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The existence problem for dynamics of dissipative systems in quantum probability
Motivated by existence problems for dissipative systems arising naturally in
lattice models from quantum statistical mechanics, we consider the following
-algebraic setting: A given hermitian dissipative mapping is
densely defined in a unital -algebra . The identity
element in is also in the domain of . Completely
dissipative maps are defined by the requirement that the induced maps,
, are dissipative on the by complex
matrices over for all . We establish the existence of different
types of maximal extensions of completely dissipative maps. If the enveloping
von Neumann algebra of is injective, we show the existence of an
extension of which is the infinitesimal generator of a quantum
dynamical semigroup of completely positive maps in the von Neumann algebra. If
is a given well-behaved *-derivation, then we show that each of the
maps and is completely dissipative.Comment: 24 pages, LaTeX/REVTeX v. 4.0, submitted to J. Math. Phys.; PACS 02.,
02.10.Hh, 02.30.Tb, 03.65.-w, 05.30.-