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Advances
in
Mathematics
www.elsevier.com/locate/aim
Amenability
and
subexponential
spectral
growth
rate
of
Dirichlet
forms
on
von
Neumann
algebras
✩ Fabio Cipriania,∗, Jean-Luc Sauvageotba DipartimentodiMatematica,PolitecnicodiMilano,piazzaLeonardodaVinci32,
20133Milano,Italy
b
InstitutdeMathématiques,CNRS-UniversitéDenisDiderot, F-75205 ParisCedex13,France
a r t i c l e i n f o a bs t r a c t
Article history:
Received9December2016 Accepted1October2017 CommunicatedbyDanVoiculescu MSC: 46L57 46L87 46L54 43A07 Keywords:
vonNeumannalgebra Amenability
HaagerupProperty(H) Dirichletformspectralgrowth Countablediscretegroup
In this work we apply Noncommutative Potential Theory to characterize (relative) amenability and the (relative) Haagerup Property (H) of von Neumann algebras interms ofthespectralgrowthofDirichletforms.Examplesdealwith (inclusionsof)countablediscretegroupsandfreeorthogonal compactquantumgroups.
©2017ElsevierInc.Allrightsreserved.
✩ This work has been supported by LYSM Laboratorio Ypatia di Scienze Matematiche I.N.D.A.M.
Italy–C.N.R.S.-Univ. Aix-Marseille France, GREFI-GENCO Groupe de Recherche Franco Italienne en Géométrie NonCommutativeI.N.D.A.M.Italy–C.N.R.S.FranceandM.I.U.R.ItalyPRIN2012Project No.2012TC7588-003.
* Correspondingauthor.
E-mailaddresses:fabio.cipriani@polimi.it(F. Cipriani),jean-luc.sauvageot@imj-prg.fr
(J.-L. Sauvageot).
https://doi.org/10.1016/j.aim.2017.10.017
1. Introductionanddescriptionoftheresults
Classical resultsrelate the metricproperties of conditionallynegative definite func-tionsonacountablediscretegroupΓ toitsapproximationproperties.Forexample,there existsaproper,conditionallynegativedefinitefunction onΓ ifandonlyifthereexists a sequence ϕn ∈ c0(Γ) of normalized, positive definite functions, vanishing at infinity
andconvergingpointwisetotheconstantfunction1.
Inacelebratedwork[27],U.Haagerupprovedthatthelengthfunctionofafreegroup Fn withn∈ {2,· · · ,∞} generatorsisnegativedefinite,thusestablishingforfreegroups
the aboveapproximation property. Since then the property is referredto as Haagerup ApproximationProperty(H)orGromova-T-menability(see[6]).
Inaddition,ifforaconditionallynegative definitefunction onacountablediscrete group Γ, the series g∈Γe−t(g) converges for all t > 0, then there exists a sequence
ϕn∈ l2(Γ) ofnormalized,positivedefinitefunctions,convergingpointwisetotheconstant
function 1 ([25, Thm 5.3]). This latter property is just one of the several equivalent appearancesof amenability, aproperty introduced by J. von Neumann in1929 [40] in ordertoexplaintheBanach–TarskiparadoxinEuclideanspacesRnexactlywhenn≥ 3.
In this note we are going to discuss extensions of the above results concerning amenabilityforσ-finite vonNeumannalgebras N .
Thedirectionalongwhich weare goingto look forsubstitutes oftheabove summa-bilityconditionrelatedtoamenability,isthatofNoncommutativePotential Theory.
Thisis suggestedbyarecent resultbyCaspers–Skalski[5]asserting thatN hasthe (suitablyformulated)HaagerupApproximationProperty (H)ifandonlyifthere exists aDirichletform (E,F) onthestandardHilbertspaceL2(N ),havingdiscrete spectrum.
Thelinkbetweenthepropernessconditionforaconditionallynegativedefinite func-tion onacountablediscretegroupsΓ andthegeneralizedoneonvonNeumannalgebras, relies onthefactthat,when thevon NeumannalgebraN = L(Γ) istheonegenerated bythe left regularrepresentation of Γ,thequadratic form E[a] =g∈Γ(g)|a(g)|2 on
thestandardspace L2(L(Γ),τ ) l2(Γ) isaDirichlet form ifand onlyifthefunction
isconditionally negativedefinite and itsspectrumis discreteif andonly if is proper. Moreover,onacountable,finitely generated,discretegroupΓ withpolynomialgrowth, there exist aconditionallynegativedefinite functions, having polynomialgrowth and growthdimensionsarbitrarilycloseto thehomogeneousdimensionofΓ (see [16]).
This point of view thus suggests that a condition providing amenability of a von Neumannalgebrawithfaithfulnormalstate(N,ω) couldbe thesubexponentialspectral growth ofaDirichletform(E,F) onthestandardspaceL2(N,ω),i.e.thediscretenessof
thespectrumof(E,F) andthesummabilityoftheseriesk≥0e−tλkforallt> 0,where
λ0,λ1,. . . aretheeigenvaluesof(E,F).
ThesecondfundamentalfactthatwillallowtouseDirichletformstoinvestigatethe amenability of a von Neumann algebra, is the possibility to express this property in terms of Connes’ correspondences: N is amenable if and only if the identity or
stan-dard N -N -correspondenceL2(N ) isweaklycontainedinthecoarseor Hilbert–Schmidt
N -N -correspondence L2(N )⊗ L2(N ) (see [31]).
In the second part of the work we provide a condition guaranteeing the relative amenability of aninclusionB⊆ N offinite vonNeumann algebrasintroducedbyPopa
[31,30], in terms of the existence of a Dirichlet form (E,F) on L2(N ) having relative
subexponentialspectral growth.Alsothisresultisbasedonthepossibilitytoexpress the relative amenabilityofavon NeumannalgebraN withrespect toasubalgebra B⊆ N
interms ofthe weakcontainmentof theidentity correspondence L2(N ) in therelative tensor productcorrespondenceL2(N )⊗
BL2(N ) introducedbySauvageot[33], [30].
Using a suitable Dirichlet form constructed in [11], whose construction uses tools developedby M.Brannanin[3], weapply theaboveresultto proveamenabilityof the vonNeumannalgebraofthefreeorthogonalquantumgroupO2+andHaagerupProperty (H) of the free orthogonal quantum groups ON+ for N ≥ 3 (see also the recent [19]), resultsfirstlyobtainedbyM.Brannan[3].
AdetaileddiscussionoftherelativeHaagerupProperty(H)forinclusionsofcountable discrete groupsintermsofconditionallynegativedefinite functionsispresented.
The paper is organized as follows: in Section 2 we provide the necessary tools on noncommutativepotentialtheoryonvonNeumannalgebraasDirichletforms,Markovian semigroupsandresolvents.
In Section3 we first recall someequivalent constructions of the coarse or Hilbert– Schmidt correspondence of avon Neumann algebra N and someconnections between themodulartheoriesofN ,ofitsoppositeNo,andoftheirspatialtensorproductN⊗No.
Then we introduce the spectral growth rate of aDirichlet form and we provethe first main result of the work about the amenability of von Neumann algebra admitting a Dirichlet form with subexponential spectral growth rate.This partterminates with an application to the amenabilityof countable discrete groups and with apartially alter-nativeapproachtotheproofofaresultofM.Brannan[3]abouttheamenabilityofthe freeorthogonalquantum groupO+2.
Section4startsrecallingsomefundamentaltool ofthebasicconstructionN,B for
inclusions B ⊆ N of finite von Neumann algebras, needed to prove the second main resultoftheworkconcerningtheamenabilityofN with respecttoitssubalgebraB.To formulatethecriterion, weintroducethespectralgrowthrateofaB-invariantDirichlet form on the standard space L2(N ) relatively to the subalgebra B, using the compact
ideal spaceJ (N,B ) ofN,B (cf.[31,30]).Thesectionterminates discussingrelative amenability for two natural subalgebras Bmin ⊆ N and Bmax ⊆ N associated to any
Dirichletform.
InSection5weextendthespectralcharacterizationoftheHaagerupProperty(H)of von Neumann algebras with countabledecomposablecenterdue to M. Caspers andA. Skalski[5]totheRelativeHaageruupProperty(H)forinclusionsoffinitevonNeumann algebras B⊆ N formulated byS.Popa[31,30].
In Section 6 we discuss the relative Haagerup Property (H) for inclusions H < G
negativedefinite functiononG whichisproperonthehomogeneousspaceG/H andin termsofquasi-normalityofH inG.
The content of the present work has been the subject of talks given in Rome II (March2015), Paris (GREFI-GENCO April2015), Berkeley (UC Seminars September 2015),Krakov(September2015), Varese(May2016).
2. Dirichletformsonσ-finitevon Neumannalgebras
RecallthatavonNeumannalgebraN isσ-finite,orcountably decomposable,ifany collectionofmutuallyorthogonalprojectionsisatmostcountableandthatthisproperty isequivalenttotheexistenceofanormal,faithfulstate.Thisisthecase,forexample,if
N actsfaithfullyonaseparableHilbertspace.
Letus consider onaσ-finite von Neumannalgebra N afixed faithful,normal state
ω∈ N∗+.Letus denoteby(N,L2(N,ω),L2+(N,ω),Jω) thestandardform ofN andby
ξω∈ L2+(N,ω) thecyclic vectorrepresentingthestate(see[26]).
Forareal vectorξ = Jωξ ∈ L2(N,ω),letus denotebyξ∧ ξω theHilbertprojection
ofthevectorξ ontotheclosedandconvexset Cω:={η ∈ L2(N,ω): η = Jωη, ξω− η ∈
L2
+(N,ω)}.
WerecallherethedefinitionofDirichletformandMarkoviansemigroup(see[8])ona genericstandardformofaσ-finitevonNeumannalgebra.Foradefinitionparticularized totheHaagerupstandardformsee[23].
Definition 2.1 (Dirichlet forms on σ-finite von Neumann algebras). A densely defined, nonnegative,lowersemicontinuousquadraticformE : L2(N,ω)→ [ 0,+∞] issaidtobe:
i)real if
E[Jω(ξ)] =E[ξ] ξ∈ L2(N, ω) ; (2.1)
ii)aDirichlet form ifitisreal andMarkovian inthesense that
E[ξ ∧ ξω]≤ E[ξ] ξ = Jωξ∈ L2(N, ω) ; (2.2)
iii) a completely Dirichlet form if all the canonical extensions En to L2(Mn(N ),
ω⊗ trn)
En[[ξi,j]ni,j=1] := n
i,j=1
E[ξi,j] [ξi,j]ni,j=1∈ L2(Mn(N ), ω⊗ trn) , (2.3)
areDirichletforms.
Bytheself-polarityofthestandardconeL2
+(N,ω),anyrealvectorξ = Jωξ∈ L2(N,ω)
decomposes uniquely as a difference ξ = ξ+ − ξ− of two positive, orthogonal vectors
positivecone).Themodulusofξ isthendefinedasthesumofthepositiveandnegative parts |ξ|:= ξ++ ξ−.
Notice that,ingeneral,thecontractionproperty
E[ |ξ| ] ≤ E[ξ] ξ = Jωξ∈ L2(A, ω)
isaconsequenceofMarkovianityandthatitisactuallyequivalenttoitwhenE[ξω]= 0.
The domain of the Dirichlet form is defined as the (dense) subspace of L2(N,ω)
where thequadraticformisfinite:F := {ξ ∈ L2(N,ω):E[ξ]< +∞}.Wewilldenote by
(L,D(L)) thedenselydefined,self-adjoint,nonnegativeoperatoronL2(A,τ ) associated
with theclosed quadraticform(E,F)
F = D(√L) and E[ξ] = √Lξ2 ξ∈ D(√L) =F .
Definition 2.2 (Markovian semigroups on standard forms of von Neumann algebras).
a) A boundedoperatorT onL2(N,ω) issaidto be
i)real if itcommuteswiththemodularconjugation:T Jω= JωT ,
ii) positive ifitleavesgloballyinvariantthepositivecone:T (L2
+(N,ω))⊆ L2+(N,ω),
iii) Markovian ifitisrealanditleavesglobally invarianttheclosed, convexsetCω:
T (Cω)⊆ Cω,
iv) completelypositive, resp.completely Markovian,ifit is realand all ofits matrix amplificationsT(n) toL2(Mn(N ),ω⊗ trn) L2(N,τ )⊗ L2(Mn(C),trn) definedby
T(n)[[ξi,j]ni,j=1] := n
i,j=1
[T ξi,j]ni,j=1 [ξi,j]i,j=1n ∈ L2(Mn(N ), ω⊗ trn) ,
are positive,resp.Markovian;
b) A stronglycontinuous, uniformlybounded, self-adjoint semigroup{Tt: t> 0} on
L2(N,ω) is said to be real (resp.positive, Markovian, completely positive, completely
Markovian)ifthe operatorsTt arereal (resp.positive,Markovian,completely positive,
completely Markovian)forallt> 0.
In literature, property in item iii) above is sometime termed submarkovian, while
markovian is meant positivitypreserving and unital. Our choiceis only dictated by a willing ofsimplicity.
NoticethatT is(completely)Markovianiffitis(completely)positiveandT ξω≤ ξω.
Notice thatifN is abelian,then positive(resp.Markovian)operators are automati-cally completelypositive(resp.completelyMarkovian).
Dirichletformsareinone-to-onecorrespondencewithMarkoviansemigroups(see[8]) throughtherelations
Tt= e−tL t≥ 0
where(L,D(L)) istheself-adjoint operatorassociated tothequadraticform(E,F). Dirichlet forms and Markovian semigroups are also in correspondence with a class of semigroupson thevon Neumann algebra. To statethis fundamentalrelation, let us considerthesymmetricembedding iω determinedbythecyclicvectorξω
iω: N → L2(N, ω) iω(x) := Δ
1 4
ωxξω x∈ N .
Here,Δωisthemodularoperatorassociatedwiththefaithfulnormalstateω (see[38]).
Wewilldenote by{σω
t : t∈ R} themodularautomorphisms groupassociated to ω and
by Nσω ⊆ N the subalgebra of elements which are analytic with respect to it. Then
(see [8]) (completely) Dirichlet forms (E,F) and (completely) Markovian semigroups
{Tt: t> 0} onL2(N,ω) areinone-to-onecorrespondencewiththoseweakly∗-continuous,
(completely) positive and contractive semigroups {St : t > 0} on the von Neumann
algebraN whicharemodularω-symmetric inthesensethat
ω(St(x)σ−i/2ω (y)) = ω(σω−i/2(x)St(y)) x, y∈ Nσω, t > 0 , (2.4)
throughtherelation
iω(St(x)) = Tt(iω(x)) x∈ N , t > 0 .
Relation(2.4)iscalledmodular symmetry anditisequivalentto
(Jωyξω|St(x)ξω) = (JωSt(y)ξω|xξω) x, y∈ N , t > 0 . (2.5)
Remark2.3.Incaseω isatrace,thesymmetricembeddingreducestoiω(x)= xξωwhile
themodularsymmetrysimplifies toω(St(x)y)= ω(xSt(y)) forx,y∈ N and t> 0.
To shorten notations, in the forthcoming part of the paper “Dirichlet form” will always mean “completely Dirichlet form” and “Markovian semigroup” will always mean “completely Markovian semigroup”.
Whenever no confusion can arise, the modular conjugation Jω will be sometime
de-noted J .
2.1. Examplesof Dirichletforms
Instancesof thenotions introduced abovemaybe found invarious frameworks. We just recall here some examples of different origins. One may consult the fundamental works[2],[22]forthecommutativecaseand[10],[9]forsurveysinthenoncommutative setting.
a) ThearchetypicalDirichletform ontheEuclidean spaceRn or,moregenerally,on any RiemannianmanifoldV ,endowedwithitsRiemannianmeasurem, istheDirichlet integral
E[a] =
V
|∇a|2dm a∈ L2(V, m) .
Inthis casethetraceonL∞(V,m) is givenbytheintegralwithrespectto themeasure
m andtheformdomainistheSobolevspaceH1(V )⊂ L2(V,m).Theassociated
Marko-vian semigroupis thefamiliar heat semigroupof theRiemannianmanifold. Interesting variationsoftheaboveDirichletintegralaretheDirichletforms oftype
E[a] :=
Rn
|∇a|2dμ a∈ L2(Rn, μ) ,
thatforsuitablechoicesofpositiveRadonmeasuresμ,aregroundstaterepresentations of HamiltonianoperatorsinQuantumMechanics.
b)Dirichletformsareafundamentaltooltointroducedifferentialcalculusandstudy Markovianstochasticprocessesonfractalsets (see[29],[15],[12,13]).
c)OnacountablediscretegroupΓ,anyconditionallynegativedefinitefunction gives
risetoaDirichletform
E[ξ] :=
s∈Γ
|ξ(s)|2(s) ,
on the Hilbert space l2(Γ), considered as the standard Hilbert space of the left von
NeumannalgebraL(Γ) generatedbytheleftregularrepresentationofΓ (see [14],[10]). Theassociated Markoviansemigroupissimplygivenbythemultiplicationoperator
Tt(a)(s) = e−t(s)a(s) t > 0 , s∈ G , a ∈ l2(Γ) .
d)OnnoncommutativetoriAθ,θ∈ [0,1] (see[18]),whichareC∗-algebrasgenerated by
two unitariesu andv,satisfyingtherelation
vu = e2iπθuv ,
theheat semigroup{Tt: t≥ 0} definedby
Tt(unvm) = e−t(n
2+m2)
unvm (n, m)∈ Z2,
is a τ -symmetric Markoviansemigroup on thevon Neumannalgebra Nθ generated by
theG.N.S.representationofthefaithful,tracialstateτ : Aθ→ C characterized by
e) There exists ageneralinterplay betweenDirichlet formsand differential calculuson tracialC∗-algebras(A,τ ) (see[34,35],[14])andthisprovidesasourceofDirichletforms onvonNeumannalgebras(generatedbyA intheG.N.S.representationofthetrace).In fact,denotingbyN thevonNeumannalgebrageneratedbytheG.N.S.representationof thetrace,if (∂,D(∂)) isadensely definedclosablederivation from L2(N,τ ) toHilbert
A-bimoduleH,then theclosure ofthequadraticform
E[a] := ∂a2
H a∈ F := D(∂)
isaDirichletformonL2(N,τ ).Viceversa,anyDirichletformonL2(N,τ ) whosedomain
isdense inA arisesin thisway from an essentiallyunique derivation onA canonically
associated with it (see [14]). Examples of this differential calculus can be found in all the situations illustrated above as well as inthe geometric framework of Riemannian foliations (see [36]) and also in the framework of Voiculescu’s Free Probability theory (see[39]).There,theDirichletformassociatedtoVoiculescu’sderivationpresentsseveral aspectsconnectedtoNoncommutativeHilbertTransform,FreeFischerInformationand FreeEntropy.
3. Amenabilityofσ-finitevon Neumannalgebras
Inthis section we relateacertain characteristic ofthe spectrumof aDirichlet form totheamenabilityofthevon Neumannalgebra.Recall thatavonNeumannalgebraN
issaidtobe amenable if,foreverynormaldualBanachN -bimoduleX,thederivations
δ : N→ X areallinner,i.e.theyhavetheform
δ(x) = xξ− ξx x∈ N
formsomevectorξ∈ X.Itisaremarkablefact,andthebyproductofatourdeforce,that this property is equivalent to several others of apparently completely different nature, such as hyperfiniteness, injectivity, semi-discreteness, Schwartz property P, Tomiyama property E.Wereferto [18,Ch. V]forareviewonthese connections.Amongthemain examples of amenable von Neumann algebras, we recall: the von Neumann algebra of a locally compact amenable group, the crossed product of an abelian von Neumann algebraby an amenablelocally compact group,the commutant von Neumannalgebra ofanycontinuousunitaryrepresentationofaconnectedlocallycompactgroup,thevon Neumannalgebragenerated byanyrepresentationofanuclearC∗-algebra.
3.1. Standardform ofthespatialtensor productof vonNeumann algebras
Here we summarize somewell known propertiesof thestandard form of thespatial tensor product of two von Neumann algebras in terms of Hilbert–Schmidt operators (details may be found in[38]), mainlywith the intention to make precise, inthe next
section, somepropertiesofthesymmetricembeddingofaproductstate.Moreprecisely we shallusethefollowing facts:
3.1.Let N ⊆ B(H) beavonNeumannalgebra.Avectorξ∈ H iscyclicforthecommutant N ifandonly if itisseparatingforN ;
3.2. Let Nk⊆ B(Hk) k = 1,2 bevon Neumannalgebras.If thevectors ξk ∈ Hk,k = 1,2
arecyclicforNk,thenthevectorξ1⊗ξ2∈ H1⊗H2iscyclicforthespatialtensorproduct
N1⊗N2;
3.3. Let Nk k = 1,2 be von Neumann algebras and L2(Nk) their standard forms. If
the vectors ξk ∈ L2+(Nk) k = 1,2 are cyclic forNk (hence separating) then the vector
ξ1⊗ξ2∈ L2(N1)⊗L2(N2) iscyclicandseparatingforthespatialtensorproductN1⊗N2.
3.2. Symmetricembeddingof tensor productofvon Neumannalgebras
Here we recall the definition and aproperty of the symmetric embedding of a von NeumannalgebrainitsstandardHilbertspace.LetN beaσ-finitevonNeumannalgebra and ω∈ N∗,+ afaithful,normalstate.
Inthestandardform(N,L2(N,ω),L2
+(N,ω)),wedenotebyξω∈ L2+(N,ω) thecyclic
vectorrepresentingthestateω andbyJωandΔω itsmodularconjugationandmodular
operator,respectively.
Thesymmetricembeddingiω: N → L2(N,ω),definedbyiω(x):= Δ
1 4
ωxξωforx∈ N,
is acompletely positivecontractionwithdenserange,whichisalsocontinuousbetween the weak∗-topology of N and the weak topology of L2(N,ω). It is also an order iso-morphism of completely ordered sets between {x = x∗ ∈ N : 0 ≤ x ≤ 1N} and
{ξ = Jωξ ∈ L2(N,ω) : 0 ≤ ξ ≤ ξω)} (see [1], [17], [26] and [4]). We shall make use
of thefollowing properties:
3.4.LetNk k = 1,2 bevonNeumannalgebrasandL2(Nk) theirstandardforms.Consider
thecyclic (hence separating)vectors ξk ∈ L2(Nk) k = 1,2 andthecyclicand separating
vector ξ1⊗ ξ2∈ H1⊗ H2 forthespatialtensor productN1⊗N2.
Let Jk,Δk be the modular conjugation and the modular operator associated to ξk ∈
Hk k = 1,2 and Jξ1⊗ξ2,Δξ1⊗ξ2 be the modular conjugation and the modular operator associated toξ1⊗ ξ2.Thenthefollowingidentificationsholdtrue
• Jξ1⊗ξ2= Jξ1⊗ Jξ2;
• N1ξ1 N2ξ2⊆ H1⊗ H2 isacorefortheclosed operatorΔ
1 2 ξ1⊗ξ2; • Δ12 ξ1⊗ξ2(η1⊗ η2)= Δ 1 2 ξ1(η1)⊗ Δ 1 2 ξ2(η2) forηk∈ Nkξk andk = 1,2.
We will denote by N◦ the opposite algebra of N : it coincides with N as a vector space but theproduct is taken inthe reverse order x◦y◦ := (yx)◦ for x◦,y◦ ∈ N◦. As
customary,weadopttheconventionthatelementsy∈ N,whenregardedas elementsof theoppositealgebraaredenotedbyy◦∈ N◦.
A linearfunctional ω onN , when consideredas alinearfunctional on theopposite algebraN◦ isdenotedbyω◦ andcalledtheopposite ofω.AsN andN◦share thesame positivecone,ifω ispositiveonN soisω◦onN◦andifω isnormalsodoesitsopposite. BythepropertiesofstandardformsofvonNeumannalgebras,itfollowsthatforthe standardform (N◦,L2(N◦,ω◦),L2
+(N◦,ω◦)) ofN◦ onehasthefollowingidentifications
L2(N◦, ω◦) = L2(N, ω) , L2+(N◦, ω◦) = L2+(N, ω) , Jω= Jω◦,
Δω◦= Δ−1ω , ξω◦ = ξω.
Using the isomorphism between N◦ and the commutant N, given by N◦ y◦ → Jωy∗Jω ∈ N, we canregard L2(N,ω) notonly as a left N -module but also as aleft
N◦-module,henceas arightN -moduleandfinally asaN -N -bimodule
y◦ξ := Jωy∗Jωξ , ξy := Jωy∗Jωξ , xξy := xJωy∗Jωξ x, y∈ N , ξ ∈ L2(N, ω) .
Thesymmetricembeddingsassociated toω andω◦ arerelatedby
iω◦(y◦) = Δ 1 4 ω◦(ξωy) = Δ 1 4 ω◦Jωy∗Jωξω= Δ− 1 4 ω Δ 1 2 ωyξω= Δ 1 4 ωyξω= iω(y) , Jω(iω(y∗)) = JωΔ 1 4 ω(y∗ξω) = JωΔ 1 4 ωJωΔ 1 2 ω(yξω) = Δ 1 4 ω(yξω) = iω(y) = iω◦(y◦) . 3.3. Coarsecorrespondence
Recall that a Hilbert–Schmidt operator T is abounded operator on L2(N,ω) such
thatTraceL2(N,ω)(T∗T )< +∞.It maybe representedas
T ξ :=
∞
k=0
μk(ηk|ξ)ξk ξ∈ L2(N, ω)
in terms of suitable orthonormal systems {ηk : k ∈ N}, {ξk : k ∈ N} ⊂ L2(N,ω)
and a sequence {μk : k ∈ N} ⊂ C such that ∞k=0|μk|2 < +∞. The set of
Hilbert–Schmidt operators HS(L2(N,ω)) is a Hilbert space under the scalar product (T1|T2):= TraceL2(N,ω)(T1∗T2).
Lemma3.5. The binormalrepresentationsπ1
co,πco2,π3co of N⊗maxN◦,characterized by
π1co: N⊗maxN◦→ B(HS (L2(N, τ )))
πco1(x⊗ yo)(T ) := xT y x, y∈ N , T ∈ HS (L2(N, τ )) ,
πco2 : N⊗maxN◦→ B(L2(N, τ )⊗ L2(N, τ ))
π3co: N⊗maxN◦→ B(L2(N, τ )⊗ L2(N, τ ))
π3co(x⊗ yo)(ξ⊗ η) := xξ ⊗ ηy x, y∈ N , ξ, η ∈ L2(N, τ ) ,
are unitarely equivalent by
U : L2(N, τ )⊗ L2(N, τ )→ L2(N, τ )⊗ L2(N, τ ) U (ξ⊗ η) := ξ ⊗ J ωη
V : L2(N, τ )⊗ L2(N, τ )→ HS(L2(N, τ )) V (ξ⊗ η)(ζ) := (η|ζ)ξ .
Theygive riseby weakclosure
(πco3(N⊗maxN◦))= N⊗N◦
of thespatialtensor productof N byitsoppositeN◦.
Lemma 3.6. The normal extension of the coarse representation πco of the C∗-algebra
N ⊗maxN◦ to thevon Neumann tensor product N⊗N◦ isthe standard representation
of N⊗N◦ (and itwillstilldenoted by thesamesymbol).
The standardpositiveconein thevariousequivalent representationsisdeterminedas
• HS(L2(N,ω))
+,theset ofallnonnegative Hilbert–Schmidtoperatorson L2(N,ω);
• (L2(N,τ )⊗ L2(N, ω))
+,generated by thevectors ξ⊗ ξ with ξ∈ L2(N,ω);
• (L2(N,τ )⊗ L2(N,ω))
+,generated by thevectors ξ⊗ Jωξ with ξ∈ L2(N,ω).
The standardHilbert spaceandthepositivecone ofN⊗N◦ will bedenoted also by L2(N⊗N◦, ω⊗ ω◦) , L2+(N⊗N◦, ω⊗ ω◦) .
Lemma 3.7. Let T : L2(N,ω) → L2(N,ω) be a bounded operator and consider on the
involutive algebra N N◦,thelinearfunctional determinedby
ΘT : N N◦→ C ΘT(x⊗ y◦) := (iω(y∗)|T iω(x)) x⊗ y◦∈ N N◦.
ThenΘT isapositivelinearfunctionalonN N◦ ifandonlyifT iscompletelypositive
(cf. Definition 2.2iv)).
Proof. i) The positive cone of N N◦ is generated by elements of type ν∗ν =
n
j,k=1x∗jxk ⊗ (yky∗j)◦ where ν =
n
k=1xk ⊗ y◦k ∈ N N◦. The result then follows
bytheidentity ΘT(ν∗ν) = n j,k=1 ΘT(x∗jxk⊗ (ykyj∗)◦) = n j,k=1 (iω(yjy∗k)|T iω(x∗jxk)) ,
the completelypositivityofthesymmetric embeddingiω: N → L2(N,τ ) and the
Lemma3.8.LetT : L2(N,ω)→ L2(N,ω) beacompletelypositiveoperatorandconsider thepositivelinearfunctional ΘT on N N◦.Then, amongtheproperties
a)ΘT is astateonN N◦
b)T isa contraction c)T ξω= ξω
wehave thatthefollowingrelations i)a)andb) implyc)andT = 1
ii)c) impliesa)and b).
Proof. i) By a) and b) we have 1 = ΘT(1N ⊗ 1N◦) = (ξω|T ξω) ≤ ξω· T ξω ≤
ξω2· T = 1 that implies T = T ξω = 1 and (ξω|T ξω) = ξω· T ξω which
provideT ξω= ξω. ii)Theproof thatc) impliesa) isimmediate whiletheproofthatc)
impliesb)canbe foundin[8]. 2
3.4. Spectral growthrate
Inthefollowingdefinition,thenotionofgrowthrateofafinitelygenerated,countable discretegroupisextendedtoσ-finitevonNeumannalgebrashavingtheHaagerup Prop-erty(H), i.e.von Neumann algebras admittingDirichlet forms withdiscrete spectrum. Theideaforthisgeneralizationresultsfrom[16](seediscussioninExample 3.11below). Definition 3.9 (Spectral growth rate of Dirichlet forms). Let (N,ω) be a σ-finite, von Neumannalgebrawithafixedfaithful,normalstateonit.Toavoidtrivialitiesweassume
N tobeinfinite dimensional.
Let(E,F) beaDirichletform onL2(N,ω) andlet(L,D(L)) be theassociated
non-negative, self-adjoint operator. Assumethat its spectrumσ(L) ={λk ≥ 0 : k ∈ N} is
discrete, i.e. its points are isolated eigenvalues of finite multiplicity (repeated in non-decreasing orderaccordingto theirmultiplicities).
Thenletusset
Λn:={k ∈ N : λk ∈ [0, n]} , βn:= (Λn) , n∈ N
anddefinethespectral growthrate of(E,F) as Ω(E, F) := lim sup
n∈N
n
βn.
TheDirichletform(E,F) issaidtohave
• exponential growth if(E,F) hasdiscretespectrumandΩ(E,F)> 1
• subexponentialgrowth if(E,F) hasdiscretespectrumandΩ(E,F)= 1
• polynomial growth if(E,F) hasdiscretespectrumand,forsomec,d> 0, βn≤ c· nd
foralln∈ N
Lemma 3.10.Setting γ0= β0 and γn:= βn− βn−1= {k ∈ N : λk ∈ (n − 1, n]} , n∈ N∗, and Ω(E, F) := lim sup n∈N∗ n √γ n we have Ω(E, F) = Ω(E, F) ≥ 1 .
Proof. On one hand, by definition, we have Ω(E,F) ≥ Ω(E,F). On the other hand, since,byassumption,N isinfinitedimensionalandσ(L) isdiscrete,wehaveΩ(E,F)≥ Ω(E,F) ≥ 1. Consider now the following identity involving analytic functions in a neighborhood of0∈ C ∞ n=0 βnzn= (1− z)−1 ∞ n=0 γnzn
and notice that the radius of convergence of the series on the left-hand side is R =
1/Ω(E,F), while theradius ofconvergence ofthe series ontheright-hand sideis R = 1/Ω(E,F) sothatR≤ R ≤ 1.Since(1− z)−1isanalyticintheopenunitdiskcentered inz = 0,theaboveidentityimpliesthatR≥ R sothatΩ(E,F)≤ Ω(E,F). 2 Example 3.11 (Spectral growth rate on countable discrete groups). i) On a countable discretegroupΓ,ifthere existsaproper,c.n.d.function,thentheassociated Dirichlet form (E,F) hasdiscretespectrumσ(L)={(g)∈ [0,+∞): g ∈ Γ}.
ii)Onafinitelygenerated,countablediscretegroupΓ,ifthelengthS corresponding
to afinitesystemofgeneratorsS⊆ Γ isnegativedefinite,thenthespectralgrowth rate Ω(ES,FS) ofthecorrespondingDirichletformcoincideswithgrowthrateof(Γ,S) (see [20, Ch. VI]).
iii) Moreover, if(Γ,S) has polynomialgrowth, it hasbeen shownin[16] thatthere exists onΓ aproper,c.n.d.function withpolynomialgrowth.Theassociated Dirichlet form (E,F) willhavepolynomialspectralgrowth rate.
Remark3.12.Byawellknownbound(see[32,Theorem 3.37]) 1≤ lim inf n βn+1 βn ≤ lim supn∈N n βn,
ifthespectralgrowth rateissubexponential,thenlim infn ββn+1n = 1 sothatthereexists
a subsequence of {βn+1
subspaces{En}n∈Ncorrespondingto theinterval[0,n]⊂ [0,+∞) admitsasubsequence suchthat lim k dim Enk+1 dim Enk = 1 .
Subexponential growth can be equivalently stated in terms of the nuclearity of the completelyMarkoviansemigroup{e−tL: t> 0} on L2(N,ω):
Lemma3.13.TheDirichletform(E,F) hasdiscretespectrumandsubexponentialspectral growthifandonlyiftheMarkoviansemigroup{e−tL: t> 0} onL2(N,ω) is nuclear,or trace-class,in thesense that:
Trace (e−tL) = k∈N e−tλk < +∞ t > 0 . Proof. Since γ0+ n∈N∗ γne−tn ≤ k∈N e−tλk ≤ γ 0+ et n∈N∗ γne−tn t > 0 ,
theseriesk∈Ne−tλk and
n∈N∗γne−tn convergeor divergesimultaneously.They
ob-viouslyconvergeforallt> 0 ifand onlyifΩ(E,F)≤ 1. 2
Example3.14. Ifon acountablediscrete groupΓ,there exists ac.n.d.function, such thatg∈Γe−t(g)< +∞ forall t> 0,then isproper,the spectrumoftheassociated Dirichlet form (E,F) coincides with {(g) ∈ [0,+∞) : g ∈ Γ} andit is thus discrete
withsubexponentialgrowth.
Thefollowingisthemainresultofthissection.
Theorem 3.15. Let (N,ω) be a σ-finite von Neumann algebra endowed with a normal, faithful stateon it. If there existsa Dirichlet form (E,F) on L2(N,ω) having
subexpo-nentialspectral growth,then N isamenable.
Proof. Recall that N is amenable if and only if the identity or standard bimodule
NL2(N )NisweaklycontainedinthecoarseorHilbert–SchmidtbimoduleHco (see[31]).
Considerthe completely positivesemigroup {Tt:= e−tL : t> 0} and assume,for
sim-plicity,thatthecyclicvectorisinvariant:Ttξω= ξωforallt> 0.Recall(cf.Lemma 3.7)
thatthecompletepositivityofTtprovidesabinormalstateonN⊗maxN◦characterized
by
Tocomputethisstate,weconsiderthespectralrepresentationTt=k≥0e−tλkPk
(con-vergingstrongly)intermsoftherank-oneprojectionsPkonL2(N,ω) associatedtoeach
eigenvalue λk (repeated according to their multiplicity). Notice thatby Markovianity,
the semigroupcommuteswith themodular conjugationJω so thateacheigenvector ξk
maybeassumed tobereal: ξk = Jωξk.Wethenhave
Φt(x⊗ y◦) = (iω(y∗)|Ttiω(x)) = ∞ k=0 e−tλk(i ω(y∗)|Pk(iω(x))) = ∞ k=0 e−tλk(i ω(y∗)|(ξk|iω(x))ξk) = ∞ k=0 e−tλk(ξ k|iω(x))(iω(y∗)|ξk) .
AstheseriesZt:=∞k=0e−tλkξk⊗ξkisnormconvergentforallt> 0 bythenuclearityof
thesemigroup,sinceJωisanantiunitaryoperatoronL2(N ),usingpropertiesinitem 3.4
abovewehave Φt(x⊗ y◦) = ∞ k=0 e−tλk(ξ k|iω(x))(Jωξk|Jωiω(y∗)) = ∞ k=0 e−tλk(ξ k|iω(x))(ξk|iω◦(y◦)) = ∞ k=0 e−tλk(ξ k⊗ ξk|iω(x)⊗ iω◦(y◦))L2(N,ω)⊗L2(N,ω) = ∞ k=0 e−tλkξ k⊗ ξkiω(x)⊗ iω◦(y◦) L2(N,ω)⊗L2(N,ω) = Ztiω⊗ω◦(x⊗ y◦) L2(N⊗N◦,ω⊗ω◦).
SincethesymmetricembeddingsofvonNeumannalgebrasarecontinuouswhenN⊗N◦
is endowedwiththe weak∗-topologyandL2(N⊗N◦,ω⊗ ω◦) isendowedwith theweak
topology,bycontinuitywehave Φt(z) = Ztiω⊗ω◦(z) L2(N⊗N◦,ω⊗ω◦) z∈ N⊗N ◦.
In other words, the linearfunctional Φtextends as a σ-weaklycontinuous linear
func-tionalonthespatialtensorproductN⊗N◦.ΦtbeingpositivebyLemma 3.7,thereexist
Φt(z) = (iω(y∗)|Ttiω(x)) = Ωt|πco(z)Ωt L2(N⊗N◦,ω⊗ω◦) z∈ N⊗N ◦
and the GNS representation of N ⊗max N◦ associated to Φt coincides with a
sub-representation of πco. Inother words, the N− N-correspondence Ht associated to the
completely positive map Tt is contained in the coarse N − N-correspondence Hco for
all t > 0. Since thesemigroup {Tt : t > 0} is strongly continuous on L2(N,ω), for all
x⊗ y◦∈ N ⊗maxN◦ wehave lim t↓0 Ωt|πco(x⊗ y◦)Ωt L2(N,ω)⊗L2(N,ω)= (iω(y ∗)|i ω(x))L2(N,ω) = (Δ14 ωy∗ξω|Δ 1 4 ωxξω) = (Δ 1 2 ωy∗ξω|xξω) = (Jωyξω|xξω) = (JωyJωξω|xξω) = (ξω|Jωy∗Jωxξω) = (ξω|xJωy∗Jωξω) = (ξω|xξωy) = (ξω|πid(x⊗ y◦)ξω) andbycontinuity lim t↓0 Ωt|πco(z)Ωt L2(N,ω)⊗L2(N,ω)= (ξω|πid(z)ξω) z∈ N ⊗maxN ◦.
ThisprovesthattheidentitycorrespondenceHidisweaklycontainedinthecoarse
corre-spondenceHco andthusN isamenableatleastifthesemigroupleavesthecyclicvector
invariant.Todealwiththegeneralcase,remarkfirstthat,bystrongcontinuity,wehave thatlimt↓0(ξω|Ttξω)=ξω2= 1 and there existt0> 0 such that(ξω|Ttξω)> 0 for all
0< t< t0.Applyingtheargumentabovetothebinormalstates
Φt(x⊗ y◦) :=
1 (ξω|Ttξω)
(iω(y∗)|Ttiω(x)) x⊗ y◦∈ N ⊗maxN◦, 0 < t < t0
wegettheamenabilityof N eveninthegeneralsituation. 2
Remark 3.16. i) The above result implies that if the von Neumann algebra N is not amenable, then any Dirichlet form (E,F) with respect to any normal, faithfulstate ω
hasexponential growth rateΩ(E,F)> 1, i.e.its sequence of eigenvalueshas exponen-tially growing distribution. ii) Conversely, it is an open question whether there exist amenablevonNeumannalgebrasonwhicheveryDirichletformhasexponentialgrowth. Theanalogywithdiscretegroupssuggeststhatthe answerislikelypositive.
Thefollowing oneisageneralizationofaresultofGuentner–Kaminker[25].
Corollary 3.17. LetΓ be acountable discretegroup, λ: Γ→ B(l2(Γ)) beits left regular representation, L(Γ) its associated von Neumann algebra and τ its trace state. If there existsaDirichletform(E,F) onL2(L(Γ),τ ) havingsubexponentialspectralgrowth, then
thegroup Γ isamenable.
Proof. Undertheassumptions,thegroupvonNeumannalgebraL(Γ) isamenablebythe abovetheorem.HencebyawellknownresultofA.Connes,thegroupΓ isamenable. 2 Example 3.18. (Free orthogonal quantum groups) On the von Neumann algebra
L∞(O+2,τ ) of the free orthogonal quantum group O+2 with respect to its Haar state
τ , ithasbeen constructedin[11] aDirichletform with anexplicitlycomputeddiscrete spectrum of polynomial growth (and spectral dimension d:= lim supnln βn
ln n = 3).
Ap-plying thetheoremaboveoneobtainsaproofoftheamenabilityofL∞(O+2,τ ),aresult whichhasbeen provedbyM.Brannan[3].
4. Relativeamenability ofinclusionsoffinitevonNeumannalgebras
Inthis sectionweextendthepreviousresulttotherelative amenability ofinclusions of finite von Neumann algebras B ⊆ N, as defined by S.Popa [31,30]. This extension is based on the properties of the relative tensor product of Hilbert bimodules and on thepropertiesofthebasicconstruction,whichwewillpresentlyrecall(see[7],[24],[28],
[37]).
4.1. Basicconstruction offinite inclusions
Let N be a von Neumann algebra admitting a normal faithful trace state τ and
1N ∈ B ⊆ N avonNeumannsubalgebrawiththesameidentity (see[7],[28],[31],[37]).
Recall that the relative tensor product L2(N,τ ) ⊗
B L2(N,τ ) over B of the
N -B-bimodule NL2(N,τ )B by the B-N -bimodule BL2(N,τ )N, constructed in [33], is
isomorphic, as anN − N-bimodule, to theN -N -correspondence HB associated to the
conditionalexpectationEB: N → N fromN ontoB.Thelatterbeinggeneratedbythe
GNS constructionappliedto thebinormalstate
ΦB : N⊗maxN◦→ C ΦB(x⊗ y◦) := τ (EB(x)y) .
According to S. Popa (see [31,30]), the inclusion B ⊆ N is said to be relatively amenable ifthestandardbimoduleNL2(N,τ )N isweaklyincludedintherelativecoarse
bimodule NL2(N,τ )⊗BL2(N ;τ )N.
LeteB be theorthogonal projectioninB(L2(N,τ )) fromL2(N,τ ) ontoL2(B,τ ) and
gen-eratedbyN andtheprojectioneB.Forexample,ifB =C1N thenN,B =B(L2(N,τ ))
andwhenB = N thenN,B = N .
Denotingbyξτ ∈ L2(N,τ ) thecyclicvectorrepresentingτ onehas
eB(xξτ) = EB(x)ξτ, eBxeB= EB(x)eB x∈ N .
Itcanbeshownthatanelement x∈ N commuteswith theprojectioneB ifandonlyif
x∈ B.Moreover, span(N eBN ) is weakly∗-dense inN,B andeBN,B eB = BeB.It
canbe shownthat
N, B = (JBJ)⊆ B(L2(N, τ ))
sothatN,B issemifinitesinceB isfinite.Inparticular,there existsauniquenormal, semifinitefaithfultraceTr characterizedby
Tr(xeBy) = τ (xy) x, y∈ N .
andthereexistsalsoauniqueN− N-bimodulemapΦ fromspan(N eBN ) intoN
satis-fying
Φ(xeBy) = xy x, y∈ N , Tr = τ◦ Φ .
The map Φ extends to a contraction between the N -N -bimodules L1(N,B ,Tr) and
L1(N,τ ) andsatisfies
eBX = eBΦ(eBX) X∈ N, B .
Moreover, Φ(eBX)∈ L2(N,B ,Tr) forall X ∈ N,B .These propertiesenableus to
provethattheidentity correspondenceL2(N,B ,Tr) ofthe algebraN,B reducesto
therelativecorrespondenceHB whenrestrictedtothesubalgebraN ⊆ N,B .
Thefollowingpropositioniswellknown,wegivetheproofforsakeofcompleteness. Proposition 4.1. The N -N -correspondences HB and L2(N,B ,Tr) are isomorphic. In
particular,thebinormalstateisgivenby
ΦB(x⊗ y◦) = (eB|xeBy)L2(N,B ,Tr) x, y∈ N sothat thecyclicvector representingthestateΦB is eB∈ L2(N,B ,Tr).
Proof. Letusconsider themapΨ definedonthedomain
D(Ψ) := span{[x ⊗ y◦]HB: x, y∈ N}
Ψ : D(Ψ)→ L2(N, B , Tr) Ψ([x⊗ y◦]HB) := xeBy x, y∈ N .
Here [x⊗ y◦]HB denotes the element of HB image of the elementary tensor product
x⊗ y◦, in the GNS construction of the state ΦB. The map is well defined because
eB2= Tr(eB)= τ (Φ(eB))= τ (1N)= 1 and xeBy2≤ x· y· eB2=x· y.
By thedefinitionof theHilbertspace HB,the map Ψ isdensely defined.For x,y ∈ N
we have
Ψ([x ⊗ y◦] HB)
2
2=xeBy22
= Tr((xeBy)∗(xeBy))
= Tr(y∗eBx∗xeBy)
= Tr(y∗eBeBx∗xeBeBy)
= Tr((eBx∗xeB)(eByy∗eB))
= Tr(EB(x∗x)eBEB(yy∗)eB)
= Tr(EB(x∗x)eBEB(yy∗))
= τ (EB(x∗x)EB(yy∗)) = τ (EB(EB(x∗x)yy∗)) = τ (EB(x∗x)yy∗) = ΦB(x∗x⊗ (yy∗)◦) = ΦB(x∗x⊗ (y∗)◦y◦) = ΦB((x∗⊗ (y∗)◦)(x⊗ y◦)) = ΦB((x⊗ y◦)∗(x⊗ y◦)) =[x ⊗ y◦]HB 2 HB.
Bypolarization,forall{xj}nj=1⊂ N wehavealso
(Ψ([xj⊗ y◦j]HB|Ψ([xk⊗ y◦k]HB)2= ([xj⊗ yj◦]HB|[xk⊗ y◦k]HB)HB j, k = 1 ,· · · , n .
Considernowν =nk=1[xk⊗ y◦k]HB ∈ D(Ψ) sothatν∗ν =
n j,k=1[x∗jxk⊗ (yky∗j)◦]HB∈ D(Ψ) and then Ψ(ν)2 2= n j,k=1 ([xj⊗ yj◦]HB|[xk⊗ y ◦ k]HB)HB = (ν|ν)HB=ν 2 HB.
Hencethe mapΨ extendsto anisometry fromHB intoL2(N,B ,Tr) whichisclearly
alsodenseinL2(N,B ,Tr).Bytheisometric propertywehavethatIm(Ψ) isclosedso thatΨ isasurjectiveisometry.Finally,forx,y∈ N wecompute
(eB|xeBy)L2(Tr)= Tr(eBxeBy)
= Tr(EB(x)eBy)
= τ (Φ(EB(x)eBy))
= τ (EB(x)y)
= ΦB(x⊗ y◦) . 2
Definition4.2(B-invariantDirichletforms).LetN beavonNeumannalgebraadmitting anormalfaithfultracialstateτ and 1N ∈ B ⊆ N avonNeumannsubalgebra.
ADirichletform(E,F) onL2(N,τ ) is saidto beaB-invariant if bF ⊆ F , E(bξ|ξ) = E(ξ|b∗ξ) b∈ B, ξ ∈ F
and
Fb ⊆ F , E(ξb|ξ) = E(ξb∗|ξ) b∈ B, ξ ∈ F .
Since, by definition, a Dirichlet form is J -real, the above two properties are in fact equivalent.
Intermsof theassociated nonnegative,self-adjointoperator(L,D(L)),B-invariance
meansthattheresolventfamily{(λ+ L)−1: λ> 0} isB-bimodularforsomeandhence allλ> 0
(λ + L)−1(bξ) = b((λ + L)−1ξ)
(λ + L)−1(ξb) = ((λ + L)−1ξ)b ξ∈ L2(N, τ ) , b∈ B ,
or that, alternatively, the semigroup {e−tL : t > 0} is for someand henceall t > 0 a B-bimodularmap
e−tL(bξ) = b(e−tLξ)
e−tL(ξb) = (e−tLξ)b ξ∈ L2(N, τ ) , b∈ B .
SincetheMarkovianityoftheDirichletformimpliesthatthesemigroupandtheresolvent commutewiththemodularconjugationJ ,wehavethattheB-invarianceoftheDirichlet formprovidesthatthesemigroupandtheresolventbelongtotherelativecommutantof
B inthebasicconstructionN,B :
Definition4.3(Relativediscretespectrum).Wesaythat(E,F) or(L,D(L)) havediscrete spectrum relative to the inclusion B ⊆ N if theMarkovian semigroup, or equivalently theresolvent,belongstothecompactidealspaceJ (N,B ) ([28],[30],[37])ofthebasic construction, generatedbyprojectionsinN,B havingfinitetrace:
e−tL∈ J (N, B ) for some and hence all t > 0 , (λ + L)−1∈ J (N, B ) for some and hence all λ > 0 .
Anotherwaytostateitisthatthespectrumof(L,D(L)) isadiscretesubsetofR+and
thateacheigenprojectionhasfiniteTr trace.
Definition4.4(RelativespectralgrowthrateofDirichletforms).LetN beavonNeumann algebra admitting a normal faithfultracial state τ and 1N ∈ B ⊆ N a von Neumann
subalgebra. ADirichletform(E,F) onL2(N,τ ) whichisB-invariant issaidtobe have • exponential spectralgrowthrelative toB⊆ N ifTr(e−tL)= +∞ forsomet> 0;
• subexponentialspectralgrowth relativetoB ⊆ N ifTr(e−tL)< +∞ forallt> 0.
Noticethat,ifTr(e−tL)< +∞ forsomet> 0, thene−tL∈ J (N,B ) sothat(L,D(L))
has discrete spectrum relative to the inclusion B ⊆ N. This applies, inparticular, to
B-invariant Dirichletforms(E,F) withsubexponentialspectral growthrelativetoB ⊆ N whichthushavenecessarilydiscretespectrumrelative totheinclusionB ⊆ N.
Remark4.5. Let EL be the spectral measure of the self-adjoint operatorL,D(L) . If
the Dirichlet form is B-invariant then EL takes its values in the class of projections of the von Neumann algebra N,B and we can consider the positive measure νBL := Tr◦ EL on [0,+∞), supported by the spectrum σ(L). In the framework of quantum
statistical mechanics, wherethe operatorL may representthetotal energyobservable, the measure νL
B acquires the meaning of “density of states” in the sense that νLB(Ω)
measures thenumber(relativelyto B)ofallowed energylevelslocated inameasurable subset Ω ⊂ σ(L). In this case the subspace L2(B,τ ) ⊂ L2(N,τ ) may represent the manifold ofgroundstatescorresponding totheminimalallowableenergylevel(seealso
Example 4.7below).ThemeasureμL
B(dλ):= λνB(dλ) hasthenthemeaningof“spectral
energydensity”inthesensethatμL
B(Ω) measurestheenergyofthesysteminasituation
where alltheallowedenergylevelsinΩ⊂ σ(L) areoccupied.
Thesubexponentialspectralgrowthcondition(relativelytoB⊆ N)canberephrased interms ofthe LaplaceTransform νˆBL sayingthatits abscissaof convergencevanishes. In this situationνˆL
B(β)= Tr (e−βL) is called thepartition function ofthe system, itis
defined for allβ > 0 andthe variable β isinterpreted as the inverse temperature. The subexponentialspectralgrowthconditionalsoallowstoconsiderthesocalledGibbs
nor-mal states,defined by Φβ(A):= Tr(Ae
−βL)
Tr(e−βL) , on the von Neumann (observable) algebra
prop-ertiesbywhichtheycanberegardedasequilibriaofthesystematthefixedvalueofthe inversetemperature. Finally, notice thatthesubexponential spectral growth condition isequivalenttotherequirementthatthemeanenergyΦβ(L):= Tr(Le
−βL)
Tr(e−βL) ofthesystem
isfiniteforany β > 0 (see[4]).
Thefollowingisthemainresultofthissection.
Theorem4.6.LetN beavonNeumannalgebraadmitting anormal faithfultracialstate τ and 1N ∈ B ⊆ N avonNeumann subalgebra.
Ifthere existsaB-invariant Dirichletform(E,F) onL2(N,τ ) having subexponential
spectralgrowth relativelytoB ⊆ N,thentheinclusionB ⊆ N isamenable.
Proof. Letuscheck firstthefollowingidentity
(T∗|xeBy)L2(Tr)= (iτ(y∗)|T (iτ(x))L2(τ )) T ∈ N, B ∩ L2(N, B , Tr) , x, y ∈ N .
Asspan(N eBN ) is weakly∗ denseinN,B ,it isenough toprovethe identity forT ∈
N eBN .IfT = ueBv forsomeu,v∈ N wehave
eByT xeB= eByueBvxeB = (eByueB)(eBvxeB) = EB(yu)eBEB(vx)eB
andthen
(T∗|xeBy)L2(Tr)= Tr(T xeBy)
= Tr(eByT xeB)
= τ (Φ(eByT xeB))
= τ (Φ(EB(yu)eBEB(vx)eB))
= τ (EB(yu)Φ(eBEB(vx)eB)) = τ (EB(yu)Φ(EB(vx)eB)) = τ (EB(yu)EB(vx)) = τ (EB(yuEB(vx))) = τ (yuEB(vx)) = (u∗y∗ξτ|EB(vx)ξτ)L2(τ ) = (iτ(y∗)|uEB(vx)ξτ)L2(τ ) = (iτ(y∗)|ueB(v(xξτ)))L2(τ ) = (iτ(y∗)|ueBv(iτ(x))L2(τ ) = (iτ(y∗)|T (iτ(x))L2(τ )
sothattheidentityholdstrue.Underthehypothesisofsubexponentialspectralgrowth, we have that Tt := e−tL ∈ L2(N,b ,Tr)∩ L1(N,b ,Tr) for all t > 0. Applying the
aboveidentity,wehavethatthebinormal states Φt: N ⊗maxN◦→ C Φt(x⊗ y◦) :=
1 (ξτ|Ttξτ)L2(N,τ )
(iτ(y∗)|Tt(iτ(x))L2(N,τ ),
well defined, bystrongcontinuityof thesemigroup,for t sufficientlycloseto zero,may be representedfort> 0 asΦt(x⊗ y◦)=(ξτ|Ttξτ1)
L2 (N,τ )(Tt|xeBy)L
2(N,b ,Tr).
By the identity above, Φt extends as a normal state on the von Neumann
al-gebra generated by the left and right representations of N in L2(N,B ,Tr). The
N -N -correspondence Ht generated by Φt is thus a sub-correspondence of a multiple
of the N -N -correspondence L2(N,B ,Tr). Since the semigroup {T
t : t > 0} strongly
convergestotheidentityoperatoronL2(N,τ ),weobtainthatthetrivialcorrespondence
from N toN isweaklycontainedintherelativecorrespondenceHB. 2
Example 4.7 (Minimal and maximal inclusions of a Dirichlet form). Let N be a von Neumann algebra and τ a normal, faithful,tracial state and let (E,F) be aDirichlet form onL2(N,τ ) withassociated self-adjointoperator(L,D(L)).
Assume that inf σ(L) = 0 and that this is an eigenvalue (not necessarily of finite multiplicity). ThespectralprojectionP0ontotheeigenspacecorrespondingtotheBorel
subset{0}⊂ [0,+∞) canberepresentedasthestronglimitP0= limt→+∞e−tL.Hence
P0isacompletelyMarkovianprojection,sothatthereexistsavonNeumannsubalgebra
Bmin ⊆ N such that P0 = eBmin. Obviously the associated Markovian semigroup is Bmin-bimodularandtheDirichletformisBmin-invariant.
Alternatively, one can consider the inclusion Bmax ⊂ N where Bmax := {Tt : t >
0}∩ N istherelativecommutantoftheMarkoviansemigroupinN .Noticethatbythe Spectral TheoremBmax={Tt}∩ N forallt > 0.Obviouslytheassociated Markovian
semigroupisBmax-bimodularandtheDirichletformisBmax-invariant.
Proposition 4.8. Let(N,τ ) be afinitevon Neumannalgebra withfaithful,normal trace. Let (E,F ) a Dirichlet form on L2(N,τ ) with generator (L,D(L)) having pure point spectrum made by distinct, isolated eigenvalues σ(L) := {λ0 < λ1 < λ2 < · · · } and
assume λ0:= inf σ(L)= 0.
Then (E,F ) has discretespectrum relative to Bmin (resp. Bmax) if and only if each
eigenspace Eλ ⊂ L2(N,τ ), λ∈ σ(L), has finite coupling constant dimBmin(Eλ)< +∞ (resp. dimBmax(Eλ)< +∞) relativetoBmin (resp.Bmax).
Remarkthatthefinite couplingconstantdimBmin(Eλ)< +∞ (resp.dimBmax(Eλ)<
+∞)relative to Bmin (resp. Bmax) is well definedforany eigenvalueλ∈ σ(L) because
anyeigenspaceEλisobviouslyaleft(andalsoright)Bmin-module(resp.Bmax-module).
Wereferto[24,Section 3.2]forthedefinitionandpropertiesoftheMurray–vonNeumann coupling constant.
Example4.9. Let K < Γ beaninclusion ofcountable, discrete groupsandlet L(K)⊂ L(Γ) be the inclusion of the finite von Neumann algebras generated by K and Γ, re-spectively. Their standard spaces coincide with l2(K) and l2(Γ) respectively and the
projectioneL(K) coincideswiththeprojectionfrom l2(Γ) ontoitssubspacel2(K).
Let : Γ → [0,+∞) be a c.n.d. function. The Dirichlet form (E,F) associated
to (introduced in Section 3.3) is L(K)-invariant if and only if vanishes on K or,
equivalently,if isarightK-invariantfunction.Inthis situationwehave:
Proposition 4.10.LetΓ be a countable, discretegroup andlet L(Γ) beits left von Neu-mann algebras. Let : Γ → [0,+∞) be a c.n.d. function and (E,F) the associated
Dirichlet form. Denote by H :={s∈ Γ: (s) = 0} the subgroup where vanishes. We thenhave
i)Bmin= Bmax= L(H);
ii)If K isasubgroupof G, then(E,F) isL(K)-invariantifandonly ifK < H.In
this case:
ii.a) isL(K)-biinvariant
ii.b)(E,F) hasdiscretespectrumrelativetoL(K)⊂ L(Γ) ifandonlyifthefunction
G/K: G/K→ [0, +∞) G/K( s) := (s)
definedfor s = sK ∈ G/K,isproper.
ii.c)If,forany t> 0, s∈G/Ke−tG/K( s)< +∞,then theinclusionL(K)⊂ L(G) is
amenable.
Proof. i)Letx=t∈Γx(t)λ(t)∈ Bmax={Tt: t> 0}∩ L(Γ).Wethen have
xδe= x(I + L)−1δe= (I + L)−1xδe , which implies 0 = L(xδe) = L( t∈Γx(t)λ(t)δe) = L( t∈Γx(t)δt) = t∈Γx(t)(t)δt.
Sothatx(t)(t)= 0 forall t∈ Γ whichinturn implies x∈ L(H) = Bmin.The reverse
inclusionisobvious.
ii) follows from the arguments of the example above. For ii) b) just notice that λ(s)eL(K)λ(s)−1 is the orthogonal projection PsK onto the subspace l2(sH).
Hence the eigenspace Eλ corresponding to the eigenvalue λ ∈ σ(L) is given by
s∈G/K , (s)=λl2(sK).Hence,L willhavediscrete spectrumrelativeto K ifand only
if each of these sums is finite (i.e. for all λ) and the set of values of is discrete, i.e.
−1({λ})/K isfiniteinG/K,i.e.G/K : G/K→ [0,+∞) isproper.
For ii.c), note that in the basic construction for B = L(K) ⊂ N = L(G), PsK =
λ(s)eBλ(s) belongsto N,B andhastrace1.HenceT r(e−tL)= s∈G/Ket(s)for all
t> 0. 2
Remark 4.11. i) If the Dirichlet form (E,F) has discrete spectrum relative to L(K)
a) thefunction G/H : G/H→ [0,+∞) is proper and(E,F) hasdiscrete spectrum
relative toL(H) ;
b) G/K being left K-invariant, hence constant on left K-cosets, and proper, left
K-cosets inG/K must befinitesets. Inotherwords,K isquasi-normalinG.
ii) On the other hand, if (E,F) has discrete spectrum relative to L(H) then the
functionG/K: G/K→ [0,+∞) willbe constantontotheright H-cosetinG/K.Thus
(E,F) hasdiscretespectrumrelativetoL(K) ifandonlyifeachrightH-cosetinG/K
is afinite unionofK-cosets, which happensifand onlyifK hasfinite index inH,i.e. when thehomogeneousspaceH/K is finite.
5. Aspectralapproachtotherelative Haagerupproperty
As alreadymentioned in the Introduction, ina recentwork [5], M. Caspers and A. Skalski characterized von Neumannalgebras having Property(H) intermsof the exis-tenceofaDirichletformwithdiscretespectrum.Inthespiritoftheprevioussection,we extendtheirresulttorelative property (H),asdefinedbyS.Popa[31,30],forinclusions of von Neumann algebras,using a completely different approach. We will make use of thefollowing wellknownproperties:
5.1. Let(N,τ ) beavonNeumann algebra endowedwith anormal,faithful trace andlet ϕ: N → N beacompletelypositive, normalcontraction suchthat τ◦ ϕ≤ τ.Then
i) thereexistsacontraction Tϕ∈ B(L2(N,τ )) characterizedby
Tϕ(xξτ) = ϕ(x)ξτ x∈ N ;
ii) there existsacompletelypositive,normal contractionϕ∗: N → N such that Tϕ∗= (Tϕ)∗
or,more explicitly,
(ϕ∗(y)ξτ|xξτ) = (yξτ|ϕ(x)ξτ) x, y∈ N .
Definition 5.2. ([31,30]) Let N be a finite von Neumann algebra and B ⊆ N a von Neumannsubalgebra.ThenN is saidto haveProperty(H) relativetoB ifthere exista normal,faithfultracialstateτ onN andanet{ϕi: i∈ I} ofnormalcompletelypositive,
B-bimodular mapsonN satisfyingtheconditions i) τ◦ ϕi≤ τ
ii) Tϕi ∈ J (N,B )
Inthis definitionJ (N,B ) is thecompact idealspace, i.e. thenorm closed ideal gen-eratedbyprojectionswithfinitetraceinN,B andTϕi istheoperatordefinedinitem
5.1above.
Byaremark ofS.Popa[30],themapsϕiinthedefinitionabovecanbe chosentobe
contractions. In the following we shall always assume this property for approximating netsoftheidentitymapofavonNeumannalgebra.
Theorem5.3.LetN beafinitevonNeumannalgebrawithcountablydecomposablecenter andfaithfultracialstateτ .LetB ⊆ N beasub-vonNeumannalgebrasuchthatL2(N,τ ),
asB-module,admits acountablebase. Thenthefollowingpropertiesareequivalent
i) N hasProperty (H)relative toB
ii) thereexistsaB-invariant Dirichlet form(E,F) onL2(N,τ ) with discretespectrum
relative toB.
Proof. Assumethatthereexists aDirichletform(E,F) onL2(N,τ ) withdiscrete
spec-trumrelative to B. Hence, the associated generator(L,D(L)) has its resolvent in the compact idealspace: (λ+ L)−1 ∈ J (N,B ).Then for all λ> 0,Sλ := λ(λ+ L)−1 ∈
J (N,B ).Moreover, anySλ is MarkovianonL2(N,τ ) which impliesthatthereexists
acompletely positive contraction ϕλ : N → N determined by Sλ(xξτ) = ϕλ(x)ξτ for
x∈ N.Since theSλ areself-adjoint onL2(N,τ ),theϕλ aresymmetric withrespectto
thetrace:τ (ϕλ(x)y)= τ (xϕλ(y)) for allx,y∈ N.Thisimpliesthat
τ (ϕλ(x)) = τ (ϕλ(1N)x)≤ τ(x) x∈ N+.
Lastconditioniii)inDefinition 5.2abovecomesfromthestrongcontinuityofthe resol-vent:
lim
λ→+∞ξ − Sλξ2= 0 ξ∈ L
2(N, τ ) .
Thetheorem isprovedinthe“if”direction.Inthereversedirection,letussuppose that
B ⊆ N is an inclusion with relative property (H) and that L2(N,τ ) is separable as
B-module.Let{ϕn: n∈ N} beasequenceofnormal,completelypositive,B-bimodular
contractions of N , satisfying the conditions of the definition above. By [30, Proposi-tion 2.2] such a sequence always exists. Each ϕn extends by Tn(xξτ) := ϕn(x)ξτ to
a B-bimodular contraction Tn of L2(N,τ ), which belongs to the compact ideal space
J (N,B ). It is also completely positive with respect to the standard positive cone
L2
+(N,τ ) and itsmatrixamplifications.It iseasytocheck thatthemapsϕ∗n appearing
in item 5.1 above, have the same properties as the ϕn’s in definition above.
Replac-ing each ϕn by (ϕn+ ϕ∗n)/2, we can suppose, without loss of generality, that the ϕn
are symmetricwith respect to τ so thatthe corresponding Tn are completely positive,
