### A ${C}^{*}$-algebraic Schoenberg theorem

Let $\U0001d504$ be a ${C}^{*}$-algebra, $G$ a compact abelian group, $\tau $ an action of $G$ by $*$-automorphisms of $\U0001d504,{\U0001d504}^{\tau}$ the fixed point algebra of $\tau $ and ${\U0001d504}_{F}$ the dense sub-algebra of $G$-finite elements in $\U0001d504$. Further let $H$ be a linear operator from ${\U0001d504}_{F}$ into $\U0001d504$ which commutes with $\tau $ and vanishes on ${\U0001d504}^{\tau}$. 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...