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

ThismeansthatthelinearspaceS=Ran(I δ)={x δ(x):x∈D(δ)}isclosedin

A.Inviewofthe(hermitian)assumptiononδwenotethatSisalsoselfadjoint,andthat11∈S.TheoperatorR:S→Ade nedbyx δ(x)→x,anddenotedby(I δ) 1,iscompletelypositive[2,Prop.1.2.8].ClearlyR(11)=11.

WenowconsiderthedoubledualtoA,denotedbyA′′,asaW -algebraM,andmaketheappropriateidenti cation(viatheuniversal -representationforA)suchthatAisregardedasaC -subalgebraofA′′,andthepre-dualofA′′isidenti edwiththedualA′ofA.(Thereaderisreferredto[35,§1.17,p.42]fordetails.)SinceM=A′′(withtheArensmultiplication)isinjectiveasaW -algebra,bytheassumption,itfollowsthatacompletelypositiveextensionmappingE:M→Mexists.IfweregardAasasubalgebraofM(asweshall),thentheextensionpropertyisgivenbytheidentity

R(s)=E(s)foralls∈S.(V.1)

NotethatS A,sothatSbecomesasubspaceofMwiththeabovementionedidenti ca-tion.

ThecompletelypositivetransformationsofMintoitselfwillbedenotedbyCP(M),andthespaceL(M)ofcompletelyboundedlineartransformationsinMgetsanorderingarisingfromtheconeCP(M).Indeed,forF∈L(M)wede neE≤FbytherequirementthatF E∈CP(M).AmongalltheparticularextensionsFofR,F∈L(M),suchthatE≤F,wechoosebyZornamaximalelementF0.(ForthebasicfactsontopologiesonCP(M)whichareneeded,thereaderisreferredto[2,Ch.1].)

ThisextensionF0,describedabove,hasthespecialpropertyofbeing1–1.We rstconsidertherestrictionofF0tothepositiveelementsinM,M+,thatis.Moreprecisely,wehavetheimplication:

x∈M+,F0(x)=0= x=0.(V.2)

Letη:M→M/Sbethecanonicallinearquotientmapping,andconsidertheconeCinthenormedquotientspaceE=M/SgivenbyC=η(M+).

Iftheelementxin(V.2)belongstoS,thentheconditionsR(x)=F0(x)=0implyx=0,sinceR=(I δ) 1.Hence,weshallassumethatxisnotinS.Thismeansthatη(x)∈Cde nesaone-dimensionalsubspace{kη(x):k∈C}inE,andthefunctionalf:kη(x)→kisnonzeroandpositive.ByKrein’stheorem[1,Thm.1,Ch.3,p.157]fextendstoapositive