**Note:** This question was already asked on Math.SE nearly a week and a half ago but did not receive any responses. To the best of my knowledge, free probability is an active topic of research, so I hope that the level of the question is appropriate for this website. If not, please let me know so I can delete the question myself.

Question:One often sees statements to the effect that "free probability is a generalization of probability theory, which is commutative, to the non-commutative case".But in what sense does classical probability theory only concern itself with commutative quantities?

If my understanding is correct, and probability theory also deals with non-commutative quantities, then in what sense is free probability a generalization of probability theory?

**Context:** The simplest random variables are real-valued, and obviously real numbers have commutative multiplication. But random variables can take values in any measurable space (at least this is my understanding and also that of Professor Terry Tao), i.e. random variables can also be random vectors, random matrices, random functions, random measures, random sets, etc. The whole theory of stochastic processes is based on the study of random variables taking values in a space of functions. If the range of the functions of that space is the real numbers, then yes we have commutative multiplication, but I don't see how that's the case if we are e.g. talking about functions into a Riemannian manifold.

**EDIT:** To clarify what I mean by "classical probability theory", here is Professor Tao's definition of random variable, which is also my understanding of the term (in the most general sense):

Let $R=(R, \mathcal{R})$ be a measurable space (i.e. a set $R$, equipped with a $\sigma$-algebra $\mathcal{R}$ of subsets of $R$). A

taking values in $R$ (or anrandom variable$R$-valued random variable) is a measurable map $X$ from the sample space to $R$, i.e. a function $X: \Omega \to R$ such that $X^{-1}(S)$ is an event for every $S \in \mathcal{R}$.

Then, barring that I am forgetting something obvious, classical probability theory is just the study of random variables (in the above sense).

**/EDIT**

To be fair though, I don't have a strong understanding of what free probability is. Reading Professor Tao's post about the subject either clarified or confused some things for me, I am not sure which.

In contrast to his other post, where he gives (what seems to me) a more general notion of random variable, in his post about free probability, Professor Tao states that there is a third step to probability theory after assigning a sample space, sigma algebra, and probability measure -- creating a commutative algebra of random variables, which supposedly allows one to define expectations. (1) How does one need a commutative algebra of random variables to define expectations? (2) Since when was defining a commutative algebra of random variables part of Kolmogorov's axiomatization of probability?

Later on his post about free probability, Professor Tao mentions that random scalar variables form a commutative algebra if we restrict to the collection of random variables for which *all moments are finite*. But doesn't classical probability theory study random variables with non-existent moments? Even in an elementary course I remember learning about the Cauchy distribution.

If so, wouldn't this make classical probability more general than free probability, rather than vice versa, since free probability isn't relevant to, e.g., the Cauchy distribution?

Professor Tao also mentions random matrices (specifically ones with entries which are random scalar variables with all moments finite, if I'm interpreting the tensor product correctly) as an example of a noncommutative algebra which is outside the domain of classical probability but within the scope of free probability. But as I mentioned before, aren't random matrices an object of classical probability theory? As well as random measures, or random sets, or other objects in a measurable space for which there is no notion of multiplication, commutative or non-commutative?

**Attempt:** Reading Professor Tao's post on free probability further, it seems like the idea might be that certain nice families of random variables can be described by commutative von Neumann algebras. Then free probability generalizes this by studying all von Neumann algebras, including non-commutative ones. The idea that certain nice families of random variables correspond to the (dual category of) commutative von Neumann algebras seems like it is explained in these two answers by Professor Dmitri Pavlov on MathOverflow (1)(2).

But, as Professor Pavlov explains in his answers, commutative von Neumann algebras only correspond to *localizable* measurable spaces, not arbitrary measurable spaces. While localizable measurable spaces seem like nice objects based on his description, there is one equivalent characterization of them which makes me suspect that they are not the most general objects studied in probability theory: any localizable measurable space "is the coproduct (disjoint union) of points and real lines". This doesn't seem to characterize objects like random functions or random measures or random sets (e.g. point processes), and maybe even not random vectors or matrices, so it does not seem like this is the full scope of classical probability theory.

Thus, if free probability only generalizes the study of localizable measurable spaces, I don't see how it could be considered a generalization of classical probability theory. By considering localizable measurable spaces in the more general framework of possibly non-commutative von Neumann algebras, it might expand the methods employed in probability theory by borrowing tools from functional analysis, but I don't see at all how it expands the scope of the subject. To me it seems like proponents of free probability and quantum probability might be mischaracterizing classical probability and measure theory. More likely I am misinterpreting their statements.

Related question. Professor Pavlov's comments on this article may be relevant.

I am clearly deeply misunderstanding at least one thing, probably several things, here, so any help in identifying where I go wrong would be greatly appreciated.