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

whichδinaC -algebraA,toaskifitisalwayspossibletoextendδtoatransformationδ

isthein nitesimalgeneratorforaquantumdynamicalsemigroup.Undertheassumptionthatδishermitian,andthattheW -algebraA′′isinjective,weestablishtheexistenceof

.OurextensionisthusanalgebraicparalleltoFriederichs’sexten-ageneratorextensionδ

sionforsemiboundedoperatorsinHilbertspace,orananalogueofPhillips’s[30]maximaldissipativeextensionofthegeneraldissipativeoperatorinHilbertspace.

Inearlierarticles[7,10,28,31]theuniquenessproblemwasconsideredforthegenerator

.But,justasisthecaseforoperatorsinHilbertspace(Friedrichs,Phillips),extension,δ δ

theextensionisgenerallynotunique,re ectingthepossibilityofdi erent“boundarycon-ditions”atin nity.

Wereferthereadertothebooks[14],[18]and[33]fordetailsonthemathematicalfoundationsofalgebraicquantumtheory.

Theissuescenteringaroundtheexistenceproblemforthedynamicalone-parametergroups,orsemigroups,ofquantumstatisticalmechanicsareperhapsbestknowninthesetupofquantumspinsystems,astheyaretreatedin[9],[25]and[34].

ExampleI.1.Themathematicalframeworkisrathergeneralsuchastoallowawideva-rietyofapplications,includingrecentonestononequilibriumstatisticalmechanics[34].Acountablyin nitesetL(sayalattice;itmaybeZνwhereνisthelatticerank,ordimension)isspeci edattheoutset.Pointss∈Laresitesatwhichquantumspinsarelocated.Foreachs∈L,letHsbea nite-dimensionalcomplexHilbertspace,i.e.,thespinvectorsatsites;andfora nitesubsetΛ L,set

HΛ:=

s∈ΛHs.

ThenletAΛbethe -algebraofall(bounded)operatorsonHΛ.Withthenaturalembedding

AΛ1 AΛ2

givenby

AΛ1 →AΛ1 1Λ2\Λ1 AΛ2,

wegettheusualinductivelimitC -algebralimΛAΛ=:A.AfunctionΛ→Φ(Λ)=Φ(Λ) ∈AΛde nedonthe nitesubsetsΛofLiscalledaninteraction,and

HΦ(Λ)=Φ(X)

X ΛforΛ1 Λ2(I.1)