Non-Commutative Disintegration Theory: from Classical Probability to Operator Algebras

Operator Algebras are incredibly interesting objects: indeed their study provides insights not only for the subject itself, but also for other areas of Mathematics and Physics. One of the key concepts is certainly non commutativity. This notion is fundamental for the inves- tigations of concepts which are generalized from classical, i.e. commutative, ones. (Non-commutative) Disintegration theory arise as a method of conditioning with respect to a given subsystem and so it brings a natural Bayesian interpretation with it, which still has to be explored (in the non commutative case). In the first part we will briefly explain what is non-commutative disintegration theory and how, from a categorical perspective, it naturally arises as a generalization of Bayes theorem in the classical (finite) probability setting. We will investigate how the structures resemble each-other, i.e. what are the similarities between the classical and quantum point of view. In the second part we will explore the connection between the dictionary of non-commutative disintegration theory and the theory of operator algebras.