The existence problem for dynamics of dissipative systems in(11)

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

′(resp.,F′)denotethetransposedoperatorstoδ (resp.,F0)withrespecttotheM–Letδ0

′isthegeneratorofτ′,andthat(I δ ′) 1=M′duality.Itfollowsbyoperatortheorythatδt

′′F0.FromtheconstructionofF0wenowdeducethatA′isinvariantunderF0.Indeed,recallthatδ′denotesthetransposedtransformationtoδwithrespecttotheA–A′duality.Byde nitionD(δ′)={a′∈A′: b′∈A′, b′,x = a′,δ(x) forallx∈D(δ)}.Butfora′∈A′

′′′′andx∈D(δ)wehave F0(a),x δ(x) = a′,x .Hence,F0(a)∈D(δ′) A′by(V.5).AnapplicationoftheNeumannexpansionto(I t

n ) 1,showsthatδ

τtiscompletelypositiveinMforallt≥0.Indeed(I

) 1.δtn

VII.THEINEQUALITYδ(x x)≥δ(x) x+x δ(x)

Itwasshownin[19]thatifδisaboundedhermitianlinearmapinaC -algebraA,thenthefollowingtwoconditionsareequivalent:

etδ(x x)≥etδ(x )etδ(x),

and

δ(x x)≥δ(x )x+x δ(x), x∈A.(VII.2) x∈A,t∈R+,(VII.1)

ForunboundedAthesituationisnotaswellunderstood.Itisthereforeofinteresttostudytheconnectionbetweentheproperty(VII.2)forδ,andtheotherconditionswhicharecustomarilyusedintheapplicationsofunboundeddissipativemappingsinoperatoralgebrastoquantumdynamics.

TheoremVII.1.LetAbeaC -algebrawithunit11,andletδbeacompletelydissipativetransformationinAwithdensedomainD(δ).Assume11∈D(δ),andδ(11)=0.(a)Letx∈D(δ)andassumethatx x∈D(δ).Then

δ(x x)≥δ(x) x+x δ(x).(VII.3)