caa a22 4500 605496404 CHVBK 20210128100536.0 cr unu---uuuuu 210128e20150801xx s 000 0 eng 10.1007/s10231-014-0407-5 doi (NATIONALLICENCE)springer-10.1007/s10231-014-0407-5 Exchangeable stochastic processes and symmetric states in quantum probability [Elektronische Daten] [Vitonofrio Crismale, Francesco Fidaleo] We analyze general aspects of exchangeable quantum stochastic processes, as well as some concrete cases relevant for several applications to Quantum Physics and Probability. We establish that there is a one-to-one correspondence between quantum stochastic processes, either preserving or not the identity, and states on free product $$C^*$$ C ∗ -algebras, unital or not unital, respectively, where the exchangeable ones correspond precisely to the symmetric states. We also connect some algebraic properties of exchangeable processes, that is the fact that they satisfy the product state or the block-singleton conditions, to some natural ergodic ones. We then specialize the investigation for the $$q$$ q -deformed Commutation Relations, $$q\in (-1,1)$$ q ∈ ( - 1 , 1 ) (the case $$q=0$$ q = 0 corresponding to the reduced group $$C^{*}$$ C ∗ -algebra $$C^*_r({\mathbb F}_\infty )$$ C r ∗ ( F ∞ ) of the free group $${\mathbb F}_\infty $$ F ∞ on infinitely many generators), and the Boolean ones. A generalization of de Finetti theorem to the Fermi CAR algebra (corresponding to the $$q$$ q -deformed Commutation Relations with $$q=-1$$ q = - 1 ) is proven, by showing that any state is symmetric if and only if it is conditionally independent and identically distributed with respect to the tail algebra. Moreover, we show that the Boolean stochastic processes provide examples for which the condition to be independent and identically distributed w.r.t. the tail algebra, without mentioning the a-priori existence of a preserving conditional expectation, is in general meaningless in the quantum setting. Finally, we study the ergodic properties of a class of remarkable states on the group $$C^{*}$$ C ∗ -algebra $$C^*({\mathbb F}_\infty )$$ C ∗ ( F ∞ ) , that is the so-called Haagerup states. Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg, 2014 Exchangeability nationallicence Noncommutative probability and statistics nationallicence $$C^{*}$$ C ∗ -algebras nationallicence States nationallicence Applications to quantum physics nationallicence Crismale Vitonofrio Dipartimento di Matematica, Università degli studi di Bari, Via E. Orabona, 4, 70125, Bari, Italy aut Fidaleo Francesco Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133, Rome, Italy aut Annali di Matematica Pura ed Applicata (1923 -) Springer Berlin Heidelberg 194/4(2015-08-01), 969-993 0373-3114 194:4<969 2015 194 10231 https://doi.org/10.1007/s10231-014-0407-5 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s10231-014-0407-5 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Crismale Vitonofrio Dipartimento di Matematica, Università degli studi di Bari, Via E. Orabona, 4, 70125, Bari, Italy aut NATIONALLICENCE 700 1- Fidaleo Francesco Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133, Rome, Italy aut NATIONALLICENCE 773 0- Annali di Matematica Pura ed Applicata (1923 -) Springer Berlin Heidelberg 194/4(2015-08-01), 969-993 0373-3114 194:4<969 2015 194 10231