Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 083, 27 pages      arXiv:2204.03595
Contribution to the Special Issue on Non-Commutative Algebra, Probability and Analysis in Action

Markovianity and the Thompson Group $F$

Claus Köstler a and Arundhathi Krishnan b
a) School of Mathematical Sciences, University College Cork, Cork, Ireland
b) Department of Pure Mathematics, University of Waterloo, Ontario, Canada

Received April 08, 2022, in final form October 07, 2022; Published online October 27, 2022

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.

Key words: noncommutative stationary Markov processes; representations of Thompson group $F$.

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