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Let $𝔄$ be a $C^{*}$-algebra, $G$ a compact abelian group, $\tau$ an action of $G$ by $*$-automorphisms of $𝔄,{𝔄}^{\tau }$ the fixed point algebra of $\tau$ and ${𝔄}_F$ the dense sub-algebra of $G$-finite elements in $𝔄$. Further let $H$ be a linear operator from ${𝔄}_F$ into $𝔄$ which commutes with $\tau$ and vanishes on ${𝔄}^{\tau }$. We prove that $H$ is a complete dissipation if and only if $H$ is closable and its closure generates a $C_0$-semigroup of completely positive contractions. These complete dissipations are classified in terms of certain twisted negative definite...