  We prove a Schoenberg-type correspondence for non-unital semigroups which
generalizes an analogous result for unital semigroup proved by Michael
Sch\"urmann. It characterizes the generators of semigroups of linear maps on
$M_n(C)$ which are $k$-positive, $k$-superpositive, or $k$-entanglement
breaking. As a corollary we reprove Lindblad, Gorini, Kossakowski, Sudarshan's
theorem. We present some concrete examples of semigroups of operators and study
how their positivity properties can improve with time.
