Dilations of semigroups on von Neumann algebras and noncommutative Lp-spaces

Abstract

We prove that any weak* continuous semigroup (Tt)t⩾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 ⁎-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 Lp-spaces and examples of semigroups to which the results of this paper can be applied. Our results imply the boundedness of the McIntosh's H∞ functional calculus of the generators of these semigroups on the associated noncommutative Lp-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 Rn.