Markov dilations of semigroups of Fourier multipliers

TL;DR

This paper constructs a Markov dilation for continuous semigroups of Fourier multipliers on group von Neumann algebras, providing a new framework for analyzing their structure and properties.

Contribution

It introduces a Markov dilation for weak* continuous semigroups of selfadjoint unital completely positive Fourier multipliers on group von Neumann algebras, expanding the theoretical toolkit.

Findings

Established a Markov dilation for the specified semigroups

Extended the understanding of Fourier multipliers in operator algebras

Provided a new approach for analyzing semigroup dynamics

Abstract

We describe a Markov dilation for any weak* continuous semigroup $(T_t)_{t \geq 0}$ of selfadjoint unital completely positive Fourier multipliers acting on the group von Neumann algebra $\mathrm{VN}(G)$ of a locally compact group $G$.