Ergodic properties of Bogoliubov automorphisms in free probability

Abstract

We show that some $C^*$--dynamical systems obtained by "quantizing" classical ones on the free Fock space, enjoy very strong ergodic properties. Namely, if the classical dynamical system $(X, T, \m)$ is ergodic but not weakly mixing, then the resulting quantized system $(\gg,\a)$ is uniquely ergodic (w.r.t the fixed point algebra) but not uniquely weak mixing. The same happens if we quantize a classical system $(X, T, \m)$ which is weakly mixing but not mixing. In this case, the quantized system is uniquely weak mixing but not uniquely mixing. Finally, a 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 GNS representation associated to the unique invariant state generates a von Neuman factor of one of the following types: $I_{\infty}$, $II_{1}$, $III_ł$ where $ł\in(0,1]$. The results listed above are extended to the $q$--commutation relations, provided $|q|<\sqrt2-1$.