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Exchangeable stochastic processes and symmetric states in quantum probability. (English) Zbl 1319.60071

Summary: 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 \(C^*\)-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 to the \(q\)-deformed commutation relations, \(q\in (-1,1)\) (the case \(q=0\) corresponding to the reduced group \(C^{*}\)-algebra \(C^*_r({\mathbb F}_\infty )\) of the free group \({\mathbb F}_\infty \) on infinitely many generators), and the Boolean ones. A generalization of the de Finetti theorem to the Fermi CAR algebra (corresponding to the \(q\)-deformed commutation relations with \(q=-1\)) is proved 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 \(C^{*}\)-algebra \(C^*({\mathbb F}_\infty )\), that is, the so called Haagerup states.

MSC:

60G09 Exchangeability for stochastic processes
46L53 Noncommutative probability and statistics
46L05 General theory of \(C^*\)-algebras
46L30 States of selfadjoint operator algebras
46N50 Applications of functional analysis in quantum physics

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References:

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