non-commutative objects. Instead of defining a probability space as a triplet $(\Omega,\mathcal{F},\mathbb{P})$ we switch to a pair $(\mathcal{A},\varphi)$ where $\mathcal{A}$ is an algebra of random variables and $\varphi\colon\mathcal{A}\to\mathbb{C}$ is a linear functional, in classical situation $\varphi=\mathbb{E}$. It is natural then to consider algebras $\mathcal{A}$ where random variables do not commute (for example $C^*$ or $W^*$–algebras). For bounded random variables independence can be equivalently understood as a rule of calculating mixed moments. It turns out that while for commuting random variables only one such rule leads to a meaningful notion of independence, the non-commutative setting is richer and one can consider several notions of independence. Free independence seems to be the one which is the most important. The precise definition of freeness is stated in Section 2 below.

Free probability emerged from questions related to operator algebras however the development of this theory showed that it is surprisingly closely related with the classical probability theory. First evidence of such relations appeared with Voiculescu’s results about asymptotic freeness of random matrices. Asymptotic freeness roughly speaking states that (classically) independent, unitarily invariant random matrices, when size goes to infinity, become free.\
Another link between free and classical probability goes via infinite divisibility. With 