Symmetries and ergodic properties in Quantum Probability

We deal with the general structure of the stochastic processes by using the standard techniques of Operator Algebras. In this context, it appears natural that in the quantum case one can exhibit a huge class of such stochastic processes: each of them is associated to a quotient of the universal object made of the free product $C^*$-algebra. The quantum (i.e. noncommutative) case describes the most general situation, and the classical (i.e. commutative) probability scheme is seen as a particular case of the quantum one. The ergodic properties of stationary and exchangeable processes are discussed in detail for many interesting cases arising from Quantum Physics and Quantum Probability.