The existence problem for dynamics of dissipative systems in(6)

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

Ifδ:X→YismerelyalineartransformationbetweenBanachspacesXandY,withdomainD(δ)denseinX,thenthetransposed(orconjugate)transformationδ′iswellde nedasalineartransformationδ′:Y′→X′withdomainD(δ′)={f∈Y′: g∈X′s.t.f(δ(x))=g(x)forallx∈D(δ)}.Forf∈D(δ′),δ′(f)=g.ThedomainD(δ′)isweak*-denseinY′i δisclosable.Itisknown[29]thatdissipativeoperatorsareclosable.

PLETELYPOSITIVESEMIGROUPS(QUANTUMDYNAMICALSEMIGROUPS)

LetMbeaW -algebrawithpredualM .LetτtbeafamilyofcompletelypositivemappingsofMintoitself,indexedbythetimeparametert∈[0,∞).Assumethatτ0istheidentitytransformationinM,andthatτt(11)=11forallt∈[0,∞),where11denotestheunitelementoftheW -algebraMinquestion.Weassumefurtherthatthesemigrouplawholds,τt1+t2=τt1 τt2fort1,t2∈[0,∞),and nallythateachτtisanormalmappinginM.Recallthatnormalityisequivalenttotherequirementthattheconjugatesemigroupτt′[16]ofM′leavesinvariantthesubspaceM .Finallywerequirecontinuityofeachscalarfunction,t→ (τt(a)),forall ∈M anda∈M.Asemigroupwhichsatis esalltherequirementsaboveiscalledacompletelypositivesemigroup.Becauseoftherelevancetoquantumdynamics,weshallalsocallitaquantumdynamicalsemigroup[21].

Thein nitesimalgeneratorofagivencompletelypositivesemigroup(τt,M)isa,generallyunbounded,transformation,denotedbyζ,inM.Thedomainofthegeneratorζisgivenby

D(ζ)={a∈M: b∈Ms.t.forallt,τt(a) a= t

0τs(b)ds}.

Byde nitionζ(a)=b.Itiseasytosee[16]thatζ(a)=d