Quantization of subgroups of the affine group

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Journal of Functional Analysis

www.elsevier.com/locate/jfa

Quantization of subgroups of the affine group

P. Bieliavsky^a, V. Gayral^b,S. Neshveyev^c,∗, L. Tuset^d

aInstitutdeRechercheenMathématiqueetPhysique,UniversitéCatholiquede Louvain,CheminduCyclotron,2,1348Louvain-la-Neuve,Belgium

bLaboratoiredeMathématiques,CNRSUMR9008,UniversitédeReims Champagne-Ardenne,Moulinde laHousse- BP1039,51687Reims,France

cDepartmentofMathematics,UniversityofOslo,P.O.Box1053Blindern, NO-0316Oslo,Norway

dDepartmentofComputerScience,OsloMet- storbyuniversitetet,P.O.Box4St.

Olavsplass,NO-0130Oslo,Norway

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received27June2019 Accepted23October2020 Availableonline10November2020 CommunicatedbyDanVoiculescu

Keywords:

Quantumgroups Dualcocycles

Kohn-Nirenbergquantization

ConsideralocallycompactgroupG=QV suchthatV is abelianandtheactionofQonthedualabeliangroupVˆ has afreeorbitoffullmeasure.WeshowthatsuchagroupGcan bequantizedinthreeequivalentways:

(1) by reflecting across the Galois object defined by the canonicalirreduciblerepresentationofGonL²(V);

(2) by twisting the coproducton thegroup von Neumann algebra of Gby a dual2-cocycle obtained fromthe G- equivariantKohn–NirenbergquantizationofV ×Vˆ; (3) byconsideringthebicrossedproductdefinedbyamatched

pairofsubgroupsofQVˆ bothisomorphictoQ.

Inthesimplest caseoftheax+b groupoverthereals, the dualcocyclein(2) isan analyticanalogueoftheJordanian twist.ItwasfirstfoundbyStachurausingdifferentideas.The equivalenceofapproaches (2)and (3)inthiscaseimpliesthat thequantumax+bgroupofBaaj–Skandalisisisomorphicto thequantumgroupdefinedbyStachura.

Along the way we prove a number of results for arbitrary locallycompactgroupsG.UsingrecentresultsofDeCommer weshowthat aclassofG-Galoisobjectsisparametrizedby

* Correspondingauthor.

E-mailaddresses:[email protected](P. Bieliavsky),[email protected] (V. Gayral),[email protected](S. Neshveyev),[email protected](L. Tuset).

https://doi.org/10.1016/j.jfa.2020.108844

0022-1236/©2020TheAuthors.PublishedbyElsevierInc.ThisisanopenaccessarticleundertheCC BYlicense(http://creativecommons.org/licenses/by/4.0/).

certaincohomologyclassesinH²(G;T).Thisextendsresults of Wassermann and Davydov in the finite group case. A newphenomenonisthatalreadytheunitclassinH²(G;T) can correspond to a nontrivial Galois object. Specifically, we show that anynontrivial locally compactgroup G with group von Neumann algebra a factor of type I admits a canonicalcohomology classofdual 2-cocyclessuch that the correspondingquantizationof Gisneithercommutativenor cocommutative.

©2020TheAuthors.PublishedbyElsevierInc.Thisisan openaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).

Contents

0. Introduction . . . . 2

1. Preliminaries . . . . 4

2. Galoisobjectsanddualcocycles . . . . 6

2.1. ProjectiverepresentationsandGaloisobjects . . . . 6

2.2. Dualcocycles . . . 11

2.3. Dualcocyclesdefinedbygenuinerepresentations . . . 16

2.4. Examples:subgroupsoftheaffinegroup . . . 18

3. Kohn–Nirenbergquantization . . . 22

3.1. Kohn–NirenbergquantizationofV ×Vˆ . . . 22

3.2. DualcocyclesonsubgroupsofAff(V) . . . 25

3.3. IdentificationoftheGaloisobjects . . . 33

3.4. Deformationofthetrivialcocycle . . . 38

3.5. Multiplicativeunitaries . . . 39

3.6. Stachura’sdualcocycle . . . 45

4. Bicrossedproductconstruction . . . 48

References . . . 51

0. Introduction

Although the problem ofquantization ofLie bialgebras wassolved infull generality morethan20yearsagobyEtingofandKazhdan [15],thelistofnoncompactPoisson–Lie groups admitting nonformal (analytic)quantizations is still quite short. The difficulty liesnotonlyinmakingsenseofcertainformalconstructions,butinthatthereexistreal obstacles indoing so.Afamous exampleis thegroupSU(1,1).AttheHopf∗-algebraic levelitsquantizationiswellunderstood,butbya“nogo”resultofWoronowiczthere is nowayofmakingsense ofitattheoperatoralgebraiclevel [36].Aswasfirstrealizedby Korogodsky [19] andthen completedby Koelink andKustermans [18], theright group to quantize inthis caseis thenonconnected groupSU(1,1)Z/2Z, thenormalizer of SU(1,1) inSL(2,C).

Thepresentpaperismotivatedbyaneveneasierexample,theax+bgroupGoverthe reals. Its Liealgebragis generated bytwo elements x,y suchthat[x,y]=y. Consider theLie bialgebra(g,δ),withthecobracketδdefinedbythetriangularr-matrix

r:=x⊗y−y⊗x.

Itcanbeexplicitlyquantizedusing theJordaniantwist

Ω := exp{x⊗log(1 +hy)} ∈(Ug⊗Ug)[[h]]

found independently by Coll–Gerstenhaber–Giaquinto [8] and Ogievetsky [27].¹ This twist and its generalizations have been extensively studied, see, e.g., [20], [21]. It is particularlypopularinphysicsliterature,asitcanbeusedtoconstructtheκ-Minkowski space [6],whichby[22] canalsobeobtainedfromabicrossedproductconstruction.

IfwewanttomakesenseofΩ asaunitaryoperatoronL²(G×G),sincetheelementsx andy areskew-adjoint,ourbestbetisto takeh∈iR,butthenwestillhaveaproblem withthelogarithm,asthespectrumofyistheentirelineiR.Acorrectanalyticanalogue of the Jordanian twist was found by Stachura [30], see formula (3.31) below, but it turnsoutthat,similarly tothecaseofSU(1,1),itisimportanttoworkwiththeentire nonconnectedax+bgroup.Whattodointheconnectedcaseremainsanopenproblem.

In fact, in an earlier paper [4] the first two authors found a universal deformation formula for the actions of the connected component of the ax+b group (and, more generally, of Kählerian Lie groups) on C^∗-algebras. Unfortunately, despite the claims in [26] and [5], this formula turned out to define a coisometric butnonunitary dual 2- cocycle,whichisthetermweprefertouseintheanalyticalsettinginsteadofthe“twist”.

SeeRemark2.12belowanderratumto [5] forfurtherdiscussion.

Ifwe doconsider thenonconnected ax+b group,then even earlier,Baajand Skan- dalis constructedits quantizationas abicrossed product oftwo copies of R^∗ [29]. One disadvantage of this constructionis that from theoutset it is notclear how justifiable itisto call theirquantum groupaquantizationof theax+bgroup.Some justification wasgivenlaterbyVaesandVainerman [33].

The present work grew out of the natural question how the constructions in [29], [4] and [30] are related. As we already said, we found out that [4] does not lead to a unitary cocycle and therefore cannot actually be used to quantize the ax+b group.

Butthe constructions in [29] and [30] turned out to be equivalent, as wasconjectured byStachura.Furthermore, wefound aninterpretationof theJordanian twist/Stachura cocycle in terms of the Kohn–Nirenberg quantization, which allowed us to construct quantumanaloguesofaclassofsemidirectproductsQV.Wealsorealizedthatthese constructions have avery natural description withinDe Commer’s analytic version of theHopf–Galoistheory [11,12].

Inmore detail,the main results and organization of thepaper are as follows. After ashortpreliminarysection, webegin bydiscussing G-Galoisobjectsforgeneral locally compactgroupsGinSection2.ThesearevonNeumannalgebrasequippedwithactions ofG thatareinan appropriatesense freeand transitive. Forcompact groupssuchac- tions are known in the operator algebraliterature as full multiplicity ergodic actions.

1 In [8], the Jordanian twist doesnotappearin exponential form. Toour knowledge,it wasfirstob- servedin [16] thatColl–Gerstenhaber–Giaquinto’stwistcanbeputintheexponentialformandtherefore itcoincideswithOgievetsky’stwist.

Using recentresultsof DeCommer [12] we showthattheG-Galoisobjects withunder- lyingalgebras factorsoftypeI areclassifiedbycertainsecondcohomologyclassesonG (Theorem2.4).Forfinitegroupssucharesultquicklyleadstoacompleteclassificationof G-Galois objectsobtainedbyWassermann [34] (althoughtheresultisnotvery explicit there, see [25]) and Davydov [10]. For infinite groups the situation is of course more complicated,asingeneralthereexist Galoisobjectsbuiltonnon-type-Ialgebras.

We show next that under extra assumptions a G-Galois object of the form (B(H),Adπ), where π is a projective representation of G on H, defines a dual uni- tary2-cocycle(Proposition2.9).Wedonotknowwhethertheseassumptionsarealways automaticallysatisfied,butweshowthattheyareifπisagenuinerepresentation(The- orem2.13).Thisimpliesthatifthegroupvon NeumannalgebraW^∗(G) ofanontrivial groupG isatypeI factor,then there existsacanonical nontrivialcohomology classof dual cocyclesonG.Thisgivesprobablytheshortestexplanationwhyaquantizationof, forexample,theax+bgroupexistsattheoperatoralgebraiclevel.AttheLie(bi)algebra level thisisrelatedtoDrinfeld’sresultonquantizationofFrobenius Liealgebras [13].

The construction of the dual cocyclein Section2 is, however, rather inexplicit and in Section 3 we find a formula for such a cocycle for the semidirect products G = QV such that V is abelian and the action of Q on the dual abelian group Vˆ has a freeorbit of full measure (Assumption2.15). It is well-knownthat producingadual 2-cocycle/twist is essentially equivalent to finding a G-equivariant deformation of an appropriate algebra of functions on G. Our assumptions on G = QV imply that we can identify L²(G) with L²(V ×Vˆ) in a G-equivariant way. The Kohh–Nirenberg quantization of V ×Vˆ provides then adeformation of L²(G) and gives rise to a dual unitary2-cocycleΩ (Theorem3.12).Thecohomologyclassofthiscocycleisexactlythe onewedefinedinSection2(Theorem3.18).

In fact, there are two versions of the Kohn–Nirenberg quantization, so we get two cohomologous dualcocycles.In thecaseoftheax+b groupwe showthatoneofthese cocyclescoincideswith Stachura’scocycle(Proposition3.28).

Finally, inSection4weconsider thebicrossed product definedbyamatchedpair of twocopiesofQinQVˆ.Weshowthatthisquantumgroupisself-dualandisomorphic to(W^∗(G),Ω ˆΔ(·)Ω^∗) (Theorem4.1andCorollary4.2).Thisisachievedbyshowingthat themultiplicativeunitaryofthetwistedquantumgroup(W^∗(G),Ω ˆΔ(·)Ω^∗) isgivenbya pentagonaltransformationonQ×Vˆ (Theorem3.26)andbyapplyingtheBaaj–Skandalis procedureofreconstructingamatched pairofgroupsfrom suchatransformation [2].

1. Preliminaries

LetGbealocallycompactgroup.Wefix aleftinvariantHaarmeasure dgonGand denote byL^p(G),p∈[1,∞],theassociatedfunctionspaces.

ThemodularfunctionΔ= ΔG isdefinedbytherelation

G

f(hg)dh= Δ(g)⁻¹

G

f(h)dh for f ∈C_c(G).

ThenΔ(g)⁻¹dgis arightinvariantHaarmeasure onG.

Ina similar way, ifq ∈ Aut(G),then themodulus |q| = |q|G of q is defined by the identity

G

f(q(h))dh=|q|⁻¹

G

f(h)dh for f ∈C_c(G).

Weletλandρbetheleftandrightregular(unitary)representationsofGonL²(G):

(λgf)(h) =f(g⁻¹h) and (ρgf)(h) = Δ(g)^1/2f(hg).

Forafunctionf onGwedefineafunctionfˇby fˇ(g) :=f(g⁻¹).

WealsoletJ =JG andJˆ= ˆJG be themodularconjugations ofL^∞(G) and W^∗(G):=

λ(G):

J f := ¯f and J fˆ := Δ^−1/2f,¯ˇ andweusetheshorthandnotation

J :=JJˆ= ˆJ J, so that Jf = Δ^−1/2f .ˇ (1.1) The multiplicative unitary W = W_G: L²(G)⊗L²(G) → L²(G)⊗L²(G) of G is defined by

(W f)(g, h) =f(g, g⁻¹h).

ThemultiplicativeunitaryWˆ = ˆWG ofthedualquantumgroupisdefinedby Wˆ =W₂₁^∗, so that ( ˆW f)(g, h) =f(hg, h).

The coproduct Δ :ˆ W^∗(G) → W^∗(G) ¯⊗W^∗(G) on the group von Neumann algebra W^∗(G) isdefinedbyΔ(λˆ _g)=λ_g⊗λ_g.Wethenhave

Δ(x) = ˆˆ W^∗(1⊗x) ˆW for x∈W^∗(G).

LetnowV bealocallycompactAbeliangroupandVˆbeitsPontryagindual.Elements ofV willbedenotedbytheLatinlettersv,vj,v. . . whileelementsofVˆ willbedenoted bytheGreeklettersξ,ξj,ξ. . .. Wewill useadditivenotationboth onV andonVˆ.

ThedualityparingVˆ×V →T will bedenotedbye^iξ,v.Thisisjustanotation,we donotclaimthatthereisanexponentialfunctionhere.Wealsolet

e^−iξ,v:=e^iξ,v=e^i−ξ,v=e^iξ,−v.

Wefix aHaarmeasure dv onV andwenormalizetheHaar measuredξ ofVˆ so that theFouriertransformFV definedby

(FVf)(ξ) :=

V

e^−iξ,vf(v)dv

becomes unitary from L²(V) to L²( ˆV). For functions in several variables only one of which is in V, we use thesame symbol FV to denote the partial Fourier transform in thatvariable.

2. Galoisobjects anddualcocycles

2.1. Projective representationsand Galoisobjects

Hopf–Galois objects is awell-studied topic inHopf algebra theory. An adaptionof thisnotiontolocallycompactquantumgroupshasbeendevelopedbyDeCommer [11].

Let usrecall themain definitions.Wewill dothis forgenuinegroups,as this ismainly thecaseweareinterestedin,butitwillbeimportantforusthatthetheoryisdeveloped at leastforlocallycompactgroupsandtheirduals.

Let G be a locally compact group and β be an action of G on a von Neumann algebraN.Suchanactioniscalledintegrableiftheoperator-valuedweight

P:N → N^β, N+a→

G

β_g(a)dg,

issemifinite.Ifβ isinadditionergodic,thenwegetanormalsemifinitefaithfulweight ˜ϕ onN suchthatP(a)= ˜ϕ(a)1.Notethat

˜

ϕ(β_g(a)) = Δ(g)⁻¹ϕ(a) for all˜ a∈ N+. (2.1) Wecanthendefineanisometric map

G:L²(N,ϕ)˜ ⊗L²(N,ϕ)˜ →L²(G;L²(N,ϕ)),˜ GΛ(a)˜ ⊗Λ(b)˜

(g) = ˜Λ(βg(a)b),

where Λ :˜ Nϕ˜ → L²(N,ϕ) denotes˜ the GNS-map.The pair (N,β) consisting of avon NeumannalgebraN andanergodicintegrableactionβ ofGonN iscalled aG-Galois object iftheGaloismapGisunitary.

ThefollowingcharacterizationofGaloisobjectsisofteneasierto use.

Proposition 2.1.A pair (N,β)consisting of a von Neumannalgebra N andan ergodic integrableaction β of a locallycompactgroup Gon N is aG-Galoisobject if and only ifN Gis afactor,whichisthennecessarilyof typeI.

Proof. Thisisasimpleconsequenceofresultsin [11,Section 2].Indeed,theintegrability assumptionimpliesthatN GhasacanonicalrepresentationηonL²(N,ϕ),˜ andthen by [11, Theorem 2.1] thepair (N,β) is aG-Galoisobject ifand onlyifη isfaithful. If N Gis afactor,thenη isfaithful,sowe getoneimplicationintheproposition.

Next, the ergodicity of the action β implies that the action on N by the auto- morphisms Adη(λ_g) is ergodic as well,that is, N∩η(W^∗(G)) = C1. It follows that η(N G) =B(L²(N,ϕ)).˜ Hence, if (N,β) is aGalois object, then N Gis a factor canonicallyisomorphictoB(L²(N,ϕ)).˜

We will be interested in G-Galois objects that are themselves factors of type I, in whichcase we say,following [12], that(N,β) isaI-factorial G-Galois object. Identify- ingN withB(H) foraHilbertspaceH,wethengetaprojectiveunitaryrepresentation π:G→P U(H) suchthatβ_g= Adπ(g).Notethattheequivalenceclassofπisuniquely determinedby(N,β).

Remark2.2.Ergodicityofβ = Adπisequivalenttoirreducibilityofπ.Assumingergodic- ity,integrabilityoftheactionAdπisequivalenttosquare-integrabilityoftheirreducible projective representation π, meaning thatthere are nonzero vectors ξ,ζ for which the function g → (π(g)ξ,ζ) is square-integrable. This observation goes back to [9, Exam- ple 2.8,Chapter III],butletus givesomedetails.

Assumefirst thattheactionAdπisintegrable.Thenthedomain ofdefinitionofthe weight ˜ϕ must contain a nonzero rank-one operator θξ,ξ. Then, for every ζ ∈ H, the functiong→((Adπ(g))(θξ,ξ)ζ,ζ)=|(π(g)ξ,ζ)|²isintegrable,so πissquare-integrable.

Conversely,assumeπissquare-integrable.Thenby [14] (forgenuinerepresentations) and by [1] (for projective representations) there exists aunique positive, possibly un- bounded,nonsingularoperatorKonH,calledtheDuflo–Mooreformal degreeoperator, suchthat

G

|(π(g)ξ, ζ)|²dg=K^1/2ξ²ζ² for all ξ∈Dom(K^1/2) and ζ∈H.

This implies that θ_ξ,ξ is in the domain of definition of the weight ϕ˜ and ϕ(θ˜ _ξ,ξ) = K^1/2ξ². ItfollowsthattheactionAdπisintegrableand

˜

ϕ= Tr(K^1/2·K^1/2).

Note forfutureusethatproperty (2.1) translatesinto

(Adπ(g))(K) = Δ(g)K. (2.2)

Since the action Adπ is ergodic, this determines K uniquely up to a scalarfactor. In particular, as was already observed in [14], if G is unimodular then K is scalar, and otherwiseK isunbounded.

Remark2.3.By [11],givenaG-Galoisobject,wegetalocallycompactquantumgroupG obtained by reflecting G across the Galois object. If G is abelian, then G = G. But if Gisanonabelian genuinelocally compact groupandourGaloisobject hastheform (B(H),Adπ) foraprojectiverepresentationπofG,thenGisagenuinequantumgroup.

Indeed,assumeGisagroup.BythegeneraltheoryweknowthatB(H) isaG-Galois objectwithrespecttoanactionβofG commutingwiththeactionofG,see [11].There exist scalarsχg(g)∈T suchthat

β_g(˜π(g)) =χg(g)˜π(g) for all g∈G, g∈G,

where ˜π(g) isany liftof π(g) to U(H). Then χ_g is acharacterof G. Furthermore, by ergodicity of the action β, if χ_g1 = χ_g2 for some g₁,g₂ ∈ G, then π(g˜ ₁) and π(g˜ ₂) coincideuptoascalarfactor,whichbysurjectivity oftheGaloismapfor(B(H),Adπ) ispossible onlywheng1=g2.Thereforethemapg→χg isaninjectivehomomorphism from G into the groupof characters of G. Hence G is abelian, which contradicts our assumption.

By a recent duality result of De Commer [12], for any locally compact quantum group G, there is a bijection between the isomorphism classes of I-factorial G-Galois objectandI-factorialG-Galoisˆ objects.Thisbijectionisconstructedasfollows.Suppose we are given aI-factorial G-Galois object (N,β). Then N∩(N G), equipped with the dual action, becomes a I-factorial Galois object for G.ˆ (More precisely, we rather get aGaloisobjectfor theoppositecomultiplication onL^∞( ˆG) and thenan additional application of modularconjugations is needed to really get aGalois object for G,ˆ but this isunnecessaryinoursettingof genuinegroupsandtheirduals.)

There isasimpleclassofG-Galoisˆ objectsconstructed asfollows. Assumefrom now onthatGissecondcountable.Letω beaT-valuedBorel2-cocycleonG.Considerthe ω-twisted leftregularrepresentationofGonλ^ω:G→B(L²(G)) defined by

(λ^ω_gf)(h) =ω(g, g⁻¹h)f(g⁻¹h),

satisfyingλ^ω_gλ^ω_h =ω(g,h)λ^ω_gh,andletW^∗(G;ω):=λ^ω(G)⊂ B(L²(G)).ThenW^∗(G;ω) equipped withthecoaction λ^ω_g →λ^ω_g ⊗λ_g ofG(orinother words,theactionofG)ˆ isa G-Galoisˆ object,see [11, Section 5] forthis statementinthesetting oflocally compact quantum groups.

In fact,this coversall possible I-factorial G-Galoisˆ objects and byduality we get a descriptionoftheI-factorialG-Galoisobjects:

Theorem2.4. Foranysecond countablelocallycompactgroup G,there isabijectionbe- tweentheisomorphismclassesofI-factorialG-Galoisobjectsandthecohomologyclasses [ω]∈H²(G;T)such that thetwisted group von Neumannalgebra W^∗(G;ω)is atype I factor.

Here H²(G;T) denotes the Moore cohomology of G, which is based on Borel cochains [23].

Explicitly,thebijectioninthetheoremisdefinedasfollows.ToaI-factorialG-Galois object(B(H),Adπ) weassociate thecohomologyclass [ωπ]∈H²(G;T) definedbythe projective representation πof G.Recall thatthis means thatwe liftπ to aBorelmap

˜

π:G→U(H) anddefine aBorelT-valued2-cocycleω_π onGbytheidentity

˜

π(g)˜π(h) =ω_π(g, h)˜π(gh).

Theinverse map associates to [ω] ∈H²(G;T) (such thatW^∗(G;ω) isatype I factor) theisomorphismclass oftheG-Galoisobject(W^∗(G;ω),Adλ^ω).

ForfinitegroupsG,thistheoremisessentiallyduetoWassermann [34] (seealso [25]), aswell astoDavydov [10] inthepurelyalgebraic setting.

Wedividetheproofof Theorem2.4 into acoupleof lemmas.Let (B(H),Adπ) bea I-factorialG-Galoisobjectandω bethecocycledefinedbyalift˜πofπ.

Lemma 2.5.The G-Galoisˆ object associated with the I-factorial G-Galois object (B(H),Adπ)isisomorphic to(W^∗(G;ω),¯ α),wherethecoaction αofG isdefinedby

α(λ^ω_g^¯) =λ^ω_g^¯⊗λ_g. (2.3) Proof. Wehaveanisomorphism

B(H)G∼=B(H) ¯⊗W^∗(G; ¯ω), B(H)T →T⊗1, λg→π(g)˜ ⊗λ^ω_g^¯.

Explicitly,thecrossedproductB(H)GisthevonNeumannsubalgebraofB(L²(G;H)) generatedbytheoperatorsλgconsideredasoperatorsonL²(G;H) andtheoperatorsT˜ forT ∈B(H) definedby

( ˜T ξ)(g) =π(g)^∗T π(g)ξ(g).

Then the required isomorphism is given by AdU, where U: L²(G;H) → L²(G;H) is defined as

(U ξ)(g) = ˜π(g)ξ(g).

This givestheresult.

Inparticular,itfollowsthatW^∗(G;ω) is¯ atypeIfactor.AsJ W^∗(G;ω)J =W^∗(G;ω),¯ the von Neumann algebra W^∗(G;ω) isa typeI factor as well.ByDe Commer’s dual- ity result [12] we conclude that the map associating [ω_π] to the isomorphism class of (B(H),Adπ) isinjectiveanditsimageiscontainedinthesetofcohomologyclasses[ω]

suchthatW^∗(G;ω) isatypeI factor.

Considernowanarbitrarycohomologyclass[ω]∈H²(G;T) suchthatW^∗(G;ω) isa type I factor.Tofinishtheproofitsufficestoestablishthefollowing.

Lemma 2.6.The pair(W^∗(G,ω),Adλ^ω)isaG-Galoisobject.

Proof. Let us start with the I-factorial G-Galoisˆ object (W^∗(G;ω),¯ α), with the coac- tion α given by (2.3). By definition, the crossed product W^∗(G,ω)¯ αGˆ is the von Neumann subalgebraofB(L²(G)⊗L²(G)) generatedbyα(W^∗(G;ω)) and¯ 1⊗L^∞(G).

Theunitaryω21Wˆ commuteswith1⊗L^∞(G) andsatisfies ω21Wˆ(λ^ω_g^¯⊗λg) = (1⊗λ^ω_g^¯)ω21W .ˆ

ThereforetheconjugationbythisunitarydefinesanisomorphismofW^∗(G,ω)αGˆonto thealgebra1⊗B(L²(G)).Thisisomorphismmapsα(W^∗(G;ω)) onto¯ 1⊗W^∗(G;ω).¯

In order to understand whathappens with the dual action, initially defined by the automorphisms Ad(1⊗ρg),itisconvenient toassumethatthecocycleω satisfies

ω(g, e) =ω(e, g) =ω(g, g⁻¹) = 1 for all g∈G,

which is alwayspossible to achieveby replacing ω byacohomologous cocycle.Then ˆJ is the modularconjugation of W^∗( ˆG;ω),¯ so thatW^∗( ˆG;ω)¯ = ˆJ W^∗( ˆG;ω) ˆ¯ J. Put ρ^ω_g = J λˆ ^ω_g^¯J.ˆ Then

(ρ^ω_g^¯f)(h) = Δ(g)^1/2ω(g, g⁻¹h⁻¹)f(hg) = Δ(g)^1/2ω(h, g)f(hg).

It isthennotdifficulttocheckthat

ω21Wˆ(1⊗ρg) ˆW^∗ω¯21= (λ^ω_g^¯⊗ρ^ω_g).

WethusseethatthevonNeumannalgebraα(W^∗(G;ω))¯ ∩(W^∗(G;ω)¯ αG),ˆ together withtherestrictionofthedualaction,isisomorphictoρ^ω(G)= ˆJ W^∗(G;ω) ˆ¯ J withthe actiongiven bytheautomorphisms Adρ^ω_g.As

ρ^ω_g = ˆJ λ^ω_g^¯Jˆ= ˆJ J λ^ω_gJJ ,ˆ

weconcludethattheG-GaloisobjectassociatedwiththeG-Galoisˆ object(W^∗(G;ω),¯ α) is isomorphicto(W^∗(G;ω),Adλ^ω).

2.2. Dualcocycles

AnimportantclassofG-Galoisobjectsarisesfromdual2-cocycles.Byadualunitary 2-cocycle onGwemeanaunitaryelement Ω∈W^∗(G) ¯⊗W^∗(G) suchthat

(Ω⊗1)( ˆΔ⊗ι)(Ω) = (1⊗Ω)(ι⊗Δ)(Ω).ˆ (2.4) Similarlyto theG-Galoisˆ objects W^∗(G;ω) consideredabove, suchcocyclesleadto G- Galois objects W^∗( ˆG;Ω) (which for notational consistency with W^∗(G;ω) we should haverather denotedbyW^∗( ˆG;Ω^∗)).Following [26,Section 4],theycanbe describedas follows.

Identifyasusual theFourier algebraA(G) withthepredualofW^∗(G).Given adual unitary 2-cocycle Ω ∈ W^∗(G) ¯⊗W^∗(G), the von Neumann algebra N := W^∗( ˆG;Ω) ⊂ B(L²(G)) isgenerated bytheoperators

πΩ(f) := (f⊗ι)( ˆWΩ^∗), f ∈A(G).

Defineanactionβ ofGonN by

βg(x) := (Adρg)(x), x∈ N,

whereweremindthatρg= ˆJ λgJˆistherightregularrepresentation.Themap πΩisthe representationofthealgebraA(G) equippedwiththenewproduct

(f1Ωf2)(g) := (f1⊗f2)Δ(λˆ g)Ω^∗

. (2.5)

TherepresentationπΩ hastheequivarianceproperty βg

πΩ(f)

=πΩ(λgf). (2.6)

By[33, Section 1.3], thecanonical weightϕ˜onN hasthe following description. Its GNS-space canbe identified withL²(G),withtheGNS-mapΛ :˜ Nϕ˜ →L²(G) uniquely determined by

Λ(π˜ Ω(f)) = ˇf for f ∈A(G) such that ˇf ∈L²(G), (2.7) whereweremindthatfˇ(g)=f(g⁻¹).Inparticular, forf asabovewehave

˜

ϕ(πΩ(f)^∗πΩ(f)) =fˇ²2. (2.8) Twodual unitary 2-cocycles Ω,Ω are called cohomologous ifthere exists aunitary u∈W^∗(G) suchthat

Ω = (u⊗u)Ω ˆΔ(u)^∗.

ThecohomologyclassesformasetH²( ˆG;T).Ingeneralthissetdoesnothaveanyextra structure.

Proposition 2.7. Two dual unitary 2-cocycles Ω, Ω on a locally compact group G are cohomologousif andonly ifthey defineisomorphic G-Galoisobjects.

Proof. AsΔ(u)ˆ = ˆW^∗(1⊗u) ˆW,itiseasyto seethatifΩ = (u⊗u)Ω ˆΔ(u)^∗ thenAdu defines aG-equivariantisomorphismofW^∗( ˆG;Ω) onto W^∗( ˆG;Ω).

Conversely,assumewehaveaG-equivariantisomorphismθ:W^∗( ˆG;Ω)→W^∗( ˆG;Ω).

Denote by Λ and˜ Λ˜ the GNS-maps for these objects as described above. Then the isomorphism θ is implemented by the unitary u defined by uΛ(x)˜ = ˜Λ(θ(x)). Since by (2.6) and (2.7) theactionsofGareimplementedinasimilarwaybytheunitariesρg:

ρgΛ(x) = Δ(g)˜ ^1/2Λ(β˜ g(x)) and ρgΛ˜(x) = Δ(g)^1/2Λ˜(β_g(x)), we concludethatuρg=ρgu, henceu∈W^∗(G).

Denote by G and G the corresponding Galois maps. Then by definition we have (1⊗u)G =G(u⊗u). On theother hand, by [11, Proposition 5.1] wehave G = ˆWΩ^∗ and G = ˆWΩ^∗.Hence

(1⊗u) ˆWΩ^∗= ˆWΩ^∗(u⊗u).

Using againthatΔ(u)ˆ = ˆW^∗(1⊗u) ˆW, weconcludethatΩ= (u⊗u)Ω ˆΔ(u)^∗. CombinedwithTheorem2.4thispropositionallowsonetodescribeapartofH²( ˆG;T) intermsofcohomologyofG.Namely,denotebyH_I²( ˆG;T) thesubsetofH²( ˆG;T) formed bytheclasses[Ω] suchthatW^∗( ˆG;Ω) isatypeIfactor.GivensuchanΩ,wecanidentify W^∗( ˆG;Ω) with B(H) for a Hilbert space H. Then the action β of G on W^∗( ˆG;Ω) is givenbyaprojectiverepresentationπofGonH,andwedenotebyc_Ωthecorresponding 2-cocycleω_π onG.Similarly,denotebyH_I²(G;T) thesubsetofH²(G;T) formedbythe classes[ω] suchthatW^∗(G;ω) isatypeI factor.

Corollary 2.8.For any second countable locally compact group G, the map Ω → [cΩ] defines an embeddingof H_I²( ˆG;T) intoH_I²(G;T).

A natural questionis whetherthis embedding isonto. We donotknow the answer, but asa steptowardsasolution ofthis problem letus explain howdual cocyclesarise from Galoismapsunderextraassumptions.

ItwillbeusefultogobeyondGaloisobjects.Assumewearegivenasquare-integrable irreducible projective representation π of G on H. Assume also that we are given a unitarymap

Op :L²(G)→HS(H) such that Op(λgf) = (Adπ(g))(Op(f)), (2.9)

whichwewillcallaquantizationmap.HereHS(H) denotestheHilbertspaceofHilbert–

Schmidtoperators onH.AsinSection2.1,considertheDuflo–MooreoperatorKonH and the weight ϕ˜ = Tr(K^1/2·K^1/2). As the GNS-space for ϕ˜ we could take HS(H), with the GNS-map Λ uniquely˜ determined by Λ(T K˜ ^−1/2) := T for T ∈ HS(H) such thatT K^−1/2is abounded operator.Butusing theunitaryOp wecantransportevery- thing to L²(G). Thus we take L²(G) as the GNS-space, with the GNS-map uniquely determined by

Λ(Op(f˜ )K^−1/2) :=f for f ∈L²(G) such that Op(f)K^−1/2∈B(H). (2.10) Considerthecorresponding GaloismapG,so G:L²(G)⊗L²(G)→L²(G)⊗L²(G),

G(f1⊗f2)(g, h) = ˜Λ

Adπ(g)

(Op(f1)K^−1/2) Op(f2)K^−1/2

(h). (2.11) Finally,define

Ω := (J ⊗ J)G^∗(1⊗ J) ˆW , (2.12) whereweremindthatJ =JJˆ.

Proposition2.9. Withtheabovesetupandnotation,theoperatorΩiscoisometric.Itlies inthealgebra W^∗(G) ¯⊗W^∗(G)andsatisfies thecocycleidentity (2.4).

In particular, Ω is a dual unitary 2-cocycle on G if and only if (B(H),Adπ) is a G-Galois object. Moreover, if Ω is indeed unitary, then (B(H),Adπ) is isomorphic to theG-Galoisobject(W^∗( ˆG;Ω),β)defined by Ω.

Proof. SincetheGaloismapsarealwaysisometric, itisclearthatΩ iscoisometric.

Next, a straightforward application of definition (2.11) together with scaling prop- erty (2.2) yieldthefollowingidentitiesforG, cf. [11,Lemma 3.2]:

G(λg⊗1) = (ρg⊗1)G, G(1⊗λg) = (λg⊗λg)G. Togetherwiththeidentities

Wˆ(ρg⊗1) = (ρg⊗1) ˆW , Wˆ(1⊗ρg) = (λg⊗ρg) ˆW , Jρg=λgJ

this implies that Ω commutes with the operators ρ_g ⊗1 and 1⊗ρ_g. Hence Ω ∈ W^∗(G) ¯⊗W^∗(G).

Turning to thecocycle identity, by [11, Proposition 3.5] the Galois map G (denoted byG˜inop. cit.)satisfiesthefollowinghybridpentagonrelation:

Wˆ12G13G23=G23G12.

(Moreprecisely,theresultin [11] isformulatedonlyfortheGaloisobjects,butaninspec- tion of theproof shows thatit remains valid for arbitrary integrableergodic actions.) PlugginginG= (1⊗ J) ˆWΩ^∗(J ⊗ J) weget

Wˆ12Wˆ13Ω^∗₁₃Wˆ23Ω^∗₂₃= ˆW23Ω^∗₂₃Wˆ12Ω^∗₁₂,

and using Δ(x)ˆ = ˆW^∗(1⊗x) ˆW and the pentagonrelation Wˆ12Wˆ13Wˆ23 = ˆW23Wˆ12 we obtaintherequiredcocycleidentity.

Finally,bythedefinitionofΩ,theGaloismapGisunitaryifandonlyifΩ isunitary.

AssumingthatΩ andG areunitary,by [11,Proposition 3.6(1)] theelements (ω⊗ι)(G) forω∈B(L²(G))_∗spanaσ-weaklydensesubspaceofπ_ϕ˜(B(H)).Recallingthedefinition of W^∗( ˆG;Ω) weconcludethat

πϕ˜(B(H)) =JW^∗( ˆG; Ω)J.

The actionof GonB(H) is implementedontheGNS-space by theunitariesλ_g,while thatonW^∗( ˆG;Ω) bytheunitariesρ_g.SinceJρ_gJ =λ_g,weseethattheGaloisobjects (B(H),Adπ) and(W^∗(G;Ω),β) areindeedisomorphic.

Remark2.10.Although[11,Proposition 3.6(1)] isformulatedonlyfortheGaloisobjects, its proofremainsvalidforany integrableergodicaction.Therefore weseethatstarting fromasquare-integrableirreducibleprojectiverepresentationπofGonH andaunitary quantization map Op : L²(G) → HS(H), we can define a dual coisometric cocycle Ω by (2.12) andthenthesetofelements(ω⊗ι)( ˆWΩ^∗) forω∈B(L²(G))_∗spanaσ-weakly dense subspace of Jπ_ϕ˜(B(H))J. Therefore Ω contains a complete information about (B(H),Adπ) independentlyofwhetherwedealwithaGaloisobjector not.

Remark2.11.Thequantizationmap Op definesaproduct onL²(G) by Op(f₁ f₂) = Op(f₁) Op(f₂).

IngeneraltheproductΩdefinedbyΩ isapparentlynotthesameasonA(G)∩L²(G).

Inorderto seethis,letusproceedabitinformally,withouttryingtofully justifyevery step.Bydefinitionwehave

Jπ_Ω(f)J =J(f ⊗ι)( ˆWΩ^∗)J = (f⊗ι)(G(J ⊗1)).

Take functionsf1,f2 ∈ L²(G) and consider the function f ∈ A(G) defined by f(g) = (λgf1,f2).Applying(·f1,f2) tothefirstlegofG(J ⊗1) wethen get

JπΩ(f)J =

G

Adπ(g)

(Op(Jf1)K^−1/2)f2(g)dg= Op(Δ^−1/2f)K^−1/2.

WethusconcludethatΩ isrelatedtothequantizationmapOp definedby Op(f) = Op(Δ^−1/2f)K^−1/2,

sothatOp(f_1Ωf₂)= Op(f₁)Op(f₂).

TheproductsdefinedbythequantizationmapsOp andOpcoincideifandonlyifthe mapf →Op⁻¹(Op(Δ^−1/2f)K^−1/2) isanautomorphismwithrespectto.Ingeneralwe seenoreasonwhythisshouldbethecase.Butthiscanhappen.ObservethatsinceΔ is theonlypositivemeasurablefunctionFonGsuchthatλgF = Δ(g)⁻¹F,anyreasonable extensionof Op toaclassof functionsincludingΔ^s shouldsatisfyOp(Δ^1/2)=cK^−1/2 foraconstantc>0.ThenOp⁻¹(Op(Δ^−1/2f)K^−1/2)=c⁻¹(Δ^−1/2f)Δ^1/2.Fromthis weseethatforthemapf →Op⁻¹(Op(Δ^−1/2f)K^−1/2) tobeanautomorphismitsuffices tohavetheidentities

Δ^sΔ^t= Δ^s+t, Δ^−1/2f =cΔ^−α f Δ^α−1/2 (2.13) forsomeα∈R.Fortheexamplesstudiedinthispaperwewillindeedhavesuchidentities, withc = 1 andα= 1/2.On otherhand,forthe examplestudiedin [5] (which isnota Galoisobject)wehadc= 1 andα= 1/4.

Wethussee thattheproblem of describingH_I²( ˆG;T) reducesto thefollowing ques- tion: it is true that for any I-factorial Galois object (B(H),Adπ) there is a unitary quantizationmap (2.9)?Thiscanbereformulatedasarepresentation-theoreticproblem asfollows.Assumewearegivena2-cocycleωonGsuchthatW^∗(G;ω) isatypeIfactor.

WeidentifyW^∗(G;ω) withB(H) andputπ_ω(g):=λ^ω_g ∈B(H).Theng→π_ω(g)⊗π^c_ω(g) isawell-definedunitaryrepresentationofGonH⊗H¯,whereπ^c_ω(g) ¯ξ:=π_ω(g)ξ.Isthis representationequivalentto theregularrepresentation?

The answer is known to be “yes” for finite groups [34,24]. Indeed, the Galois map givesaunitaryequivalence

π_ω⊗π_ω^c ⊗ε_H⊗εH¯ ∼ρ⊗ε_H⊗εH¯,

whereεL denotesthetrivialrepresentationofGontheHilbertspaceL.Thisimpliesthe requiredequivalenceπω⊗π_ω^c ∼λforfinitegroupsG,butfallsshortofwhatweneedfor generalG.

Remark 2.12.In order to stress that it can be dangerous to rely too much on analo- gieswiththefinitegroupcase,notethatforanysquare-integrableirreducibleprojective representationπ ofGonH,theGaloismap alwaysdefines anembeddingoftherepre- sentation Adπ⊗ε_HS(H) on HS(H)⊗HS(H) into ρ⊗ε_HS(H). It follows thatfor finite groupsexistenceof aunitaryquantization map (2.9) is equivalent to (B(H),Adπ) be- ingaGalois object.Thisiscertainly(but untilrecentlyunexpectedly!)notthecasefor generalgroups.Forexample,fortheconnectedcomponentGoftheax+bgroupoverR

therearetwoinequivalentinfinitedimensionalirreducibleunitaryrepresentations.They arebothsquare-integrableandbothadmitunitaryquantizationmaps [4].ButGhasno I-factorial Galoisobjects,sinceH²(G;T) istrivialandW^∗(G) isthesumof twotypeI factors.

2.3. Dualcocyclesdefined by genuinerepresentations

Wenow turntoGaloisobjects(B(H),Adπ) definedbygenuinerepresentations.

Theorem 2.13.Forany nontrivial second countablelocallycompactgroup G,a(square- integrable, irreducible) unitary representation π:G→ B(H) such that (B(H),Adπ) is a G-Galois object exists if and only if W^∗(G) isa type I factor. Moreover, if π exists, then

(i) π is unique upto unitary equivalence; explicitly, by identifying W^∗(G) with B(H) wecantake π(g)=λg;

(ii) theGaloisobject (B(H),Adπ)isdefined byadual unitary 2-cocycle ΩonG.

Recall thatby Proposition 2.7 the cohomology class of Ω is determined by the iso- morphism classofthecorresponding Galoisobject.Thereforetheabovetheorem shows thatifW^∗(G) isatypeIfactor,thenwegetacanonicalclass[Ω]∈H²( ˆG;T).Interms of Corollary2.8,thisclass correspondstotheunitofH²(G;T).

Note alsothatthe conditionthatW^∗(G) is atypeI factorimpliesthatGisneither compact nor discrete, so the situation described in the theorem is apurely analytical phenomenon.

Proof of Theorem2.13. ThefirststatementisanimmediateconsequenceofTheorem2.4 applied to genuine representations and, correspondingly, to the trivial 2-cocycle ω = 1 on G. Furthermore, that theorem implies that π is unique up to equivalence as a projective representation. Thereforeto provepart(i)we onlyhavetoshow theslightly strongerstatementthatπis alsouniqueupto equivalenceas agenuinerepresentation.

Thus,weidentifyW^∗(G) withB(H),takeπ(g)=λ_g,andwehavetoshowthatforany characterη:G→T therepresentationsηπandπareequivalent.Therepresentationsηλ and λare unitarily equivalent, e.g.,byFell’sabsorption principle. Itfollows thatthere exists an automorphism θ of W^∗(G) such that θ(λg) = η(g)λg for all g. As θ is an automorphism of B(H) = W^∗(G), it is unitarily implemented, which means exactly thatηπandπareunitarily equivalent.

In order to provepart (ii), byour resultsin Section2.2 itsuffices to show that the representation π⊗π^c isequivalent totheregularrepresentation.

Since we can identify W^∗(G) with B(H) in such a way that π(g) = λ_g, the rep- resentation λ is a multiple of π, so we can write λ ∼ π⊗ε_L, where ε_L is the trivial representationonaseparableHilbertspaceL.SinceGisnontrivial,theHilbertspaceH

must be infinite dimensional. But then the multiplicity of the square-integrable rep- resentation π in λ must be infinite as well [14], so the Hilbert space L is infinite dimensional.

Bypassingtotheconjugaterepresentationsweget π^c⊗εL∼λ^c∼λ∼π⊗εL.

Thisimpliesthattheirreducible representationsπ^c andπareequivalent.

Next,usingFell’sabsorption principleweget

π⊗π⊗ε_L∼π⊗λ∼λ⊗ε_H ∼π⊗ε_L⊗ε_H.

From thiswe seethatthe representationπ⊗π isamultipleof π,and inorderto con- clude thatπ⊗π∼λ it sufficesto show that themultiplicity ofπ inπ⊗π is infinite.

In other words, we have to check that the commutant of (π⊗π)(G) inB(H ⊗H) is infinitedimensional.Equivalently,thatthecommutantof(λ⊗λ)(G) inW^∗(G) ¯⊗W^∗(G) isinfinitedimensional.

Moregenerally,letusshowthatifG₁isaclosednonopensubgroupofasecondcount- able locally compact group G2 such thatW^∗(G1) is atype I factor, then the relative commutantW^∗(G₁)∩W^∗(G₂) isinfinitedimensional.

AssumeW^∗(G1)∩W^∗(G2) isfinitedimensional.DenotebyΔithemodularfunction ofG_i, byμ_i the Haarmeasure on G_i andby ϕˆ_i thestandardHaar weightonW^∗(G_i).

The modular group of ϕˆ2 is given by σt(λg) = Δ2(g)^itλg. It preserves W^∗(G1), and since W^∗(G₁) is a type I factor, there exists a normal semifinite faithful weight ˆϕ₁ on W^∗(G1) with the same modular group. Consider the unique normal semifinite operator-valued weight P: W^∗(G₂) → W^∗(G₁) such that ϕˆ₁P = ˆϕ₂. Since W^∗(G₁) isatypeIfactorandW^∗(G1)∩W^∗(G2) isfinite dimensional,itfollows,e.g., from [32, Corollary 12.12] appliedto M=W^∗(G₁) thatsuchanoperator-valuedweightmustbe bounded, hence it is ascalar multiple of a conditional expectation. In particular, the weightϕˆ₂|W^∗(G1) is semifinite. This, inturn, implies, thatϕˆ₂|W^∗(G1) is aHaar weight on W^∗(G1), hence ϕˆ2|W^∗(G1) =cϕˆ1 for a constantc > 0. We canalso conclude that Δ₂|G1= Δ₁.

DenotebyEtheconditionalexpectationobtainedbyrescalingP,sothatcϕˆ1E= ˆϕ2. Using the identity ϕˆ2(xy) = cϕˆ1(E(x)y) for appropriate elements x ∈ W^∗(G2) and y ∈ W^∗(G1), it is not difficult to compute E on a dense set of elements. Namely, if x=

G2F(g)λgdμ2(g),withF =f∗f (convolutioninL¹(G2))forsomef,f∈Cc(G2), thenwemusthave

E(x) =c⁻¹

G1

F(g)λ_gdμ₁(g).