Ergodic properties of bogoliubov automorphisms in free probability

Francesco Fidaleo, Farrukh Mukhamedov

Research output: Contribution to journalArticlepeer-review

8 Citations (Scopus)

Abstract

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 (, 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 GelfandNaimarkSegal representation associated to the unique invariant state generates a von Neumann factor of one of the following types: I, II1, IIIλ 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 $. The last result has a self-contained 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 < 2-1.

Original languageEnglish
Pages (from-to)393-411
Number of pages19
JournalInfinite Dimensional Analysis, Quantum Probability and Related Topics
Volume13
Issue number3
DOIs
Publication statusPublished - Sep 2010
Externally publishedYes

Keywords

  • Bogoliubov automorphisms
  • C*-dynamical systems
  • Unique ergodicity
  • free probability
  • mixing

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Mathematical Physics
  • Applied Mathematics

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