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