The existence problem for dynamics of dissipative systems in quantum probability

Motivated by existence problems for dissipative systems arising naturally in lattice models from quantum statistical mechanics, we consider the following $C^{\ast}$-algebraic setting: A given hermitian dissipative mapping $\delta$ is densely defined in a u

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aTheexistenceproblemfordynamicsofdissipativesystemsinquantumprobabilityPalleE.T.Jorgensen DepartmentofMathematics,TheUniversityofIowa,14MacLeanHall,IowaCity,IA52242-1419U.S.A.(Dated:February1,2008)AbstractMotivatedbyexistenceproblemsfordissipativesystemsarisingnaturallyinlatticemodelsfromquantumstatisticalmechanics,weconsiderthefollowingC -algebraicsetting:Agivenhermitiandissipativemappingδisdenselyde nedinaunitalC -algebraA.TheidentityelementinAisalsointhedomainofδ.Completelydissipativemapsδarede nedbytherequirementthattheinducedmaps,(aij)→(δ(aij)),aredissipativeonthenbyncomplexmatricesoverAforalln.Weestablishtheexistenceofdi erenttypesofmaximalextensionsofcompletelydissipativemaps.IftheenvelopingvonNeumannalgebraofAisinjective,weshowtheexistenceofanextensionofδwhichisthein nitesimalgeneratorofaquantumdynamicalsemigroupofcompletelypositivemapsinthevonNeumannalgebra.Ifδisagivenwell-behaved -derivation,thenweshowthateachofthemaps±δiscompletelydissipative.PACSnumbers:02.,02.10.Hh,02.30.Tb,03.65.-w,05.30.-dKeywords:spinsystems,dynamics,evolutionsemigroup,dissipative,C -algebra,noncommutativeproba-bility

Motivated by existence problems for dissipative systems arising naturally in lattice models from quantum statistical mechanics, we consider the following $C^{\ast}$-algebraic setting: A given hermitian dissipative mapping $\delta$ is densely defined in a u

I.INTRODUCTIONRecentapplicationsoftheoperator-theoreticapproachtodissipativequantumsystemsinclude[22]and[37].Foramoresystematicapproach,see[36].Supposewearegivenaone-parametergroupofautomorphismsαt:a→eitHae itHwhichactsonsomesetofobservablesa,speci edasadense“local”subalgebraofacompletedC -algebra.Ifwethendi erentiateatt=0,wegetthederivationδ:a→i[H,a]=i(Ha aH)whichtakestheformofaformalcommutator.TheissueiscomplicatedbythefactthattheHamiltonianHistypicallyanunboundedoperatorinstatisticalmodels,sayin nitelatticespinsystems.Inapplications,itisHthatisgiven,andtheprocessmustberuninreverse.Byanalogytoboundaryvalueproblemsfrompartialdi erentialequations,wethenexpecttoencounteranexistenceproblemforreconstructingthedynamicsofthesystemfromknowingonlyaformulaforH.

WeadopttheC -W -formalismforthedynamicsofin nitequantumsystems[10,14,18,20,21,24,31,33].ForthespecialcaseofquantumspinsystemsitisbelievedthatthedynamicsinthetimereversiblecaseisgivenbyanunboundedderivationofasuitablealgebraAofobservables[31].Dependingontherangeoftheinteraction,andthenumberofdimensionsofthespinlattice,itispossibletoexponentiatethein nitesimalderivationtoaone-parametergroupofautomorphismsαt( ∞<t<∞)ofA,oroftheenvelopingW -algebraA′′(see[35]),ortheW -algebrageneratedbyagiveninvariantstate[10,17,26,29,32,33].

Itisknownthat(open)irreversiblesystemsmaybeobtainedasrestrictionsoftime-reversiblesystems,anditfollows[20]thatthedynamicsoftheopensystemisgivenmathe-maticallybyasemigroupτt(0≤t<∞)ofcompletelypositivemappingsoftheC -algebraA,orW -algebraA′′.Thecorrespondingin pletelypositivesemigroupsalsoplayaroleinquantumcomputingalgorithms[27].Thephilosophyisthatnoiseinthequantumprocessesdictatesthedissipativesystems,asopposedtotheconservativeones(whicharegovernedbyone-parametergroupsofautomorphisms).Butinhighlattice-dimensions,orforlong-rangeinteraction,therearedi cultiesinexpo-nentiatingthein nitesimalgenerators.Thedeterminationofthetime-evolutionsαt(resp.,τt)seemstorequire“extraboundaryconditions”[7,10,26,32,33].Itisthereforeamean-ingfulfoundationalquestion,foragivencompletelydissipativein nitesimaltransformation

Motivated by existence problems for dissipative systems arising naturally in lattice models from quantum statistical mechanics, we consider the following $C^{\ast}$-algebraic setting: A given hermitian dissipative mapping $\delta$ is densely defined in a u

whichδinaC -algebraA,toaskifitisalwayspossibletoextendδtoatransformationδ

isthein nitesimalgeneratorforaquantumdynamicalsemigroup.Undertheassumptionthatδishermitian,andthattheW -algebraA′′isinjective,weestablishtheexistenceof

.OurextensionisthusanalgebraicparalleltoFriederichs’sexten-ageneratorextensionδ

sionforsemiboundedoperatorsinHilbertspace,orananalogueofPhillips’s[30]maximaldissipativeextensionofthegeneraldissipativeoperatorinHilbertspace.

Inearlierarticles[7,10,28,31]theuniquenessproblemwasconsideredforthegenerator

.But,justasisthecaseforoperatorsinHilbertspace(Friedrichs,Phillips),extension,δ δ

theextensionisgenerallynotunique,re ectingthepossibilityofdi erent“boundarycon-ditions”atin nity.

Wereferthereadertothebooks[14],[18]and[33]fordetailsonthemathematicalfoundationsofalgebraicquantumtheory.

Theissuescenteringaroundtheexistenceproblemforthedynamicalone-parametergroups,orsemigroups,ofquantumstatisticalmechanicsareperhapsbestknowninthesetupofquantumspinsystems,astheyaretreatedin[9],[25]and[34].

ExampleI.1.Themathematicalframeworkisrathergeneralsuchastoallowawideva-rietyofapplications,includingrecentonestononequilibriumstatisticalmechanics[34].Acountablyin nitesetL(sayalattice;itmaybeZνwhereνisthelatticerank,ordimension)isspeci edattheoutset.Pointss∈Laresitesatwhichquantumspinsarelocated.Foreachs∈L,letHsbea nite-dimensionalcomplexHilbertspace,i.e.,thespinvectorsatsites;andfora nitesubsetΛ L,set

HΛ:=

s∈ΛHs.

ThenletAΛbethe -algebraofall(bounded)operatorsonHΛ.Withthenaturalembedding

AΛ1 AΛ2

givenby

AΛ1 →AΛ1 1Λ2\Λ1 AΛ2,

wegettheusualinductivelimitC -algebralimΛAΛ=:A.AfunctionΛ→Φ(Λ)=Φ(Λ) ∈AΛde nedonthe nitesubsetsΛofLiscalledaninteraction,and

HΦ(Λ)=Φ(X)

X ΛforΛ1 Λ2(I.1)

Motivated by existence problems for dissipative systems arising naturally in lattice models from quantum statistical mechanics, we consider the following $C^{\ast}$-algebraic setting: A given hermitian dissipative mapping $\delta$ is densely defined in a u

istheassociatedlocalHamiltonian,wherein(I.1),thesummationisoverall nitesubsetsXofΛ.SinceAΛ1andAΛ2commutewhenΛ1∩Λ2= ,itfollowsthat

δ(a)=lim[H(Λ),a]Λ(I.2)

iswellde nedforalllocalobservablesainthedense -subalgebra

A0=

Λ nAΛinA

where[·,·]in(I.2)denotestheusualcommutator[b,a]:=ba ab.Ruelleprovedthat,ifΦistranslationallyinvariant,andif,forsomeλ>0,

∞

n=0enλsups∈L s∈X ncardX=n+1 Φ(X) <∞,(I.3)

thenthe -derivationδde nedin(I.2)isthein nitesimalgeneratorofaone-parametersubgroupof -automorphisms{αt}t∈R Aut(A),whichthensatis es

αt(a)=limeitH(Λ)ae itH(Λ)

Λ L(I.4)

foralla∈Aandt∈R,i.e.,itisapproximatelyinner.Thismeansthat,ifa∈A0,then

limt 1(αt(a) a)=δ(a).t→0t=0(I.5)

Moreover,δis,whenextendedfromA0,aclosed -derivation,inthesensethatthegraphofδisclosedinA×A.ButifΦisnottranslationallyinvariant,orif(I.3)isnotknowntohold,thennosuchconclusioniswithinreach,andtheissueofextensionsofδarises.We

ofδtoageneratorofaone-parametergroupofautomorphisms,thenaskifsomeextensionδ

orasemigroupofdissipations(seedetailsbelow),exists.

II.DEFINITIONSANDTERMINOLOGY

LetXandYbeBanachspaces.ThenthespaceofboundedlinearoperatorsfromXtoYisdenotedL(X,Y).Theconjugate(i.e.,dual)BanachspacetoXisL(X,C),andisdenotedX′.IfHisaHilbertspace,theC -algebraofallboundedoperatorsonHisdenotedB(H).LetLbealinearsubspaceofB(H)whichisself-adjointandcontainstheidentityoperatorI.WiththeorderinheritedfromB(H),thesubspaceLgetsthestructureofan

Motivated by existence problems for dissipative systems arising naturally in lattice models from quantum statistical mechanics, we consider the following $C^{\ast}$-algebraic setting: A given hermitian dissipative mapping $\delta$ is densely defined in a u

operatorsystem,intheterminologyofE ros[11].ThefullmatrixalgebraMnofallcomplexn-by-nmatricesisalsoanoperatorsystem,andsoisLn=L Mn.TheelementsinLnmayberealizedasn-by-nmatriceswithentriesfromL,(aij)ni,j=1,aij∈L.IfLandRareoperatorsystemsand :L→Risalinearmapping,thentheinducedmap(aij)→( (aij))ofLnintoRnisdenoted n.Itis,infact, idn.Wesay[2]that iscompletelypositive(resp.,completelycontractive)if nispositive(resp.,contractive)foralln.WesaythatRisinjectiveifforeverypairofoperatorsystems,L L1,andeverycompletelypositivemap :L→R,thereisacompletelypositiveextensionψ:L1→R.Thatis,ψ(x)= (x)forallx∈L.IfRisavonNeumannalgebrainaHilbertspaceH,itisknown[13,38]thatRisinjectivei thereisanorm-oneprojectionofB(H)ontoR.

IfAisaC -algebra,itisknown[12]thatAisnucleari thedoubleconjugate(dual)A′′isinjectiveasaW -algebra.Connesshowed[13]thatafactorRonaseparableHilbertspaceisinjectivei itismatricial.

III.DISSIPATIVETRANSFORMATIONS

AnoperatorδinaBanachspaceXissaidtobedissipative[29]ifoneofthefollowingthreeequivalentconditionsissatis ed:

(i)ForallxinthedomainD(δ)ofδ,thereisanelementf∈X′,dependingonx,such

that f =1,f(x)= x ,andRef(δ(x))≤0.

(ii)ForallxinD(δ),andallf∈X′satisfying f =1,andf(x)= x ,theinequality

Ref(δ(x))≤0isvalid.

(iii)ForallxinD(δ),andallα∈R+,theinequality x αδ(x) ≥ x holds.

Theproofoftheequivalencecanbefound,forexample,in[7],buttheequivalencecanalsobeshowntobeaconsequenceoftheapproximationideainSectionIXandPropositionX.2inthepresentpaper.

IfXisanoperatorsystem,wesaythatδiscompletelydissipativeiftheinducedmappingδninXnisdissipativeforalln=1,2,....RecallthatXn=X Mn,andδn:(xij)→(δ(xij)),withdomainD(δn)={(xij)∈Xn:xij∈D(δ)}.

FinallywesaythatthetransformationδishermitianifthedomainD(δ),intheoperatorsystemX,isinvariantunderthe -involutionofX,andifδ(x )=δ(x) forallx∈D(δ).

Motivated by existence problems for dissipative systems arising naturally in lattice models from quantum statistical mechanics, we consider the following $C^{\ast}$-algebraic setting: A given hermitian dissipative mapping $\delta$ is densely defined in a u

Ifδ:X→YismerelyalineartransformationbetweenBanachspacesXandY,withdomainD(δ)denseinX,thenthetransposed(orconjugate)transformationδ′iswellde nedasalineartransformationδ′:Y′→X′withdomainD(δ′)={f∈Y′: g∈X′s.t.f(δ(x))=g(x)forallx∈D(δ)}.Forf∈D(δ′),δ′(f)=g.ThedomainD(δ′)isweak*-denseinY′i δisclosable.Itisknown[29]thatdissipativeoperatorsareclosable.

PLETELYPOSITIVESEMIGROUPS(QUANTUMDYNAMICALSEMIGROUPS)

LetMbeaW -algebrawithpredualM .LetτtbeafamilyofcompletelypositivemappingsofMintoitself,indexedbythetimeparametert∈[0,∞).Assumethatτ0istheidentitytransformationinM,andthatτt(11)=11forallt∈[0,∞),where11denotestheunitelementoftheW -algebraMinquestion.Weassumefurtherthatthesemigrouplawholds,τt1+t2=τt1 τt2fort1,t2∈[0,∞),and nallythateachτtisanormalmappinginM.Recallthatnormalityisequivalenttotherequirementthattheconjugatesemigroupτt′[16]ofM′leavesinvariantthesubspaceM .Finallywerequirecontinuityofeachscalarfunction,t→ (τt(a)),forall ∈M anda∈M.Asemigroupwhichsatis esalltherequirementsaboveiscalledacompletelypositivesemigroup.Becauseoftherelevancetoquantumdynamics,weshallalsocallitaquantumdynamicalsemigroup[21].

Thein nitesimalgeneratorofagivencompletelypositivesemigroup(τt,M)isa,generallyunbounded,transformation,denotedbyζ,inM.Thedomainofthegeneratorζisgivenby

D(ζ)={a∈M: b∈Ms.t.forallt,τt(a) a= t

0τs(b)ds}.

Byde nitionζ(a)=b.Itiseasytosee[16]thatζ(a)=d

Motivated by existence problems for dissipative systems arising naturally in lattice models from quantum statistical mechanics, we consider the following $C^{\ast}$-algebraic setting: A given hermitian dissipative mapping $\delta$ is densely defined in a u

elements{uξ|ξ∈H\{0}}suchthat

i

uξuη=e

Motivated by existence problems for dissipative systems arising naturally in lattice models from quantum statistical mechanics, we consider the following $C^{\ast}$-algebraic setting: A given hermitian dissipative mapping $\delta$ is densely defined in a u

ThismeansthatthelinearspaceS=Ran(I δ)={x δ(x):x∈D(δ)}isclosedin

A.Inviewofthe(hermitian)assumptiononδwenotethatSisalsoselfadjoint,andthat11∈S.TheoperatorR:S→Ade nedbyx δ(x)→x,anddenotedby(I δ) 1,iscompletelypositive[2,Prop.1.2.8].ClearlyR(11)=11.

WenowconsiderthedoubledualtoA,denotedbyA′′,asaW -algebraM,andmaketheappropriateidenti cation(viatheuniversal -representationforA)suchthatAisregardedasaC -subalgebraofA′′,andthepre-dualofA′′isidenti edwiththedualA′ofA.(Thereaderisreferredto[35,§1.17,p.42]fordetails.)SinceM=A′′(withtheArensmultiplication)isinjectiveasaW -algebra,bytheassumption,itfollowsthatacompletelypositiveextensionmappingE:M→Mexists.IfweregardAasasubalgebraofM(asweshall),thentheextensionpropertyisgivenbytheidentity

R(s)=E(s)foralls∈S.(V.1)

NotethatS A,sothatSbecomesasubspaceofMwiththeabovementionedidenti ca-tion.

ThecompletelypositivetransformationsofMintoitselfwillbedenotedbyCP(M),andthespaceL(M)ofcompletelyboundedlineartransformationsinMgetsanorderingarisingfromtheconeCP(M).Indeed,forF∈L(M)wede neE≤FbytherequirementthatF E∈CP(M).AmongalltheparticularextensionsFofR,F∈L(M),suchthatE≤F,wechoosebyZornamaximalelementF0.(ForthebasicfactsontopologiesonCP(M)whichareneeded,thereaderisreferredto[2,Ch.1].)

ThisextensionF0,describedabove,hasthespecialpropertyofbeing1–1.We rstconsidertherestrictionofF0tothepositiveelementsinM,M+,thatis.Moreprecisely,wehavetheimplication:

x∈M+,F0(x)=0= x=0.(V.2)

Letη:M→M/Sbethecanonicallinearquotientmapping,andconsidertheconeCinthenormedquotientspaceE=M/SgivenbyC=η(M+).

Iftheelementxin(V.2)belongstoS,thentheconditionsR(x)=F0(x)=0implyx=0,sinceR=(I δ) 1.Hence,weshallassumethatxisnotinS.Thismeansthatη(x)∈Cde nesaone-dimensionalsubspace{kη(x):k∈C}inE,andthefunctionalf:kη(x)→kisnonzeroandpositive.ByKrein’stheorem[1,Thm.1,Ch.3,p.157]fextendstoapositive

Motivated by existence problems for dissipative systems arising naturally in lattice models from quantum statistical mechanics, we consider the following $C^{\ast}$-algebraic setting: A given hermitian dissipative mapping $\delta$ is densely defined in a u

onE,andwemayde nefunctionalf

(η(y))1F1(y)=F0(y)+f1fory∈M.(V.3)

WeclaimthatF1isoneoftheextensionsconsideredintheZorn-processwhichwasdescribedabove.ButF0≤F1,andF0=F1,contradictingthemaximalityofF0—andso,wemusthavex=0,concludingtheproofof(V.2).(Notethatin(V.3),insteadoftheidentityelement11ontheright-handsideoftheequation,wecouldhaveusedanynonzeroelementinM+.ThecorrespondingF1-transformationwouldproperlymajorizeF0,andhaveitsrangecontainedinM,sincetherangeofF0fallsinM.)

SinceF0iscompletelypositive,wehave,inparticular,F0(x )=F0(x) .So,toestablishtheidentityN(F0)={x∈M:F0(x)=0}=0,itisenoughtoshowthatthehermitianpartofN(F0)iszero.Sincewehavealreadyconsideredpositiveelements,itonlyremainstoconsiderx=x ∈N(F0)satisfyingx∈/S.Chooseapositiverealnumberksuchthatxk=x+k11∈M+.WethenhaveF0(xk)=kandxk∈/S.Itispossible,therefore,by

onE=M/Ssatisfyingf (η(xk))=l>0.Krein’stheorem,tochooseapositivefunctionalf

(η(y))1Thende neF2(y)=F0(y)+f1fory∈M.Itisasimplemattertocheckthat

ischosenpositive.FinallyF2isoneoftheZorn-extensions.Indeed,F0≤F2sincef

F2(xk)=F0(xk)+l11>F0(xk).ThiscontradictiontothemaximalityofF0concludesthe

1isde nedonF0(M)={F0(x):x∈M}.proof.SinceN(F0)=0,theinverseF0

WeproceedtoshowthatF0(M)isinfactdenseintheσ(M,A′)-topologyofM:Firstnotethattheextensionproperty(V.1)forF0translatesinto:

F0(x δ(x))=xforx∈D(δ),(V.4)

andthecorrespondingtransposedmappingsinA′thereforesatisfy:

′(I δ′)F0=I(theidentityoperatorinA′).(V.5)

′HenceF0is1–1,andthedesireddensityofF0(M)followsfromthebi-polartheoremappliedtotheA′–Mduality.NotethatinfacteveryextensionofRhasdenserange,becausecondition(V.5)issatis edforthemostgeneralsuchextension.

=I F 1isthereforeanextensionSinceF0isanextensionof(I δ) 1itisclearthatδ0

ofδ.

Motivated by existence problems for dissipative systems arising naturally in lattice models from quantum statistical mechanics, we consider the following $C^{\ast}$-algebraic setting: A given hermitian dissipative mapping $\delta$ is densely defined in a u

VI. GENERATIONPROPERTIESOFδ

isclosedanddenselyde nedintheσ-topologyofM.But(I δ ) 1=F0,Theoperatorδ

(x) ≥ x forallx∈D(δ ).Weproceedtoshowthatinfactsowealsohave x δ

(x) ≥k x kx δ(VI.1)

).Indeed,letΛdenotethesetofk>0suchthattheinequalityforallk>0andx∈D(δ

).Thenwehaveseenthatk=1belongstoΛ.Itturnsout(VI.1)issatis edforallx∈D(δ

thatΛisbothopenandclosedasasubsetofR+,andourresultfollowsbyconnectedness.Toshowopenness,suppose rstthatk0∈Λ,andthatk∈R+satis es|k k0|<k0.We

) 1,thanuse(VI.1),fork0,inestimatingthetermsintheNeumannexpansionfor(kI δ

takenaroundthepointk0.Duetotheassumption|k k0|<k0,theNeumannseriesis

)tokI δ .Termwiseestimationconvergent,anddoesindeedde neaboundedinverseR(k,δ

) ≤k 1,anditfollowsthat(VI.1)issatis edinaneighborhoodofk0.gives R(k,δ

Considernextaconvergentsequenceofpointskn→k0withkn∈Λandk0∈R+.By

)=(knI δ ) 1exist,andtheythereforesatisfyassumptiontheresolventoperatorsR(kn,δ

theresolventidentity:

) R(km,δ )=(kn km)R(kn,δ )R(km,δ ),R(kn,δ

)∈ ) ≤k 1.Itfollowsthatthenorm-limitR =limnR(kn,δaswellastheestimate R(kn,δn

.The de nesaboundedinversetok0I δL(M)exists,anditistrivialtocheckthatR

≤k 1.HenceΛisclosed,andtheestimate(VI.1)fork0isnowimpliedinthelimitby R0

argumentiscompleted.

inMisdissipativeandclosedintheσ(M,A′)-Wehaveshownthattheoperatorδ

)topology.Itis,ofcourse,alsoclosedinthenorm-topology,anditcanbeshowrthatD(δ

isthein nitesimalgeneratorisnorm-dense.Itfollowsbysemigrouptheory[23,29]thatδ

ofastronglycontinuoussemigroupτt(0≤t<∞)ofcontractionoperatorsintheBanachspaceM.

Toshowthateachτtisanormaltransformationweconsidertheadjointsemigroupτt′(cf.[16])inthenorm-dualM′andshowthatτt′leavesA′invariant.NotethatA′isbeingidenti edwiththepredualoftheW -algebraM,sothatwemayregarditasasubspaceofM′.

Motivated by existence problems for dissipative systems arising naturally in lattice models from quantum statistical mechanics, we consider the following $C^{\ast}$-algebraic setting: A given hermitian dissipative mapping $\delta$ is densely defined in a u

′(resp.,F′)denotethetransposedoperatorstoδ (resp.,F0)withrespecttotheM–Letδ0

′isthegeneratorofτ′,andthat(I δ ′) 1=M′duality.Itfollowsbyoperatortheorythatδt

′′F0.FromtheconstructionofF0wenowdeducethatA′isinvariantunderF0.Indeed,recallthatδ′denotesthetransposedtransformationtoδwithrespecttotheA–A′duality.Byde nitionD(δ′)={a′∈A′: b′∈A′, b′,x = a′,δ(x) forallx∈D(δ)}.Butfora′∈A′

′′′′andx∈D(δ)wehave F0(a),x δ(x) = a′,x .Hence,F0(a)∈D(δ′) A′by(V.5).AnapplicationoftheNeumannexpansionto(I t

n ) 1,showsthatδ

τtiscompletelypositiveinMforallt≥0.Indeed(I

) 1.δtn

VII.THEINEQUALITYδ(x x)≥δ(x) x+x δ(x)

Itwasshownin[19]thatifδisaboundedhermitianlinearmapinaC -algebraA,thenthefollowingtwoconditionsareequivalent:

etδ(x x)≥etδ(x )etδ(x),

and

δ(x x)≥δ(x )x+x δ(x), x∈A.(VII.2) x∈A,t∈R+,(VII.1)

ForunboundedAthesituationisnotaswellunderstood.Itisthereforeofinteresttostudytheconnectionbetweentheproperty(VII.2)forδ,andtheotherconditionswhicharecustomarilyusedintheapplicationsofunboundeddissipativemappingsinoperatoralgebrastoquantumdynamics.

TheoremVII.1.LetAbeaC -algebrawithunit11,andletδbeacompletelydissipativetransformationinAwithdensedomainD(δ).Assume11∈D(δ),andδ(11)=0.(a)Letx∈D(δ)andassumethatx x∈D(δ).Then

δ(x x)≥δ(x) x+x δ(x).(VII.3)

Motivated by existence problems for dissipative systems arising naturally in lattice models from quantum statistical mechanics, we consider the following $C^{\ast}$-algebraic setting: A given hermitian dissipative mapping $\delta$ is densely defined in a u

(b)Supposebothxandx belongtoD(δ).Thenδ(x )=δ(x) .

ThefollowingresultsarecorollariestotheproofsofTheoremsIV.2andVII.1.CorollaryVII.2.LetAbeaC -algebrawithunit11,andletδbecompletelydissipativeinAwithdensedomainD(δ),11∈D(δ),δ(11)=0.

(a)IfA B(H)forsomeHilbertspaceH,thenthereisasequenceofcompletelypositive

mapsEn:A→B(H),En(11)=11,suchthatthefollowingnorm-convergenceholds:

En(x) →x

and

n(En(x) x) →δ(x)forx∈D(δ).(ii)forx∈A,(i)

(b)IfD(δ)ishermitian,thenδishermitianaswell,i.e.,δ(x )=δ(x) forallx∈D(δ),

anditisthenpossible,foreachn,tochooseEntobe1–1withdenserange.

(c)LetδandAbeasin(a),andletπ:A→B(K)bearepresentationofAinaHilbert

spaceK.ThenthereexistsasequenceEn∈CP(A,B(K))suchthatthefollowingnormconvergenceholds:

En(x) →π(x)

and

n(En(x) π(x)) →π(δ(x))forx∈D(δ).(ii′)forx∈A,(i′)

Proofs.WeconsideragaintherangesubspaceS=Ran(I δ)={x δ(x):x∈D(δ)}.AsintheproofofTheoremIV.2notethatR=(1 δ) 1:S→Aiscompletelycontractive,andR(11)=11.IfAisconsideredasasubalgebraofB(H),whereHistheHilbertspaceoftheuniversalrepresentation,thenthereis,byArveson’sextensiontheorem[2,Theorem

1.2.9]acompletelypositivemappingE:A→B(H)suchthat

R(s)=E(s)foralls∈S.(VII.4)

Ifforeachn=1,2,...theoperatorδisreplacedbyn 1δ,thentheaboveargumentyieldsacompletelypositivemapEn:A→B(H)suchthatEnisanextensionofthepartiallyde nedoperator(I n 1δ) 1.

Motivated by existence problems for dissipative systems arising naturally in lattice models from quantum statistical mechanics, we consider the following $C^{\ast}$-algebraic setting: A given hermitian dissipative mapping $\delta$ is densely defined in a u

Weclaimthatthesequence(En)satis esconditions(i)and(ii)whicharelistedinCorollaryVII.2(a).Indeed,forxindenseD(δ)wehaveEn(x n 1δ(x))=x,andtherefore

En(x)=n 1En(δ(x))+x,

and

En(δ(x))=n(En(x) x).(VII.6)(VII.5)

Passingtothelimitin(VII.5),weget(i)forthespecialcasex∈D(δ),butthenalsoforallxinAbya3-εargumentsinceeachEniscontractive.Theresult(ii)ofCorollaryVII.2(a)isnowanimmediateconsequenceof(VII.6).

ReturningtotheproofofTheoremVII.1,wenotethat(b)istrivialfrom(ii).Indeed,forxandx inD(δ)wehave

δ(x )=limn(En(x ) x )=lim(n(En(x) x)) =δ(x) .n

TheproofofTheoremVII.1(a)isbasedonboth(i)and(ii),togetherwiththeKadison-SchwarzinequalityforEn:Supposex∈D(δ)andx x∈D(δ).

limn(En(x x) x x).Foreachtermontheright-handsidewehave:

n(En(x x) x x)≥n(En(x) En(x) x x)

=1(VII.7)Thenδ(x x)=

2(δ(x) (2x)+(2x) δ(x))=δ(x) x+x δ(x),

wherethelastconvergence →isbasedon(i)and(ii)fromCorollaryVII.2(a).Sinceδ(x x)isobtainedinthelimitontheleft,thedesiredinequality(VII.3)in(a)ofTheoremVII.1follows.

Onlypart(b)ofthecorollaryremains.ThetechniquefromtheproofofTheoremIV.2isappliedhere.WegobacktotheextensionEfrom(VII.4)inthebeginningofthepresentproof.ConsidertheorderingonalltheextensionsFofR,F∈L(A,B(H)),whichisinducedbytheconeCP(A,B(H)),andchoosebyZornaparticularextensionF,E≤F,whichismaximal.TheargumentfromtheproofofTheoremIV.2thenshowsthatFis1–1,andthe

=I F 1:Ran(F)→AexistsandrangeRan(F)isdense.Itfollowsthattheoperatorδ

(x)=δ(x)forallx∈D(δ).satis esδ

Motivated by existence problems for dissipative systems arising naturally in lattice models from quantum statistical mechanics, we consider the following $C^{\ast}$-algebraic setting: A given hermitian dissipative mapping $\delta$ is densely defined in a u

Ifαisapositiverealnumber,thenthesameconstructionmaybecarriedoutforthetransformationαδ,insteadofδ.HencewegetcompletelypositiveunitalmapsFαsuch

1 1thattheinverseFαexistsforeachα,andthedomainofI FαcontainsD(δ).Moreover

α=I F 1satis esδ α(x)=δ(x)forx∈D(δ).Togetasequenceofmappingssatisfyingtheδα

conditionsinCorollaryVII.2(b),weneedonlytakeEn=Fn 1inthespecialcaseα=n 1.Theproofofpart(c)inthecorollaryisparallelto(a)withthefollowingmodi cation:Arveson’sextensiontheoremisnowappliedtothemappingπ (I δ) 1:S→B(K).VIII.THEIMPLEMENTATIONPROBLEM

Theconclusion(ii′)inCorollaryVII.2(c)isofinterestwhenonewantstoimplementthetransformationδbyadissipativeoperatorinHilbertspace.Inparticular,oneisinterestedinimplementingacompletelydissipativeδ-operatorbyadissipativeHilbert-spaceoperator.Weshallestablishacleartwo-wayconnectionbetweenthedissipativenotionforδ,andfortheimplementingHilbert-spaceoperator.

TheoremVIII.1.LetAbeaC -algebrawithunit11,andletδbeacompletelydissipativetransformationinAwithdensedomainD(δ).Assume11∈D(δ)andδ(11)=0.LetωbeastateofA,andlet(πω,Kω, )bethecorrespondingGNSrepresentationofA.Letω bethevectorstateonB(Kω)givenbythecyclicvector ,i.e.,ω (X)= X | forX∈B(Kω),andassumethatitispossibletochoosethesequence(En) CP(A,B(Kω))fromCorollaryVII.2(c)insuchamannerthat

ω (En(x))=ω(x)forallx∈A.(VIII.1)

ThenthereisadissipativeoperatorLωinKωsuchthat

πω(δ(x)) =Lω(πω(x) )forallx∈D(δ).(VIII.2)

Proof.Letπ=πω,K=Kω,andlet(En) CP(A,B(K))beasequencewhich,alongwiththeconditionslistedinCorollaryVII.2(c),alsoful llstheinvariancerestriction(VIII.1)ofthepresenttheorem.Foreachnde neanoperatorCninKasfollows:

Cn(π(x) )=En(x) ,x∈A.

Motivated by existence problems for dissipative systems arising naturally in lattice models from quantum statistical mechanics, we consider the following $C^{\ast}$-algebraic setting: A given hermitian dissipative mapping $\delta$ is densely defined in a u

Then

Cnπ(x) 2= En(x) 2=ω (En(x) En(x))

≤ω (En(x x))=ω(x x)= π(x x) | = π(x) 2,

wherethenormisthatofK,andwheretheSchwarzinequalityisappliedtoEn.ItfollowsthatCniswellde ned,andthatitextendsbylimits(inK)toacontractionoperator,Cn∈B(K), Cn ≤1.

ByCorollaryVII.2(c)(ii′),wethenhave

π(δ(x)) =limn(En(x) π(x) )

=limn(Cn(π(x) ) π(x) )

=limn(Cn I)π(x)

Asaconsequence,thefollowingquadraticformonK:

π(x) ,π(y) →lim n(Cn I)π(x) |π(y) K

iswellde ingthecontractivepropertyofCn,itiseasytoshowthatthisquadraticformisgivenbyadissipativeoperatorL;thatistosay

lim n(Cn I)π(x) |π(y) = Lπ(x) |π(y) .

Sincethelimitontheleftisalsoequaltotheinnerproduct

π(δ(x)) |π(y) ,

theidentity(VIII.2)ofthetheoremfollows.forx∈D(δ).

IX.ACONDITIONFORCOMPLETEDISSIPATIVENESS

Inapplications[18,24,33]itisoftenpossibletodeterminethederivationδinaparticularrepresentation.IfmoreoverthederivationisknowntobeimplementedbyadissipativeoperatorinthecorrespondingHilbertspace,thenitfollowsinspecialcasesthatδitselfiscompletelydissipative.

1andletδbeadenselyde nedtransforma-TheoremIX.1.LetAbeaC -algebrawithunit1

tioninAsuchthat11∈D(δ)andδ(11)=0.LetωbeastateonAsuchthatδisimplemented

Motivated by existence problems for dissipative systems arising naturally in lattice models from quantum statistical mechanics, we consider the following $C^{\ast}$-algebraic setting: A given hermitian dissipative mapping $\delta$ is densely defined in a u

byadissipativeHilbert-spaceoperatorLintherepresentationπω.Assumemoreoverthatπωisfaithful,andthatL =0,where denotesthecyclicvectorintheGNSrepresentation.Thenδiscompletelydissipativeonitsdomain.

Proof.LetH=HωbetheHilbertspaceofthefaithfulrepresentationπωandletLbetheoperatorinHwhichisassumedtoexist,satisfyingconditions(i)and(ii)below:(i)ThedomainofLisπω(D(δ)) ,andLisadissipativeoperatorintheHilbertspaceH;(ii)Limplementsδintherepresentationπω,whichisequivalenttotherequirementthat

L isde nedonπω(D(δ)) ,andthatonthisdomainthefollowingoperatoridentityisvalid:

π(δ(a))=Lπ(a)+π(a)L foralla∈D(δ).(IX.1)

Weshow rstthatδmustnecessarilybeadissipativeoperator.Indeed,byPhillips’s

ofLexistswhichisthein nitesimalgeneratoroftheorem[30,Thm.1.1.3]anextensionL

astronglycontinuoussemigroupS(t)ofcontractionoperatorsintheHilbertspaceH.WenotethatS(t)implementsasemigroupσ(t)ofpositivemappingsinB(H),givenby

σ(t)(A)=S(t)AS(t) (IX.2)

forallt∈[0,∞)andA∈B(H).Bysemigrouptheorywenotethatthegenerator(ζsay)ofσ(t)isdissipative,sothefollowingestimateholds:

A αζ(A) ≥ A

forallα∈[0,∞)andA∈D(ζ).

Ifδωdenotestheoperatorπω(a)→πω(δ(a))withdomainπω(D(δ)),thenweclaim(easyproof)that

δω(A)=ζ(A)forallA∈D(δω),(IX.4)(IX.3)

andtheknownestimate(IX.3)abovethenimplies

πω(a) απω(δ(a)) ≥ πω(a) (IX.5)

fora∈D(δ)andα∈[0,∞).Butπωisfaithful(andhenceisometric),so(IX.5)isinfactequivalenttothedissipationestimate

a αδ(a) ≥ a

Motivated by existence problems for dissipative systems arising naturally in lattice models from quantum statistical mechanics, we consider the following $C^{\ast}$-algebraic setting: A given hermitian dissipative mapping $\delta$ is densely defined in a u

fortheoperatorδitself.

Foreachn=1,2,...,wenowconsiderthetensor-productconstructionoftheC -algebraAwiththen-by-ncomplexmatricesMn;andwede neAn=A Mn,δn=δ idn,theoperatorobtainedbyapplicationofδtoeachentryaijinthematrixrepresentationofelementsinAn,ωn=ω trnwheretrndenotesthenormalizedtraceonMn,πωn:theGNSrepresentationofAnassociatedtoωn.

Theproblemistoshowthateachoftheoperatorsδnisdissipative.WeshowthatinfactδnisimplementedbyadissipativeHilbert-spaceoperatorintherepresentationπωn.Hence,the rstpartoftheproofappliesandyieldstheconclusionoftheclaimsinceeachrepresentationπωnisfaithful,beingthetensorproductoffaithfulrepresentations.

LetHndenotetherepresentationHilbertspaceofπωn.Weproceedto ndadissipative

operatorLninHnsuchthatδnisimplementedbyLn.Inviewof(IX.1)thismeansthatπωn(δn(a))=Lnπωn(a)+πωn(a)L n

foralla∈D(δn)=D(δn) Mn(algebraictensorproduct) Anasanoperatoridentityonπωn(D(δn)) n Hn.Here ndenotesthecyclicvectorfortherepresentationπωn,i.e.,

ωn(a)= πωn(a) n| n

Ournextstepistheveri cationofthefollowing:

Reωn(a δn(a))≤0

Ln n=0,

ωn(a δn(a))= Lnπωn(a) n|πωn(a) n fora∈D(δn).foralla∈D(δn),(IX.7)(IX.8)(IX.9)foralla∈An.(IX.6)

Itwillfollowfrom(IX.7)and(IX.9)thatanimplementingoperatorLnsatisfying(IX.8)mustnecessarilybedissipative.

Notethat(IX.8)isveri edforn=1byassumption.

intothisidentityyieldsidentity(IX.9)forthecasen=1.

LetTndenotethetrace-vectorforthetracerepresentativeτnofMn.Thenπωn=π τn,andtherefore

πωn(a b) Tn| Tn = π(a) τn(b)Tn| Tn

= π(a) | τn(b)Tn|Tn =ω(a)trn(b)

=ω trn(a b)=ωn(a b)

Henceω(a δ(a))= π(δ(a)) |π(a) = Lπ(a) +π(a)L |π(a) .SubstitutionofL = L =0

Motivated by existence problems for dissipative systems arising naturally in lattice models from quantum statistical mechanics, we consider the following $C^{\ast}$-algebraic setting: A given hermitian dissipative mapping $\delta$ is densely defined in a u

foralla∈Aandb∈Mn.Hence n= Tn.IfwecanshowthatasimpletensoroperatorLnimplementsδninπωn,thenidentity(IX.9),forarbitraryn,followsfromthecasen=1

whichwasestablishedabove.

However,itiseasytoseethattheoperatorLn=L Insatis estherequirementswhichwerelistedabove.Indeed

πωn(δn(a b))=πω(δ(a)) τn(b)

=(Lπω(a)+πω(a)L ) τn(b)

=Lnπω(a) τn(b)+πω(a) τn(b)L n

=Lnπωn(a b)+πωn(a b)L n

foralla∈Aandb∈Mn.ItfollowsthatLnimplementsδninπωn.

Onlytheveri cationof(IX.7)forn>1thenremains.Letaij∈A

insomea∈An=A Mn.Thenthe(i,j)’thentrycijina δn(a)is bethematrixentries

n

k=1a kiδ(akj).Hence

ωn(a δn(a))=(ω trn)(cij)

= n

ω(cii)= ω(a kiδ(aki)).

i=1ik

SinceReω(a kiδ(aki))≤0,(IX.7)follows.

RemarkIX.2.Inthefoundationsofirreversiblestatisticalthermodynamics[14,20,21,24,28],themostconclusiveresultshavebeenobtainedfordynamicalsemigroupswhicharede-scribedmathematicallyasstronglycontinuous,completelypositive,contractionsemigroupsTtontheBanachspaceT(H)ofalltrace-classoperatorsonagivenseparable∞-dimensionalHilbertspaceH.Lindblad[28]foundaformulaforthein nitesimalgenerator

W=d

Motivated by existence problems for dissipative systems arising naturally in lattice models from quantum statistical mechanics, we consider the following $C^{\ast}$-algebraic setting: A given hermitian dissipative mapping $\delta$ is densely defined in a u

However,TheoremIV.2combinedwiththeaboveresultssuggeststhataW -algebra,properlysmallerthanB(H),issuitableforquantumdynamics.Ontheonehand,B(H)(orT(H)intheconjugate(dual)formulation)istoobigtoaccomodatetheextensions;and,ontheotherhand,therequirementthatC(H)containthedomainofthegeneratoralsoappearstobetoorestrictive.

X.UNBOUNDED*-DERIVATIONS

LetAbeaunitalC -algebra,andletD(δ)beadense -subalgebracontainingtheidentity11.Alineartransformationδ:D(δ)→Aissaidtobea(unbounded) -derivationifδ(ab)=δ(a)b+aδ(b)fora,b∈D(δ),andδ(a )=δ(a) fora∈D(δ).

Since,for -derivations,oneisprimarilyinterestedinextensionswhicharealso -derivations,itisnaturaltoworkwithatwo-sidedconditioninplaceofthedissipativenotionswhichwerestudiedintheprevioussectionsformoregeneraloperators.Thefollow-ingsuchtwo-sidedconditionwassuggestedbySakai[36],andadoptedbyseveralauthorsinsubsequentresearchonunbounded -derivations.

De nitionX.1.A -derivationδ:D(δ)→Aissaidtobewellbehavedifforallpositivea∈D(δ)thereisastateφonAsuchthatφ(a)= a andφ(δ(a))=0.

Theargumentintheprevioussectionyields:

PropositionX.2.Letδ:D(δ)→Abea -derivation.Thenthefollowingfourconditionsareequivalent:

(i)δiswellbehaved.

(ii)Forallpositivea∈D(δ),andforallstatesφonAsatisfyingφ(a)= a ,wehave

φ(δ(a))=0.

(iii)Eachoftheoperators±δisdissipative.

(iv) a+αδ(a) ≥ a forallα∈Randalla∈D(δ).

De nitionX.3.A -derivationδ:D(δ)→Aissaidtobewellbehavedinthematricialsenseif,foreachn=1,2,...,the -derivationδn=δ idn:D(δ) Mn→A Mniswellbehaved.Recallthatδnmayberegardedasatransformationonn-by-nmatriceswithentriesinA.Forsuchamatrixa=(aij),i,j=1,...,n,wehaveδn(a)=(δ(aij)).

Motivated by existence problems for dissipative systems arising naturally in lattice models from quantum statistical mechanics, we consider the following $C^{\ast}$-algebraic setting: A given hermitian dissipative mapping $\delta$ is densely defined in a u

TheoremX.4.Everywell-behaved -derivationisalsowellbehavedinthematricialsense(i.e.,completelywellbehaved).

LemmaX.5.Letδ:D(δ)→Abeawell-behaved -derivation,andleta∈D(δ)bepositive.ThenthereisastateφonAsuchthatφ(a)= a ,andφ(δ(b))=0foradensesetofelementsb∈C (a)∩D(δ).(HereC (a)denotestheabelianC -subalgebrageneratedbya;andeveryelementinC (a)canbeapproximatedinnormbyasequenceofelementsbsatisfyingtheconclusionofthelemma.)

Proofs.Theimplication(i) (ii)inPropositionX.2isthekeytotheproofofLemmaX.5.Sincefunctionalcalculusisalsoapplied,weshallassumeinfactthatδisclosed.ByaresultofKishimoto-Sakai[36]thisisnolossofgenerality.LetabeapositiveelementinD(δ).NotethattheGelfand-transformsetsupanisomorphismbetweentheC -algebrasC (a)andC(sp(a)),continuousfunctionsonthespectrumofa.Letλ0=l.u.b.sp(a).Thenthestatec→c(λ0)onC(sp(a))correspondstoastateonC (a)viatheGelfand-transform.ThelatterstateisthenextendedtoAbyKrein’stheorem,andtheextendedstateisdenotedbyφ.Ithasthemultiplicativeproperty:φ(b1b2)=φ(b1)φ(b2)forb1,b2∈C (a).

Nowletgbeanon-decreasing(monotone)continuousrealfunctionde nedonsp(a).ThentheGelfand-transformofg(a)achievesitsmaximumatthepointλ0sincethetransformofadoes.ButitisknownthatifgisalsoofclassC2(twocontinuousderivatives)theng(a)∈D(δ)∩C (a).Henceφ(g(a))= g(a) .AnapplicationofPropositionX.2,(i) (ii),thenyieldstheconclusion

φ(δ(g(a)))=0.

Therestrictionofanarbitrarymonomialλntosp(a)satis estheconditionslistedforg.Hence,byStone-Weierstrassthereisadensesetofelementsb∈C (a)∩D(δ)satisfyingtheconclusionofthelemma.(Alternatively,everypositivefunctionfinC4maybewrittenintheformf=g1 g2,withg1andg2bothhavingthepropertieslistedaboveforg,weconcludethatφ(δ(f(a)))=φ(δ(g1(a))) φ(δ(g2(a)))=0.)

Now,foreach xedelementa∈D(δ)+wechooseastateφ=φaandadense -subalgebraB=BaofC (a)accordingtoLemmaX.5;i.e.,werequirethatφa(δ(b))=0forb∈Ba,aswellasφa(a)= a .ConsidertheGNSrepresentationofthealgebraB,resp.A,withrepresentationspaceHφ,resp.Kφ,andde ne:

H= Hφ,resp.,K=

Kφ.(X.1)

Motivated by existence problems for dissipative systems arising naturally in lattice models from quantum statistical mechanics, we consider the following $C^{\ast}$-algebraic setting: A given hermitian dissipative mapping $\delta$ is densely defined in a u

ThenHisaclosedsubspaceoftheHilbertspaceK,andwecanthende neanoperatorSwithdensedomainfromHtoKasfollows:

Sπφ(b) φ=πφ(δ(b)) φ

Forvectorsξ1andξ2inthedomainofSwehave

Sξ1|ξ2 + ξ1|Sξ2 =0.(X.3)forb∈Bφ.(X.2)

Theveri cationof(X.3)maybebasedonthedirect-sumdecomposition(X.1)above.If ξi=φπ(bi) φfori=1,2andbi∈Bφ,thenidentity(X.3)reducesto

πφ(δ(b1)) φ|πφ(b2) φ + πφ(b1) φ|πφ(δ(b2)) φ =0.

Theindividualtermsworkouttobe:

φ(b 2δ(b1))+φ(δ(b2)b1)=φ(δ(b2b1))=0.

Hence,thesymmetrycondition(X.3)isherebyreducedtotheconclusionofLemmaX.5foragivenwell-behavedderivationδ.

IfPdenotestheorthogonalprojectioninKwithrangeH,identity(X.3)impliesthattheoperatorξ→PSξmayinfactberegardedasaskewsymmetricoperatorintheHilbertspaceH,withdensedomainthere.WeshallalsodenotethisoperatorbyS.Theveri cationoftheidentity

π(δ(b))=Sπ(b) π(b)S

islefttothereader.

Followingtheideaof§IX,wenowconsiderthe -derivationsδn=δ idn(foreachn=1,2,...)introducedinDe nitionX.3.Foragiven -algebraCwedenotebyCnthe -algebraC Mn.Correspondingly, -algebrasD(δ)n,An,andBnarede nedforeachn.ApplicationoftheGNSrepresentationtoeachφn=φ trnyieldssequencesofHilbertspaces

H(n) K(n)

asin(X.1)witheachH(n),resp.,K(n),adirectsumofGNSrepresentationspacesassociatedtoφn.

Thecalculationsin§IXshowthattheoperatorSn=S Insatis esthen’th-orderversionof(X.2),thatis,(X.2)holdswiththequadrupleS,π,B,δreplacedbySn,πn,Bn,δn.

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