Ergodic properties of Bogoliubov automorphisms in free probability

Abstract

We show that some $C^{\ast }$--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 $C^{\ast }$--dynamical systems whose GNS representation associated to the unique invariant state generates a von Neuman factor of one of the following types: $I_{\infty }$, $I{I}_1$, $II{I}_{ł}$ where $ł\in (0,1]$. The results listed above are extended to the $q$--commutation relations, provided $|q|<\sqrt{2}-1$.

16 pages

References in corpus (3)

• Unique ergodicity of free shifts and some other automorphisms of C*-algebras
• Strict weak mixing of some C*-dynamical systems based on free shifts
• Unique mixing of the shift on the C*--algebras generated by the q--canonical commutation relations

Cited by in corpus (5)

• Ergodic theorems in quantum probability: an application to the monotone stochastic processes
• Ergodic properties of the Anzai skew-product for the noncommutative torus
• Relative ergodic properties of C*-dynamical systems
• Ergodic theory for quantum semigroups
• On tensor products of weak mixing vector sequences and their applications to uniquely $E$-weak mixing $C^{\ast }$- dynamical systems