Markovianity and the Thompson Group $F$

TL;DR

This paper explores the connection between the Thompson group F and bilateral stationary noncommutative Markov processes, establishing a correspondence through representations and tensor dilations, with applications to classical probability.

Contribution

It introduces a novel link between Thompson group F representations and bilateral stationary noncommutative Markov processes, expanding understanding of their interplay.

Findings

Representations of F produce a large class of bilateral stationary noncommutative Markov processes.

Bilateral stationary Markov processes in tensor dilation form yield representations of F.

A canonical association between F and bilateral stationary Markov processes in classical probability is established.

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 K\"ummerer, we canonically associate a representation of $F$ to a bilateral stationary Markov process in classical probability.