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Noncommutative probability

The traditional mathematical axiomatization of probability, due to Kolmogorov, begins with a probability space $P$ and constructs random variables as certain functions $P \to \mathbb{R}$. But start doing any probability and it becomes clear that the space $P$ is de-emphasized as much as possible; the real focus of probability theory is on the algebra of random variables. It would be nice to have an approach to probability theory that reflects this.

Moreover, in the traditional approach, random variables necessarily commute. However, in quantum mechanics, the random variables are self-adjoint operators on a Hilbert space $H$, and these do not commute in general. For the purposes of doing quantum probability, it is therefore also natural to look for an approach to probability theory that begins with an algebra, not necessarily commutative, which encompasses both the classical and quantum cases.

Happily, noncommutative probability provides such an approach. Terence Tao’s notes on free probability develop a version of noncommutative probability approach geared towards applications to random matrices, but today I would like to take a more leisurely and somewhat scattered route geared towards getting a general feel for what this formalism is capable of talking about.

Definition: A subgroup $N$ of a group $G$ is normal if $gNg^{-1} \subset N$ for all $g \in G$.
It is then proven that normal subgroups are precisely the kernels $N = \phi^{-1}(e)$ of surjective group homomorphisms $\phi : G \to G/N$. In other words, they are precisely the subgroups you can quotient by and get another group. This strikes me as backwards. The motivation to construct quotient groups should come first.