Spreadability for quantum stochastic processes, with an application to boolean commutation relations

In order to manage spreadability for quantum stochastic processes, we study in detail the structure of the involved monoids acting on the index-set of all integers Z, that is that generated by left and right hand-side partial shifts, the monoid of all strictly increasing maps whose range has finite complement, and finally the collection of all strictly increasing maps of Z. We show that such three monoids are strictly ordered, and the second-named one is the semidirect product between the first and the action of Z generated by the one-step shift. Even if the definition of a spreadable stochastic process is provided in terms of the invariance of the finite joint distributions under the natural action of the last monoid on the indices, we see that spreadability can be directly stated in terms of invariance with respect to the action of the first monoid. Concerning the stochastic processes involving the concrete boolean C*-algebra generated by the annihilators acting on the boolean Fock space (i.e., the concrete C*-algebra satisfying the boolean commutation relations), we study their spreadability directly in terms of the invariance under the monoid generated by all strictly increasing maps whose range has finite complement because, for this case, such an investigation appears more direct and manageable. Finally, we present the version of the Ryll-Nardzewski theorem for the boolean case, establishing that spreadable, exchangeable and stationary stochastic processes coincide, and describing their common structure.