The existence problem for dynamics of dissipative systems in(12)

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

(b)Supposebothxandx belongtoD(δ).Thenδ(x )=δ(x) .

ThefollowingresultsarecorollariestotheproofsofTheoremsIV.2andVII.1.CorollaryVII.2.LetAbeaC -algebrawithunit11,andletδbecompletelydissipativeinAwithdensedomainD(δ),11∈D(δ),δ(11)=0.

(a)IfA B(H)forsomeHilbertspaceH,thenthereisasequenceofcompletelypositive

mapsEn:A→B(H),En(11)=11,suchthatthefollowingnorm-convergenceholds:

En(x) →x

and

n(En(x) x) →δ(x)forx∈D(δ).(ii)forx∈A,(i)

(b)IfD(δ)ishermitian,thenδishermitianaswell,i.e.,δ(x )=δ(x) forallx∈D(δ),

anditisthenpossible,foreachn,tochooseEntobe1–1withdenserange.

(c)LetδandAbeasin(a),andletπ:A→B(K)bearepresentationofAinaHilbert

spaceK.ThenthereexistsasequenceEn∈CP(A,B(K))suchthatthefollowingnormconvergenceholds:

En(x) →π(x)

and

n(En(x) π(x)) →π(δ(x))forx∈D(δ).(ii′)forx∈A,(i′)

Proofs.WeconsideragaintherangesubspaceS=Ran(I δ)={x δ(x):x∈D(δ)}.AsintheproofofTheoremIV.2notethatR=(1 δ) 1:S→Aiscompletelycontractive,andR(11)=11.IfAisconsideredasasubalgebraofB(H),whereHistheHilbertspaceoftheuniversalrepresentation,thenthereis,byArveson’sextensiontheorem[2,Theorem

1.2.9]acompletelypositivemappingE:A→B(H)suchthat

R(s)=E(s)foralls∈S.(VII.4)

Ifforeachn=1,2,...theoperatorδisreplacedbyn 1δ,thentheaboveargumentyieldsacompletelypositivemapEn:A→B(H)suchthatEnisanextensionofthepartiallyde nedoperator(I n 1δ) 1.