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I was recently asked why Voiculescu developed free probability theory. I am not very expert in this and the only answer I was able to provide is the classical one: he was challenging the isomorphism problem of whether $II_1$-factors associated to two different free groups are isomorphic or not. First: is this true or just legend? Were there any other motivations? In particular I would be interested in more down-to-earth motivations, something that could theoretically be explained to someone with basic knowledge in probability theory and operator theory (without necessarily knowing what a $II_1$-factor is).

Thanks in advance!

Valerio

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    $\begingroup$ As for the first part... I don't have it in front of my but at the beginning of his book Free Random Variables I'm pretty sure that he says studying the free group factors (though I'm not sure if he says that solving the isomorphism problem per se) was the initial motivation, largely because, after the hyperfinite $II_1$ factor, this is the next natural (and historical) example. Also I have heard from other people that know more free probability than I do that to do this he spent years trying to calculate moments and this led him to think of freeness as an analog of independent. $\endgroup$ – Owen Sizemore Jun 27 '13 at 15:46
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Here's what Dan Voiculescu himself gave as motivation:

Around 1982, I realized that the right way to look at certain operator algebra problems was by imitating some basic probability theory. More precisely, in noncommutative probability theory a new kind of independence can be defined by replacing tensor products with free products and this can help understand the von Neumann algebras of free groups. The subject has evolved into a kind of parallel to basic probability theory, which should be called free probability theory.

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Here is a more detailed account on the history by Dan Voiculescu himself. This is from his article "Background and Outlook" in the Lectures Notes "Free Probability and Operator Algebras", see http://www.ems-ph.org/books/book.php?proj_nr=208

Just before starting in this new direction, I had worked with Mihai Pimsner, computing the K-theory of the reduced $C^*$-algebras of free groups. From the K-theory work I had acquired a taste for operator algebras associated with free groups and I became interested in a famous problem about the von Neumann algebras $L(\mathbb{F}_n)$ generated by the left regular representations of free groups, which appears in Kadison's Baton-Rouge problem list. The problem, which may have already been known to Murray and von Neumann, is:are $L(\mathbb{F}_m)$ and $L(\mathbb{F}_n)$ non-isomorphic if $m \not= n$?

This is still an open problem. Fortunately, after trying in vain to solve it, I realized it was time to be more humble and to ask: is there anything I can do, which may be useful in connection with this problem? Since I had come across computations of norms and spectra of certain convolution operators on free groups (i.e., elements of $L(\mathbb{F}_n)$), I thought of finding ways to streamline some of these computations and perhaps be able to compute more complicated examples. This, of course, meant computing expectations of powers of such operators with respect to the von Neumann trace-state $\tau(T) = \langle T e_e,e_e\rangle$, $e_g$ being the canonical basis of the $l^2$-space.

The key remark I made was that if $T_1$, $T_2$ are convolution operators on $\mathbb{F}_m$ and $\mathbb{F}_n$ then the operator on $\mathbb{F}_{m+n} = \mathbb{F}_m \ast \mathbb{F}_n$ which is $T_1 + T_2$, has moments $\tau((T_1 + T_2)^p)$ which depend only on the moments $\tau(T_j^k)$, $j = 1, 2$ , but not on the actual $T_1$ and $T_2$. This was like the addition of independent random variables, only classical independence had to be replaced by a notion of free independence, which led to a free central limit theorem, a free analogue of the Gaussian functor, free convolution, an abstract existence theorem for one variable free cumulants, etc.

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Here is an even more classical motivation: If you consider vNa as the non-commutative analogue of measure theory, then free probability theory formalizes the concept of independent random variables.

Here is a reference: http://www.uni-math.gwdg.de/mitch/free.pdf

I do not know anything about the history, sorry:(

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