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We analyze general aspects of exchangeable quantum stochastic processes, aswell as some concrete cases relevant for several applications to QuantumPhysics and Probability. We establish that there is a one-to-one correspondencebetween 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. Wealso connect some algebraic properties of exchangeable processes, that is thefact that they satisfy the product state or the block-singleton conditions, tosome natural ergodic ones. We then specialize the investigation for theq-deformed Commutation Relations, q in(-1,1) (the case q=0 corresponding to thereduced group C*-algebra of the free group on infinitely many generators), andthe Boolean ones. We also provide a generalization of De Finetti Theorem to theFermi CAR algebra (corresponding to the q-deformed Commutation Relations withq=-1), by showing that any state is symmetric if and only if it isconditionally independent and identically distributed with respect to the tailalgebra. The Boolean stochastic processes provide examples for which thecondition to be independent and identically distributed w.r.t. the tailalgebra, without mentioning the {\it a-priori} existence of a preservingconditional expectation, is in general meaningless in the quantum setting.Finally, we study the ergodic properties of a class of remarkable states on thefull group C*-algebra of the free group on infinitely many generators, that isthe so-called Haagerup states.

Exchangeable stochastic processes and symmetric states in quantum probability