  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 unital $C^{\ast}$-algebra $\mathfrak{A}$. The identity
element in ${\frak A}$ is also in the domain of $\delta$. Completely
dissipative maps $\delta$ are defined by the requirement that the induced maps,
$(a_{ij})\to (\delta (a_{ij}))$, are dissipative on the $n$ by $n$ complex
matrices over ${\frak A}$ for all $n$. We establish the existence of different
types of maximal extensions of completely dissipative maps. If the enveloping
von Neumann algebra of ${\frak A}$ is injective, we show the existence of an
extension of $\delta$ which is the infinitesimal generator of a quantum
dynamical semigroup of completely positive maps in the von Neumann algebra. If
$\delta$ is a given well-behaved *-derivation, then we show that each of the
maps $\delta$ and $-\delta$ is completely dissipative.
