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

However,TheoremIV.2combinedwiththeaboveresultssuggeststhataW -algebra,properlysmallerthanB(H),issuitableforquantumdynamics.Ontheonehand,B(H)(orT(H)intheconjugate(dual)formulation)istoobigtoaccomodatetheextensions;and,ontheotherhand,therequirementthatC(H)containthedomainofthegeneratoralsoappearstobetoorestrictive.

X.UNBOUNDED*-DERIVATIONS

LetAbeaunitalC -algebra,andletD(δ)beadense -subalgebracontainingtheidentity11.Alineartransformationδ:D(δ)→Aissaidtobea(unbounded) -derivationifδ(ab)=δ(a)b+aδ(b)fora,b∈D(δ),andδ(a )=δ(a) fora∈D(δ).

Since,for -derivations,oneisprimarilyinterestedinextensionswhicharealso -derivations,itisnaturaltoworkwithatwo-sidedconditioninplaceofthedissipativenotionswhichwerestudiedintheprevioussectionsformoregeneraloperators.Thefollow-ingsuchtwo-sidedconditionwassuggestedbySakai[36],andadoptedbyseveralauthorsinsubsequentresearchonunbounded -derivations.

De nitionX.1.A -derivationδ:D(δ)→Aissaidtobewellbehavedifforallpositivea∈D(δ)thereisastateφonAsuchthatφ(a)= a andφ(δ(a))=0.

Theargumentintheprevioussectionyields:

PropositionX.2.Letδ:D(δ)→Abea -derivation.Thenthefollowingfourconditionsareequivalent:

(i)δiswellbehaved.

(ii)Forallpositivea∈D(δ),andforallstatesφonAsatisfyingφ(a)= a ,wehave

φ(δ(a))=0.

(iii)Eachoftheoperators±δisdissipative.

(iv) a+αδ(a) ≥ a forallα∈Randalla∈D(δ).

De nitionX.3.A -derivationδ:D(δ)→Aissaidtobewellbehavedinthematricialsenseif,foreachn=1,2,...,the -derivationδn=δ idn:D(δ) Mn→A Mniswellbehaved.Recallthatδnmayberegardedasatransformationonn-by-nmatriceswithentriesinA.Forsuchamatrixa=(aij),i,j=1,...,n,wehaveδn(a)=(δ(aij)).