Schur tests and estimates for positive semigroups on non commutative $L_p$

We extend the recent results of Arnold on estimates for Kreiss positive semigroups on $L_p$-spaces to the non commutative setting, $1<p<\infty$. We also provide a sharp example. All the results are achieved thanks to the use of the Schur test, including a new version for von Neumann algebras.