The existence problem for dynamics of dissipative systems in(18)

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