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Watching (the begining of) a lecture on free probability theory by Dimitri Shlyakhtenko https://www.youtube.com/watch?v=F8Urtr39jM0, I'm led to consider the following question

Question. How can one build classical probability theory (measurable functions on measure space, expectations, etc.) from the framework of "von Neumann algebras"?

Note. I'd really appreciate a bottom-up answer (not just a dry stack of formal encyclopedic statements, except perhaps when strictly needed).

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    $\begingroup$ There are several sources for that. My personal favorite - very vivid exposition in "Noncommutative geometry" by Connes. $\endgroup$ – მამუკა ჯიბლაძე Feb 22 at 10:34
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    $\begingroup$ I guess that would be a ref on NCG. I'm interested in deriving classical probability from NCG. I really doubt that the works of Alain Connes would be a reference on the latter question. So I fail to see how your comment helps my question, but I may be missing something here... $\endgroup$ – dohmatob Feb 22 at 10:56
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    $\begingroup$ Sorry if I misunderstand your question but topics that you list (measurable functions on measure space, expectations, etc.) are considered in NCG in the context of von Neumann algebras. The whole classification of these algebras is explained in detail, and it is carefully explained how they provide a noncommutative version of measure theory. $\endgroup$ – მამუკა ჯიბლაძე Feb 22 at 15:43
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At the beginning of his talk What actually is free probability theory? Tobias Mai explains how classical probability theory fits into the context of non-commutative probability theory.

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  • $\begingroup$ Thanks, indeed I came across his lecture on youtube yesterday, and it answered my question in essence. He talks about replacing $L^{\infty-}(\Omega,\mathcal F,\mathbb P)$ with a general unital complex algebra $\mathcal A$, and replacying the expectation functional with a linear functional $\varphi:\mathcal A \to \mathbb C$ such that $\varphi(1_{\mathcal A})=1$ and $\varphi(aa^\star) \ge 0$ for all $a \in \mathcal A$ (aka a generalized expectation or state). For example, one can consider $(M_N(\mathbb C) \otimes L^{\infty-}(\Omega,\mathcal F,\mathbb P), \mbox{tr}_N \otimes \mathbb E)$. $\endgroup$ – dohmatob Feb 23 at 10:25
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I am not sure how far you want to go, but some basics are explained in this answer: Is there an introduction to probability theory from a structuralist/categorical perspective?

In particular, you have mentioned conditional expectations, which are implemented as pushforwards of measures (see also Conditional Expectation for $\sigma$-finite measures), and in the language of von Neumann algebras these correspond to the predual of the corresponding homomorphism of von Neumann algebras.

The paper arXiv:2005.05284 gives a precise formulation of the equivalence between measurable spaces and commutative von Neumann algebras, and, in particular, in Section 3 it discusses how preduals of homomorphisms correspond to pushforwards of measures.

This answer contains a list of further writings on these topics: Is there a category structure one can place on measure spaces so that category-theoretic products exist?

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  • $\begingroup$ Thanks for the useful answer. Upvoted. $\endgroup$ – dohmatob Feb 23 at 10:29
  • $\begingroup$ BTW, your first link might also be relevant to this question I asked some time ago mathoverflow.net/q/362449/78539. $\endgroup$ – dohmatob Feb 23 at 10:41
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Below I present an answer (too long to be a comment) in the same spirit as (the video linked in) the accepted answer by Roland Speicher. It is a synthesis of material presented in the tutorial (in French) Probabilités libres by Pierre Tarrago. I hope someone else finds this useful.


Algebraic view of classical probability theory

Let's recall that a classical probability space is a triple $(\Omega,\mathcal F,\mathbb P)$, where $\Omega$ is an abstract set (called the sample space), $\mathcal F$ is a collection of subsets of $\Omega$, and $\mathbb P:\mathcal F \to [0,1]$, and these satisfy certain axioms (not recalled here). One then constructs an expectation operator $\mathbb E[f] := \int_\Omega f\mbox{d}P$, acting on $(\Omega,\mathcal F)$-measurable functions. When the dust has settled, this construction is in fact equivalent to the specification of an algebra $\mathcal A \subseteq \mathbb R^\Omega$, namely the set of $\mathcal F$-measurable functions on $\Omega$, and of a linear form $\varphi:\mathcal A \to \mathbb C$ such that

$$ \begin{split} \varphi(a) &\ge 0\text{ if }a \in \mathcal A\text{ is "positive"},\\ \varphi[1_{\mathcal A}] &= 1. \end{split} $$

In the same way, one can abstractly define a noncommutative probability space from a von Neumann algebra. We recall that a von Neumann algebra is roughly an operator algebra on a Hilbert space, which contains the identity element (i.e is unital), and is closed in strong topology.

Definition. A noncommutative probability space is a pair $(\mathcal A,\varphi)$, where $\mathcal A$ is a von Neumann algebra and $\varphi:\mathcal A \to \mathbb C$ is a state, i.e a linear form such that $$ \begin{split} \varphi(aa^\star) &= 1\;\forall a \in \mathcal A,\\ \varphi(1_{\mathcal A}) &= 1. \end{split} $$

"Random variables" and their law

Given an noncommutative probability space $(\mathcal A,\varphi)$, the state $\varphi$ plays an analogous role to the role of the expectation operator $\mathbb E$ in the classical theory of probability. A normal element $a \in \mathcal A$ (i.e $aa^\star = a^\star a$) will be called a random variable. The law of any such $a \in \mathcal A$ is the mapping $\mathbb C[X] \to \mathbb C$, $P \mapsto \varphi(P(a))$. Note that because $\mathcal A$ is an algebra $P(a) \in \mathcal A$ for every complex polynomial $P$, and so it actually makes sense to write $\varphi(P(a))$. One verifies (thanks to the spectral theorem) that such an application induces a measure $\mu_a$ on $\mathbb C$ with support contained in the centered ball of radius $\|a\|$. In this way, one says a sequence $(a_n)_n \subseteq \mathcal A$ of random variables converges to a random variable $a \in \mathcal A$ if $\mu_{a_n} \to \mu_{a}$, or equivalently, if $$ \varphi(P(a_n)) \to \varphi(P(a))\;\forall P \in \mathbb C[X]. $$

Examples of noncommutative probability spaces

We now provide some examples of noncommutative probability space $(\mathcal A,\varphi)$.

  • If $(\Omega,\mathcal F,\mathbb P)$ is a probability space in the usual / classical space and $\mathbb E$ is the induced expectation operator, then $(L^\infty(\Omega,\mathcal F),\mathbb E)$ is noncommutatve probability space which is ... commutative! Recall that $L^\infty(\Omega,\mathcal F)$ is standard notation for all bounded random variables on $(\Omega,\mathcal F)$, i.e bounded $\mathcal F$-measurable functions $X:\Omega \to \mathbb C$.

  • Let $L^{\infty-}(\Omega,\mathcal F)$ the random variables on $(\Omega,\mathcal F)$ which have all finite moments and let $M_n(\mathbb C)$ be the vector space of all $n \times n$ complex matrices. Then $(M_n(\mathbb C) \otimes L^{\infty-}(\Omega,\mathcal F),(1/n)\mathbb E \otimes \mbox{Tr})$ is a non-commutative probability space (of random matrices!), one of the most important in all of the theory. Note that in this example, every random variable is an $n\times n$ random complex matrix $M=(M_{ij})$ whose entries $M_{ij}$ are $L^{\infty-}$ random variables on $(\Omega,\mathcal F)$.

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