Motivated by existence problems for dissipative systems arising naturally inlattice models from quantum statistical mechanics, we consider the following$C^{\ast}$-algebraic setting: A given hermitian dissipative mapping $\delta$ isdensely defined in a unital $C^{\ast}$-algebra $\mathfrak{A}$. The identityelement in ${\frak A}$ is also in the domain of $\delta$. Completelydissipative maps $\delta$ are defined by the requirement that the induced maps,$(a_{ij})\to (\delta (a_{ij}))$, are dissipative on the $n$ by $n$ complexmatrices over ${\frak A}$ for all $n$. We establish the existence of differenttypes of maximal extensions of completely dissipative maps. If the envelopingvon Neumann algebra of ${\frak A}$ is injective, we show the existence of anextension of $\delta$ which is the infinitesimal generator of a quantumdynamical semigroup of completely positive maps in the von Neumann algebra. If$\delta$ is a given well-behaved *-derivation, then we show that each of themaps $\delta$ and $-\delta$ is completely dissipative.
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