The existence problem for dynamics of dissipative systems in(4)

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