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

Abstract

We give sufficient conditions ensuring the strong ergodic property of unique mixing for $C^{\ast }$-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 $C^{\ast }$-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

31 pages, thm 5.13 in the old version has been replaced with prop 5.13, accepted on: Annali della Scuola Normale Superiore di Pisa, Classe di Scienze

References in corpus (3)

• De Finetti theorem on the CAR algebra
• A note on Boolean stochastic processes
• A Nonconventional Ergodic Theorem for quantum "diagonal measures"

Cited by in corpus (5)

• Weakly Monotone Fock Space and Monotone Convolution of the Wigner Law
• Distributions for Nonsymmetric Monotone and Weakly Monotone Position Operators
• On the Thermodynamics of the q-Particles
• Limits of some weighted Cesaro averages
• From discrete to continuous monotone $C^{\ast }$-algebras via quantum central limit theorems