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

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aTheexistenceproblemfordynamicsofdissipativesystemsinquantumprobabilityPalleE.T.Jorgensen DepartmentofMathematics,TheUniversityofIowa,14MacLeanHall,IowaCity,IA52242-1419U.S.A.(Dated:February1,2008)AbstractMotivatedbyexistenceproblemsfordissipativesystemsarisingnaturallyinlatticemodelsfromquantumstatisticalmechanics,weconsiderthefollowingC -algebraicsetting:Agivenhermitiandissipativemappingδisdenselyde nedinaunitalC -algebraA.TheidentityelementinAisalsointhedomainofδ.Completelydissipativemapsδarede nedbytherequirementthattheinducedmaps,(aij)→(δ(aij)),aredissipativeonthenbyncomplexmatricesoverAforalln.Weestablishtheexistenceofdi erenttypesofmaximalextensionsofcompletelydissipativemaps.IftheenvelopingvonNeumannalgebraofAisinjective,weshowtheexistenceofanextensionofδwhichisthein nitesimalgeneratorofaquantumdynamicalsemigroupofcompletelypositivemapsinthevonNeumannalgebra.Ifδisagivenwell-behaved -derivation,thenweshowthateachofthemaps±δiscompletelydissipative.PACSnumbers:02.,02.10.Hh,02.30.Tb,03.65.-w,05.30.-dKeywords:spinsystems,dynamics,evolutionsemigroup,dissipative,C -algebra,noncommutativeproba-bility