Dilations of semigroups on von Neumann algebras and noncommutative ${\mathrm{L}}^p$-spaces

We prove that any weak* continuous semigroup $(T_t{)}_{t\ge 0}$ of factorizable Markov maps acting on a von Neumann algebra $M$ equipped with a normal faithful state can be dilated by a group of Markov $\ast$-automorphisms analogous to the case of a single factorizable Markov operator, which is an optimal result. We also give a version of this result for strongly continuous semigroups of operators acting on noncommutative ${\mathrm{L}}^p$-spaces and examples of semigroups to which the results of this paper can be applied. Our results implies the boundedness of the McIntosh’s ${\mathrm{H}}^{\mathrm{\infty }}$ functional calculus of the generators of these semigroups on the associated noncommutative ${\mathrm{L}}^p$-spaces generalising some previous work from Junge, Le Merdy and Xu. Finally, we also give concrete dilations for Poisson semigroups which are even new in the case of ${\mathbb{R}}^n$.