Dilations of markovian semigroups of Fourier multipliers on locally compact groups

We prove that any weak* continuous semigroup $(T_t{)}_{t\ge 0}$ of Markov Fourier multipliers acting on a group von Neumann algebra $\mathrm{V}\mathrm{N}(G)$ associated to a locally compact group $G$ can be dilated by a weak* continuous group of Markov $\ast$-automorphisms on a bigger von Neumann algebra. Our construction relies on probabilistic tools and is even new for the group ${\mathbb{R}}^n$. Our results imply 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.