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

ThenHisaclosedsubspaceoftheHilbertspaceK,andwecanthende neanoperatorSwithdensedomainfromHtoKasfollows:

Sπφ(b) φ=πφ(δ(b)) φ

Forvectorsξ1andξ2inthedomainofSwehave

Sξ1|ξ2 + ξ1|Sξ2 =0.(X.3)forb∈Bφ.(X.2)

Theveri cationof(X.3)maybebasedonthedirect-sumdecomposition(X.1)above.If ξi=φπ(bi) φfori=1,2andbi∈Bφ,thenidentity(X.3)reducesto

πφ(δ(b1)) φ|πφ(b2) φ + πφ(b1) φ|πφ(δ(b2)) φ =0.

Theindividualtermsworkouttobe:

φ(b 2δ(b1))+φ(δ(b2)b1)=φ(δ(b2b1))=0.

Hence,thesymmetrycondition(X.3)isherebyreducedtotheconclusionofLemmaX.5foragivenwell-behavedderivationδ.

IfPdenotestheorthogonalprojectioninKwithrangeH,identity(X.3)impliesthattheoperatorξ→PSξmayinfactberegardedasaskewsymmetricoperatorintheHilbertspaceH,withdensedomainthere.WeshallalsodenotethisoperatorbyS.Theveri cationoftheidentity

π(δ(b))=Sπ(b) π(b)S

islefttothereader.

Followingtheideaof§IX,wenowconsiderthe -derivationsδn=δ idn(foreachn=1,2,...)introducedinDe nitionX.3.Foragiven -algebraCwedenotebyCnthe -algebraC Mn.Correspondingly, -algebrasD(δ)n,An,andBnarede nedforeachn.ApplicationoftheGNSrepresentationtoeachφn=φ trnyieldssequencesofHilbertspaces

H(n) K(n)

asin(X.1)witheachH(n),resp.,K(n),adirectsumofGNSrepresentationspacesassociatedtoφn.

Thecalculationsin§IXshowthattheoperatorSn=S Insatis esthen’th-orderversionof(X.2),thatis,(X.2)holdswiththequadrupleS,π,B,δreplacedbySn,πn,Bn,δn.