a nondegenerate ∗-homomor-
phismfromAintoM(A⊗A).Supposethatthefollowingconditionshold:
(1) (∆⊗ι)∆=(ι⊗∆)∆,
(2) thelinearmappingsdefinedbytheassignmentsa⊗b(cid:1)∆(a)(b⊗1)anda⊗b(cid:1)
∆(a)(1⊗b)arebijectionsfromA⊗Aontoitself.
Thenthepair(A,∆)iscalledamultiplierHopf∗-algebra.
In condition (1), we are regarding both maps as maps into M(A⊗A⊗A), so that
theirequalitymakessense.Itfollowsfromcondition(2),bytakingadjoints,thatthe
582 ERIK BÉDOS ET AL.
mapsdefinedbytheassignmentsa⊗b(cid:1)(b⊗1)∆(a)anda⊗b(cid:1)(1⊗b)∆(a)arealso
bijectionsfromA⊗Aontoitself.
Let(A,∆)beamultiplierHopf∗-algebraandletωbealinearfunctionalonAand
aanelementinA.Thereisauniqueelement(ω⊗ι)∆(a)inM(A)forwhich
(cid:1) (cid:2) (cid:1) (cid:2)
(ω⊗ι) ∆(a) b=(ω⊗ι) ∆(a)(1⊗b) ,
(cid:1) (cid:2) (cid:1) (cid:2) (3.1)
b(ω⊗ι) ∆(a) =(ω⊗ι) (1⊗b)∆(a) ,
for all b ∈ A. The element (ι⊗ω)∆(a) in M(A) is determined similarly. Thus, ω
induceslinearmaps(ω⊗ι)∆and(ι⊗ω)∆fromAtoM(A).
There exists a unique nonzero ∗-homomorphism ε from A to C such that, for all
a∈A,
(ε⊗ι)∆(a)=(ι⊗ε)∆(a)=a. (3.2)
Themapεiscalledthecounit of(A,∆).Also,thereexistsauniqueantimultiplicative
linearisomorphismS onAthatsatisfiestheconditions
(cid:1) (cid:2)
m(S⊗ι) ∆(a)(1⊗b) =ε(a)b,
(cid:1) (cid:2) (3.3)
m(ι⊗S) (b⊗1)∆(a) =ε(a)b,
foralla,b∈A.Herem:A⊗A→Adenotesthelinearizationofthemultiplicationmap
A×A→A.ThemapSiscalledtheantipodeof(A,∆).NotethatS(S(a∗)∗)=a,forall
a∈A.
Let π and π be nondegenerate homomorphisms from A into some algebras B
1 2
and C, respectively. Clearly, the homomorphism π ⊗π :A⊗A→B⊗C is then non-
1 2
degenerate.Hence,wemayformtheproductπ π :A→M(B⊗C)definedbyπ π =
1 2 1 2
(π ⊗π )∆,whereπ ⊗π isextendedtoM(A⊗A)bynondegeneracy.Obviously,π π
1 2 1 2 1 2
isanondegeneratehomomorphismanditis∗-preservingwheneverbothπ andπ
1 2
are∗-preserving.Thisproductiseasilyseentobeassociative,withεasaunit.
For later use, we remark that the set of nonzero multiplicative linear functionals
ωonAisagroupunderthisproduct,withinverseoperationgivenbyω−1=ωS.To
see this, note that multiplicativity implies ω⊗ω=ω◦m. Therefore, if a,b∈A, we
haveω(b)(ω(ωS))(a)=(ω⊗ω)((b⊗1)(ι⊗S)(∆(a)))=ω◦m(ι⊗S)((b⊗1)∆(a))=
ω(ε(a)b)=ω(b)ε(a).Hence,(ω(ωS)(a))=ε(a),foralla∈A,asrequired.
If ω∈A(cid:4), we say ω is left invariant if (ι⊗ω)∆(a)=ω(a)1, for all a∈A. Right
invarianceisdefinedsimilarly.Ifanonzeroleft-invariantlinearfunctionalonAexists,
itisunique,uptomultiplicationbyanonzeroscalar.Similarly,foranonzeroright-
invariantlinearfunctional.Ifϕisaleft-invariantfunctionalonA,thefunctionalψ=
ϕS isrightinvariant.
IfAadmitsanonzero,left-invariant,positivelinearfunctionalϕ,wecall(A,∆)an
algebraicquantumgroupandwecallϕaleftHaarintegral on(A,∆).Faithfulnessof
ϕisautomatic.
Notethatalthoughψ=ϕS isrightinvariant,itmaynotbepositive.Ontheother
hand,itisprovedin[13]thatanonzero,right-invariant,positivelinearfunctionalon
A—arightHaarintegral—necessarilyexists.AsforaleftHaarintegral,arightHaar
integralisnecessarilyfaithful.
AMENABILITY AND COAMENABILITY OF ALGEBRAIC QUANTUM GROUPS 583
Thereisauniquebijectivehomomorphismρ:A→Asuchthatϕ(ab)=ϕ(bρ(a)),
foralla,b∈A.Moreover,ρ(ρ(a∗)∗)=a.
Wenowdiscussdualityofalgebraicquantumgroups.If(A,∆)isanalgebraicquan-
tum group, denote by Aˆ the linear subspace of A(cid:4) consisting of all functionals ϕa,
where a∈A. Since ϕa=ρ(a)ϕ, we have Aˆ={aϕ|a∈A}. If ω ,ω ∈Aˆ, one can
1 2
definealinearfunctional(ω ⊗ω )∆onAbysetting
1 2
(cid:1) (cid:2) (cid:1)(cid:1) (cid:2) (cid:2)
ω ⊗ω ∆(a)=(ϕ⊗ϕ) a ⊗a ∆(a) , (3.4)
1 2 1 2
whereω =ϕa andω =ϕa .Usingthis,thespaceAˆcanbemadeintoanondegen-
1 1 2 2
erate∗-algebra.Themultiplicationisgivenbyω ω =(ω ⊗ω )∆andtheinvolution
1 2 1 2
is given by setting ω∗(a)=ω(S(a)∗)−, for all a∈A and ω ,ω ,ω∈Aˆ; it is clear
1 2
thatω ω ,ω∗∈A(cid:4) butwecanshowthat,infact,ω ω ,ω∗∈Aˆ.
1 2 1 2
OnecanrealizeM(Aˆ)asalinearspacebyidentifyingitasthelinearsubspaceofA(cid:4)
consistingofallω∈A(cid:4)forwhich(ω⊗ι)∆(a)and(ι⊗ω)∆(a)belongtoA.(Itisclear
thatAˆbelongstothissubspace.)InthisidentificationofM(Aˆ),themultiplicationand
involutionaredeterminedby
(cid:1) (cid:2) (cid:1)(cid:1) (cid:2) (cid:2) (cid:1)(cid:1) (cid:2) (cid:2)
ω ω (a)=ω ι⊗ω ∆(a) =ω ω ⊗ι ∆(a) ,
1 2 1 2 2 1
(cid:1) (cid:2) (3.5)
ω∗(a)=ω S(a)∗ −,
foralla∈Aandω ,ω ,ω∈M(Aˆ).
1 2
NotethatthecounitεofAistheunitofthe∗-algebraM(Aˆ).
Thereisaunique∗-homomorphism∆ˆfromAˆtoM(Aˆ⊗Aˆ)suchthatforallω ,ω ∈
1 2
Aˆanda,b∈A,
(cid:1)(cid:1) (cid:2) (cid:1) (cid:2)(cid:2) (cid:1) (cid:2)(cid:1) (cid:2)
ω ⊗1 ∆ˆ ω (a⊗b)= ω ⊗ω ∆(a)(1⊗b) ,
1 2 1 2
(cid:1) (cid:1) (cid:2)(cid:1) (cid:2)(cid:2) (cid:1) (cid:2)(cid:1) (cid:2) (3.6)
∆ˆ ω 1⊗ω (a⊗b)= ω ⊗ω (a⊗1)∆(b) .
1 2 1 2
Ofcourse,wearehereidentifyingA(cid:4)⊗A(cid:4)asalinearsubspaceof(A⊗A)(cid:4)intheusual
way,sothatelementsofAˆ⊗AˆcanberegardedaslinearfunctionalsonA⊗A.
Thepair(Aˆ,∆ˆ)isanalgebraicquantumgroup,calledthedual of(A,∆).Itscounit
εˆandantipodeSˆaregivenbyεˆ(aϕ)=ϕ(a)andSˆ(aϕ)=(aϕ)◦S,foralla∈A.
There is an algebraic quantum group version of Pontryagin’s duality theorem for
locally compact Abelian groups which asserts that (A,∆) is canonically isomorphic
to the dual of (Aˆ,∆ˆ); that is, (A,∆) is isomorphic to its double dual (Aˆˆ,∆ˆˆ). This is
statedmorepreciselyinthefollowingresult.
Theorem3.1. Supposethat(A,∆)isanalgebraicquantumgroupwithdoubledual
(Aˆˆ,∆ˆˆ).Letπ:A→Aˆˆbethecanonicalmapdefinedbyπ(a)(ω)=ω(a),foralla∈A
and ω∈Aˆ. Then π is an isomorphism of the algebraic quantum groups (A,∆) and
(Aˆˆ,∆ˆˆ);thatis,π isa∗-algebraisomorphismofAontoAˆˆforwhich(π⊗π)∆=∆ˆˆπ.
Wewillneedtoconsideranobjectassociatedtoanalgebraicquantumgroupcalled
its analytic extension. (See [13] for full details.) We need first to recall the concept
of a GNS pair. Suppose given a positive linear functional ω on a ∗-algebra A. Let
H be a Hilbert space, and let Λ:A→H be a linear map with dense range for which
584 ERIK BÉDOS ET AL.
(Λ(a),Λ(b))=ω(b∗a),foralla,b∈A.Thenwecall(H,Λ)aGNSpair associatedto
ω.Asiswellknown,suchapairalwaysexistsandisessentiallyunique.For,if(H(cid:4),Λ(cid:4))
isanotherGNS pairassociatedtoω,themapΛ(a)(cid:1)Λ(cid:4)(a)extendstoaunitary U :
H→H(cid:4).
Ifϕ isaleftHaarintegralonanalgebraicquantumgroup(A,∆),and(H,Λ)isan
associated GNS pair, then it can be shown that there is a unique ∗-homomorphism
π:A→B(H) such that π(a)Λ(b)=Λ(ab), for all a,b∈A. Moreover, π is faithful
andnondegenerate.WeletA denotethenormclosureofπ(A)inB(H).Thus,A is
r r
a nondegenerate C∗-subalgebra of B(H). The representation π:A→B(H) is essen-
tially unique, for if (H(cid:4),Λ(cid:4)) is another GNS pair associated to ϕ, and π(cid:4):A→B(H(cid:4))
is the corresponding representation, then, as we observed above, there exists a uni-
tary U:H→H(cid:4) such that UΛ(a)=Λ(cid:4)(a), for all a∈A, and consequently, π(cid:4)(a)=
Uπ(a)U∗.
Nowobservethatthereexistsauniquenondegenerate∗-homomorphism∆ :A →
r r
M(A ⊗A )suchthat,foralla∈Aandallx∈A⊗A,wehave
r r
(cid:1) (cid:2) (cid:1) (cid:2)
∆ π(a) (π⊗π)(x)=(π⊗π) ∆(a)x ,
r
(cid:1) (cid:2) (cid:1) (cid:2) (3.7)
(π⊗π)(x)∆ π(a) =(π⊗π) x∆(a) .
r
We observe also that if ω∈A∗ and x ∈A , then the elements (ω⊗ι)(∆ (x)) and
r r r
(ι⊗ω)(∆ (x))bothbelongtoA .
r r
First,supposethatωisgivenasω=τ(π(a)·),forsomeelementa∈Aandfunc-
tionalτ∈A∗.Forx=π(b),whereb∈A,wehave
r
(cid:1) (cid:2) (cid:1) (cid:1) (cid:2) (cid:2)(cid:2)
(ω⊗ι) ∆ (x) =π (τπ⊗ι) (a⊗1 ∆(b) ∈π(A). (3.8)
r
By continuity, we get (ω⊗ι)(∆ (x)) ∈ A for all x ∈ A . It now follows that (ω⊗
r r r
ι)(∆ (x))∈A ,forarbitraryω∈A∗andx∈A ,byTaylor’sresultonlinearfunction-
r r r r
alsmentionedearlierandacontinuityargument.That(ι⊗ω)(∆ (x))∈A isproved
r r
inasimilarway.
WealsorecallthattheBanachspaceA∗becomesaBanachalgebraundertheproduct
r
inducedfrom∆ ,thatis,definedbyτω=(τ⊗ω)∆ ,forallτ,ω∈A∗.
r r r
Sincethesets∆(A)(1⊗A)and∆(A)(A⊗1)spanA⊗A,∆ (A )(1⊗A )and∆ (A )
r r r r r
(A ⊗1)havedenselinearspaninA ⊗A .Wegetfromthisthefollowingcancellation
r r r
laws,foragivenfunctionalω∈A∗:
r
(1) ifτω=0,forallτ∈A∗,thenω=0;
r
(2) ifωτ=0,forallτ∈A∗,thenω=0.
r
Usingthesecancellationproperties,itfollowseasilythat
(cid:3) (cid:1) (cid:2) (cid:4) (cid:3) (cid:1) (cid:2) (cid:4)
A = (ω⊗ι) ∆ (x) |x∈A , ω∈A∗ = (ι⊗ω) ∆ (x) |x∈A , ω∈A∗ . (3.9)
r r r r r r r
Notethatweuse[·]todenotetheclosedlinearspan.
WealsoneedtorecallthatthereisauniqueunitaryoperatorW onH⊗Hsuchthat
(cid:1) (cid:1) (cid:2)(cid:2)
W (Λ⊗Λ) ∆(b)(a⊗1) =Λ(a)⊗Λ(b), (3.10)
AMENABILITY AND COAMENABILITY OF ALGEBRAIC QUANTUM GROUPS 585
foralla,b∈A.Thisunitarysatisfiestheequation
W W W =W W ; (3.11)
12 13 23 23 12
thus,itisamultiplicativeunitary,saidtobeassociatedto (H,Λ).Herewehaveused
thelegnumberingnotationof[1].
WecanshowthatW ∈M(A ⊗B (H)),soespeciallyW ∈(A ⊗B (H))(cid:4)(cid:4)=M⊗¯B(H),
r 0 r 0
whereM denotesthevonNeumannalgebrageneratedbyA .Further,A isthenorm
r r
closureofthelinearspace{(ι⊗ω)(W)|ω∈B (H)∗}.Also,∆ (a)=W∗(1⊗a)W,for
0 r
alla∈A .
r
Thepair(A ,∆ )isareducedlocallycompactquantumgroupinthesenseof[12,
r r
Definition4.1];wecallittheanalyticextensionof(A,∆)associatedtoϕ.
Considernowthealgebraicdual(Aˆ,∆ˆ)of(A,∆).Aright-invariantlinearfunctional
ψˆ is defined on Aˆ by setting ψˆ(aˆ)=ε(a), for all a∈A. Here aˆ=aϕ and ε is the
counitof(A,∆).Sincethelinearmap,A→Aˆ,a(cid:1)aˆ,isabijection(byfaithfulnessof
ϕ), the functional ψˆ is well defined. Now define a linear map Λˆ :Aˆ→H by setting
Λˆ(aˆ)=Λ(a),foralla∈(cid:1).Sinceψˆ(bˆ∗aˆ)=ϕ(b∗a)=(Λ(a),Λ(b)),foralla,b∈A,it
followsthat(H,Λˆ)isaGNSpairassociatedtoψˆ.Itcanbeshownthatitisunitarily
equivalenttotheGNSpairforaleftHaarintegralϕˆ of(Aˆ,∆ˆ).Hence,wecanuse(H,Λˆ)
todefinearepresentationoftheanalyticextension(Aˆ,∆ˆ)of(Aˆ,∆ˆ)onthespaceH.
r r
Thereisaunique∗-homomorphismπˆ :Aˆ→B(H)suchthatπˆ(a)Λˆ(b)=Λˆ(ab),for
all a,b∈Aˆ. Moreover, πˆ is faithful and nondegenerate. Let Aˆ be the norm closure
r
ofπˆ(A)inB(H),soAˆ isanondegenerateC∗-subalgebraofB(H).ThevonNeumann
r
algebrageneratedbyAˆ willbedenotedbyMˆ.WecanshowthatW ∈M(B (H)⊗Aˆ)
r 0 r
andthatAˆ isthenormclosureofthelinearspace{(ω⊗ι)(W)|ω∈B (H)∗}.Define
r 0
alinearmap∆ˆ :Aˆ →M(Aˆ ⊗Aˆ)bysetting∆ˆ (a)=W(a⊗1)W∗,foralla∈Aˆ.Then
r r r r r r
∆ˆ is the unique ∗-homomorphism ∆ˆ :Aˆ →M(Aˆ ⊗Aˆ) such that, for all a∈Aˆand
r r r r r
x∈Aˆ⊗Aˆ,
(cid:1) (cid:2) (cid:1) (cid:2)
∆ˆ πˆ(a) (πˆ⊗πˆ)(x)=(πˆ⊗πˆ) ∆ˆ(a)x ,
r
(cid:1) (cid:2) (cid:1) (cid:2) (3.12)
(πˆ⊗πˆ)(x)∆ˆ πˆ(a) =(πˆ⊗πˆ) x∆ˆ(a) .
r
NotethatwecanshowthatW ∈M(A ⊗Aˆ)and(∆ ⊗ι)(W)=W W .
r r r 13 23
Analgebraicquantumgroup(A,∆)isofcompacttypeifAisunital,andofdiscrete
typeifthereexistsanonzeroelementh∈Asatisfyingah=ha=ε(a)h,foralla∈A.
Proposition3.2. Let(A,∆)beanalgebraicquantumgroup.Ifitisofcompacttype,
itsanalyticalextension(A ,∆ )isacompactquantumgroupinthesenseofWoronowicz.
r r
Ifitisofdiscretetype,itsanalyticalextension(A ,∆ )isadiscretequantumgroupin
r r
thesenseofWoronowiczandVanDaele.
The duality of discrete and compact quantum groups is stated precisely in the
followingresult.
Proposition3.3. Analgebraicquantumgroup(A,∆)isofcompacttype(resp.,of
discretetype)ifandonlyifitsdual(Aˆ,∆ˆ)isofdiscretetype(resp.,ofcompacttype).
586 ERIK BÉDOS ET AL.
Example3.4. Wefinishthissectionwithabriefdiscussionofthealgebraicquan-
tumgroupsassociatedtoadiscretegroupΓ.Thisillustratestheideasoutlinedabove
andprovidesthemotivationforconceptsweintroducelater.
First,considerthe∗-algebraK(Γ).Thisisprovidedwithacomultiplication∆ˆmaking
itanalgebraicquantumgroupbysetting∆ˆ(f)(x,y)=f(xy),forallf ∈K(Γ).Here
weareidentifyingK(Γ)⊗K(Γ)withK(Γ×Γ)byidentifyingthetensorproductg⊗hof
twoelementsg,h∈K(Γ)withthefunctioninK(Γ×Γ)definedby(x,y)(cid:1)g(x)h(y).
We then identify M(K(Γ)⊗K(Γ)) with F(Γ×Γ). The reason for using the notation ∆ˆ
willbeapparentshortly.
Now, let A=C(Γ) be the group algebra of Γ. Recall that, as a linear space, A has
canonical linear basis the elements of Γ and that the multiplication on A extends
thatofΓ andtheadjointoperationisdeterminedbyx∗=x−1,forallx∈Γ.Wecan
make A into an algebraic quantum group by providing it with the comultiplication
∆:A→A⊗A determined on the elements of Γ by setting ∆(x)=x⊗x. We will now
sketchtheproofthatthedual(Aˆ,∆ˆ)isthealgebraicquantumgroup(K(Γ),∆ˆ).
First,observethataleftHaarintegralfor(A,∆)isgivenbytheuniquelinearfunc-
tional ϕ on A for which ϕ(x)=δ , for all x ∈Γ, where 1 is the unit of Γ and δ
x1
is the usual Kronecker delta function. If x,y ∈Γ, then (xϕ)(y)=ϕ(yx)=δx−1,y.
It follows that the functionals xϕ (x ∈Γ) provide a linear basis for Aˆ. Hence, if e
x
(x∈Γ) is the canonical linear basis for K(Γ) given by e (y)=δ , we have a linear
x xy
isomorphismfromAˆtoK(Γ)givenbymappingxϕontoex−1.Usingthisisomorphism
asanidentification,itisstraightforwardtocheckthatthemultiplications,adjointop-
erationsandcomultiplicationsonAˆandK(Γ)arethesame;thus,(Aˆ,∆ˆ)=(K(Γ),∆ˆ),
asclaimed.
AGNSpair(H,Λ)associatedtoϕisgivenbytakingH=(cid:12)2(Γ)andΛ(x)=ex−1,for
allx∈Γ.Wechooseex−1 ratherthanexinthisformulasoastoensureΛˆ(ex)=ex.We
needthistogetthecorrectformforπˆ:itfollowseasilynowthattherepresentation,
πˆ:Aˆ→B(H),istheoneobtainedbyleftmultiplicationbyelementsofAˆ;ittherefore
extendsfromAˆ=K(Γ)toa∗-isomorphismπˆ from(cid:12)∞(Γ)ontoavonNeumannsub-
algebraofB(H).ItistriviallyverifiedthatthisvonNeumannistheonegeneratedby
πˆ(Aˆ);hence,πˆ((cid:12)∞(Γ))=Mˆ.
Of course, the representation π : A → B(H) is the one associated to the (right)
regular representation of Γ on (cid:12)2(Γ). Hence, the analytic extension A associated to
r
(A,∆) is the reduced group C∗-algebra C∗(Γ) and the corresponding von Neumann
r
algebraM issimplythegroupvonNeumannalgebraofΓ.
Wewillreturntothismotivatingsetupinthesequel.
4. Amenabilityandcoamenability. WewillretainallthenotationfromSection3.
If (A,∆) is an algebraic quantum group, recall that we use the symbol M to denote
thevonNeumannalgebrageneratedbyA .Ofcourse,A andπ(A)areweaklydense
r r
in M. Since the map ∆ is unitarily implemented, it has a unique weakly continuous
r
extension to a unital ∗-homomorphism ∆ :M→M⊗¯M, given explicitly by ∆ (a)=
r r
W∗(1⊗a)W,foralla∈M.TheBanachspaceM∗ maythenberegardedasaBanach
algebra when equipped with the canonical multiplication induced by ∆ ; thus, the
r
productoftwoelementsωandσ isgivenbyωσ =(ω⊗¯σ)◦∆ .
r