Ergodic properties of bogoliubov automorphisms in free probability - Fidaleo, Francesco; Mukhamedov, Farrukh (World Scientific Publishing, 2010)
We show that some C∗-dynamical systems obtained by free Fock quantization of classical ones, enjoy ergodic properties much stronger than their boson or fermion analogous. Namely, if the classical dynamical system (X, T, μ) is ergodic but not weakly mixing, then the resulting free quantized system (G, α) is uniquely ergodic (w.r.t. the fixed point algebra) but not uniquely weak mixing. The same happens if we quantize a classical system (X, T, μ) which is weakly mixing but not mixing. In this case, the free quantized system is uniquely weak mixing but not uniquely mixing. Finally, a free quantized system arising from a classical mixing dynamical system, will be uniquely mixing. In such a way, it is possible to exhibit uniquely weak mixing and uniquely mixing C∗-dynamical systems whose Gelfand–Naimark–Segal representation associated to the unique invariant state generates a von Neumann factor of one of the following types: I∞, II 1, III λ where λ ∈ (0, 1]. The resulting scenario is then quite different from the classical one. In fact, if a classical system is uniquely mixing, it is conjugate to the trivial one consisting of a singleton. For the sake of completeness, the results listed above are extended to the q-Commutation Relations, provided |q| < √ 2 − 1. The last result has a selfcontained meaning as we prove that the involved C∗-dynamical systems based on the q-Commutation Relations are conjugate to the corresponding one arising from the free case (i.e. q = 0), at least if |q| < √ 2 − 1.