The existence problem for dynamics of dissipative systems in(17)

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