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

TL;DR

This paper proves that weak* continuous semigroups of factorizable Markov maps on von Neumann algebras can be dilated by groups of automorphisms, extending to noncommutative L^p-spaces and providing new concrete dilations, including for Poisson semigroups.

Contribution

It establishes a dilation theory for semigroups of Markov maps on von Neumann algebras and noncommutative L^p-spaces, generalizing previous results and providing explicit examples.

Findings

Dilations of semigroups by automorphisms are possible for von Neumann algebras.

The results imply boundedness of McIntosh's H-infinity functional calculus for generators.

Concrete dilations for Poisson semigroups are constructed, including new cases in R^n.

Abstract

We prove that any weak* continuous semigroup $(T_t)_{t \geq 0}$ 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 $\mathrm{L}^p$-spaces and examples of semigroups to which the results of this paper can be applied. Our results implies the boundedness of the McIntosh's $\mathrm{H}^\infty$ 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 $\mathbb{R}^n$.