The existence problem for dynamics of dissipative systems in(20)

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)