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

onE,andwemayde nefunctionalf

(η(y))1F1(y)=F0(y)+f1fory∈M.(V.3)

WeclaimthatF1isoneoftheextensionsconsideredintheZorn-processwhichwasdescribedabove.ButF0≤F1,andF0=F1,contradictingthemaximalityofF0—andso,wemusthavex=0,concludingtheproofof(V.2).(Notethatin(V.3),insteadoftheidentityelement11ontheright-handsideoftheequation,wecouldhaveusedanynonzeroelementinM+.ThecorrespondingF1-transformationwouldproperlymajorizeF0,andhaveitsrangecontainedinM,sincetherangeofF0fallsinM.)

SinceF0iscompletelypositive,wehave,inparticular,F0(x )=F0(x) .So,toestablishtheidentityN(F0)={x∈M:F0(x)=0}=0,itisenoughtoshowthatthehermitianpartofN(F0)iszero.Sincewehavealreadyconsideredpositiveelements,itonlyremainstoconsiderx=x ∈N(F0)satisfyingx∈/S.Chooseapositiverealnumberksuchthatxk=x+k11∈M+.WethenhaveF0(xk)=kandxk∈/S.Itispossible,therefore,by

onE=M/Ssatisfyingf (η(xk))=l>0.Krein’stheorem,tochooseapositivefunctionalf

(η(y))1Thende neF2(y)=F0(y)+f1fory∈M.Itisasimplemattertocheckthat

ischosenpositive.FinallyF2isoneoftheZorn-extensions.Indeed,F0≤F2sincef

F2(xk)=F0(xk)+l11>F0(xk).ThiscontradictiontothemaximalityofF0concludesthe

1isde nedonF0(M)={F0(x):x∈M}.proof.SinceN(F0)=0,theinverseF0

WeproceedtoshowthatF0(M)isinfactdenseintheσ(M,A′)-topologyofM:Firstnotethattheextensionproperty(V.1)forF0translatesinto:

F0(x δ(x))=xforx∈D(δ),(V.4)

andthecorrespondingtransposedmappingsinA′thereforesatisfy:

′(I δ′)F0=I(theidentityoperatorinA′).(V.5)

′HenceF0is1–1,andthedesireddensityofF0(M)followsfromthebi-polartheoremappliedtotheA′–Mduality.NotethatinfacteveryextensionofRhasdenserange,becausecondition(V.5)issatis edforthemostgeneralsuchextension.

=I F 1isthereforeanextensionSinceF0isanextensionof(I δ) 1itisclearthatδ0

ofδ.