In the process of identifying a suitable distributional symmetry to describe Markovianity, it has been conjectured by C. Köstler that there is a certain correspondence between unilateral Markov shifts and representations of the Thompson monoid $F^+$.

After having illustrated this correspondence in the context of tensor products of $W^*$-algebraic probability spaces, I will present the following two general results. A representation of the Thompson monoid $F^+$ in the endomorphisms of a $W^*$-algebraic probability space yields a noncommutative Markov process (in the sense of Kümmerer). Conversely, such a representation is obtained from a noncommutative Markov process which is given as a coupling to a so-called spreadable noncommutative Bernoulli shift.

As time permits I will discuss how partial spreadability manifests itself when the underlying noncommutative probability space comes from the group von Neumann algebra L(F).

This is joint work with C. Köstler and S. Wills.