Symmetries and ergodic properties in quantum probability

Abstract

Abstract. We deal with the general structure of (noncommutative) stochastic processes by using the standard techniques of Operator Algebras. Any stochastic process is associated to a state on a universal object, i.e. the free product C * -algebra in a natural way. In this setting one recovers the classical (i.e. commutative) probability scheme and many others, like those associated to the Monotone, Boolean and the q-deformed canonical commutation relations including the Bose/Fermi and Boltzmann cases. Natural symmetries like stationarity and exchangeability, as well as the ergodic properties of the stochastic processes are reviewed in detail for many interesting cases arising from Quantum Physics and Probability.

Wick Order, Spreadability and Exchangeability for Monotone Commutation Relations

“…In Theorem 2.3 of [7], it was proven that there is a one-to-one correspondence between states on * J A and quantum stochastic processes. More in detail, one sees that the quadruple A, H, {ι j } j∈J , Ω determines a unique state ϕ ∈ S * J A , and a representation π of * J A on the Hilbert space H such that (π, H, Ω) is the Gelfand-Naimark-Segal (GNS fir short) representation of the state ϕ. Conversely, each state ϕ ∈ S * J A defines a unique stochastic process, up to unitary equivalence, just by looking at its GNS representation.…”

“…After arguing that stochastic processes labelled by the index set J = Z, and involving the algebra of the sample A, correspond to states on the free product algebra * J A, we also note (cf. Theorem 2.3 of [7]) that exchangeable or stationary stochastic processes give rise to symmetric or shift invariant states on * J A, respectively.…”

“…The interest in noncommutative stochastic processes invariant under some distributional symmetries had a huge increase in the last years, and the references listed in [7,10] offer a rich but not exhaustive account. This development is certainly related to the possibility of introducing de Finetti-type theorems (see e.g.…”

“…Coming back to general features of quantum probability, very recently (cf. [6,7]) it was shown a one-to-one correspondence between unitarily equivalent classes of quantum stochastic processes on the index set J, for the sample C * -algebra A, and states on the free product C * -algebra * J A. Suppose further that g : J → J is a general permutation of J (i.e.…”

From discrete to continuous monotone C*-algebras via quantum central limit theorems

“…1}. The C * -algebra B could be actually obtained by factoring out the free product C * -algebra by the ideal generated by a "concrete" commutator, according to the results established in [6,7].…”