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

Ifαisapositiverealnumber,thenthesameconstructionmaybecarriedoutforthetransformationαδ,insteadofδ.HencewegetcompletelypositiveunitalmapsFαsuch

1 1thattheinverseFαexistsforeachα,andthedomainofI FαcontainsD(δ).Moreover

α=I F 1satis esδ α(x)=δ(x)forx∈D(δ).Togetasequenceofmappingssatisfyingtheδα

conditionsinCorollaryVII.2(b),weneedonlytakeEn=Fn 1inthespecialcaseα=n 1.Theproofofpart(c)inthecorollaryisparallelto(a)withthefollowingmodi cation:Arveson’sextensiontheoremisnowappliedtothemappingπ (I δ) 1:S→B(K).VIII.THEIMPLEMENTATIONPROBLEM

Theconclusion(ii′)inCorollaryVII.2(c)isofinterestwhenonewantstoimplementthetransformationδbyadissipativeoperatorinHilbertspace.Inparticular,oneisinterestedinimplementingacompletelydissipativeδ-operatorbyadissipativeHilbert-spaceoperator.Weshallestablishacleartwo-wayconnectionbetweenthedissipativenotionforδ,andfortheimplementingHilbert-spaceoperator.

TheoremVIII.1.LetAbeaC -algebrawithunit11,andletδbeacompletelydissipativetransformationinAwithdensedomainD(δ).Assume11∈D(δ)andδ(11)=0.LetωbeastateofA,andlet(πω,Kω, )bethecorrespondingGNSrepresentationofA.Letω bethevectorstateonB(Kω)givenbythecyclicvector ,i.e.,ω (X)= X | forX∈B(Kω),andassumethatitispossibletochoosethesequence(En) CP(A,B(Kω))fromCorollaryVII.2(c)insuchamannerthat

ω (En(x))=ω(x)forallx∈A.(VIII.1)

ThenthereisadissipativeoperatorLωinKωsuchthat

πω(δ(x)) =Lω(πω(x) )forallx∈D(δ).(VIII.2)

Proof.Letπ=πω,K=Kω,andlet(En) CP(A,B(K))beasequencewhich,alongwiththeconditionslistedinCorollaryVII.2(c),alsoful llstheinvariancerestriction(VIII.1)ofthepresenttheorem.Foreachnde neanoperatorCninKasfollows:

Cn(π(x) )=En(x) ,x∈A.