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

operatorsystem,intheterminologyofE ros[11].ThefullmatrixalgebraMnofallcomplexn-by-nmatricesisalsoanoperatorsystem,andsoisLn=L Mn.TheelementsinLnmayberealizedasn-by-nmatriceswithentriesfromL,(aij)ni,j=1,aij∈L.IfLandRareoperatorsystemsand :L→Risalinearmapping,thentheinducedmap(aij)→( (aij))ofLnintoRnisdenoted n.Itis,infact, idn.Wesay[2]that iscompletelypositive(resp.,completelycontractive)if nispositive(resp.,contractive)foralln.WesaythatRisinjectiveifforeverypairofoperatorsystems,L L1,andeverycompletelypositivemap :L→R,thereisacompletelypositiveextensionψ:L1→R.Thatis,ψ(x)= (x)forallx∈L.IfRisavonNeumannalgebrainaHilbertspaceH,itisknown[13,38]thatRisinjectivei thereisanorm-oneprojectionofB(H)ontoR.

IfAisaC -algebra,itisknown[12]thatAisnucleari thedoubleconjugate(dual)A′′isinjectiveasaW -algebra.Connesshowed[13]thatafactorRonaseparableHilbertspaceisinjectivei itismatricial.

III.DISSIPATIVETRANSFORMATIONS

AnoperatorδinaBanachspaceXissaidtobedissipative[29]ifoneofthefollowingthreeequivalentconditionsissatis ed:

(i)ForallxinthedomainD(δ)ofδ,thereisanelementf∈X′,dependingonx,such

that f =1,f(x)= x ,andRef(δ(x))≤0.

(ii)ForallxinD(δ),andallf∈X′satisfying f =1,andf(x)= x ,theinequality

Ref(δ(x))≤0isvalid.

(iii)ForallxinD(δ),andallα∈R+,theinequality x αδ(x) ≥ x holds.

Theproofoftheequivalencecanbefound,forexample,in[7],buttheequivalencecanalsobeshowntobeaconsequenceoftheapproximationideainSectionIXandPropositionX.2inthepresentpaper.

IfXisanoperatorsystem,wesaythatδiscompletelydissipativeiftheinducedmappingδninXnisdissipativeforalln=1,2,....RecallthatXn=X Mn,andδn:(xij)→(δ(xij)),withdomainD(δn)={(xij)∈Xn:xij∈D(δ)}.

FinallywesaythatthetransformationδishermitianifthedomainD(δ),intheoperatorsystemX,isinvariantunderthe -involutionofX,andifδ(x )=δ(x) forallx∈D(δ).