The existence problem for dynamics of dissipative systems in(15)

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

Then

Cnπ(x) 2= En(x) 2=ω (En(x) En(x))

≤ω (En(x x))=ω(x x)= π(x x) | = π(x) 2,

wherethenormisthatofK,andwheretheSchwarzinequalityisappliedtoEn.ItfollowsthatCniswellde ned,andthatitextendsbylimits(inK)toacontractionoperator,Cn∈B(K), Cn ≤1.

ByCorollaryVII.2(c)(ii′),wethenhave

π(δ(x)) =limn(En(x) π(x) )

=limn(Cn(π(x) ) π(x) )

=limn(Cn I)π(x)

Asaconsequence,thefollowingquadraticformonK:

π(x) ,π(y) →lim n(Cn I)π(x) |π(y) K

iswellde ingthecontractivepropertyofCn,itiseasytoshowthatthisquadraticformisgivenbyadissipativeoperatorL;thatistosay

lim n(Cn I)π(x) |π(y) = Lπ(x) |π(y) .

Sincethelimitontheleftisalsoequaltotheinnerproduct

π(δ(x)) |π(y) ,

theidentity(VIII.2)ofthetheoremfollows.forx∈D(δ).

IX.ACONDITIONFORCOMPLETEDISSIPATIVENESS

Inapplications[18,24,33]itisoftenpossibletodeterminethederivationδinaparticularrepresentation.IfmoreoverthederivationisknowntobeimplementedbyadissipativeoperatorinthecorrespondingHilbertspace,thenitfollowsinspecialcasesthatδitselfiscompletelydissipative.

1andletδbeadenselyde nedtransforma-TheoremIX.1.LetAbeaC -algebrawithunit1

tioninAsuchthat11∈D(δ)andδ(11)=0.LetωbeastateonAsuchthatδisimplemented