The existence problem for dynamics of dissipative systems in(13)

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

Weclaimthatthesequence(En)satis esconditions(i)and(ii)whicharelistedinCorollaryVII.2(a).Indeed,forxindenseD(δ)wehaveEn(x n 1δ(x))=x,andtherefore

En(x)=n 1En(δ(x))+x,

and

En(δ(x))=n(En(x) x).(VII.6)(VII.5)

Passingtothelimitin(VII.5),weget(i)forthespecialcasex∈D(δ),butthenalsoforallxinAbya3-εargumentsinceeachEniscontractive.Theresult(ii)ofCorollaryVII.2(a)isnowanimmediateconsequenceof(VII.6).

ReturningtotheproofofTheoremVII.1,wenotethat(b)istrivialfrom(ii).Indeed,forxandx inD(δ)wehave

δ(x )=limn(En(x ) x )=lim(n(En(x) x)) =δ(x) .n

TheproofofTheoremVII.1(a)isbasedonboth(i)and(ii),togetherwiththeKadison-SchwarzinequalityforEn:Supposex∈D(δ)andx x∈D(δ).

limn(En(x x) x x).Foreachtermontheright-handsidewehave:

n(En(x x) x x)≥n(En(x) En(x) x x)

=1(VII.7)Thenδ(x x)=

2(δ(x) (2x)+(2x) δ(x))=δ(x) x+x δ(x),

wherethelastconvergence →isbasedon(i)and(ii)fromCorollaryVII.2(a).Sinceδ(x x)isobtainedinthelimitontheleft,thedesiredinequality(VII.3)in(a)ofTheoremVII.1follows.

Onlypart(b)ofthecorollaryremains.ThetechniquefromtheproofofTheoremIV.2isappliedhere.WegobacktotheextensionEfrom(VII.4)inthebeginningofthepresentproof.ConsidertheorderingonalltheextensionsFofR,F∈L(A,B(H)),whichisinducedbytheconeCP(A,B(H)),andchoosebyZornaparticularextensionF,E≤F,whichismaximal.TheargumentfromtheproofofTheoremIV.2thenshowsthatFis1–1,andthe

=I F 1:Ran(F)→AexistsandrangeRan(F)isdense.Itfollowsthattheoperatorδ

(x)=δ(x)forallx∈D(δ).satis esδ