Markovianity and the Thompson Group 𝐹

Abstract

We show that representations of the Thompson group 𝐹 in the automorphisms of a noncommutative probability space yield a large class of bilateral stationary noncommutative Markov processes. As a partial converse, bilateral stationary Markov processes in tensor dilation form yield representations of 𝐹. As an application, and building on a result of Kümmerer, we canonically associate a representation of 𝐹 to a bilateral stationary Markov process in classical probability.