Ergodic theorems in quantum probability: an application to the monotone stochastic processes

TL;DR

This paper investigates ergodic properties of quantum dynamical systems derived from Yang-Baxter-Hecke quantisation, focusing on monotone and Boolean processes, and establishes conditions for strong ergodic behavior and unique mixing.

Contribution

It provides sufficient conditions for strong ergodic properties in quantum systems and compares monotone and Boolean cases, highlighting their differences in mixing behavior.

Findings

Boolean processes exhibit unique mixing with respect to fixed point subalgebra.

Monotone processes do not have the property of unique mixing.

Stationary stochastic processes form a segment in both cases.

Abstract

We give sufficient conditions ensuring the strong ergodic property of unique mixing for $C^*$-dynamical systems arising from Yang-Baxter-Hecke quantisation. We discuss whether they can be applied to some important cases including monotone, Boson, Fermion and Boolean $C^*$-algebras in a unified version. The monotone and the Boolean cases are treated in full generality, the Bose/Fermi cases being already widely investigated. In fact, on one hand we show that the set of stationary stochastic processes are isomorphic to a segment in both the situations, on the other hand the Boolean processes enjoy the very strong property of unique mixing with respect to the fixed point subalgebra and the monotone ones do not