# Another blog on “Free Probability” by Teo Banica

Teo Banica got a bit bored by the lockdown and started to write a series of blogs on various topics, close to his heart and his knowledge – one of them is also one free probability. Check it out here. It’s written in Teo’s personal style, which might seem annoying or provocative to some, but in any case it’s interesting …

Update (September 2020): It seems that Teo got also bored or annoyed of his own blog, so the link above does not work any more … but much of the material has actually been moved to lecture notes and videos. In particular, Teo has the goal of trying to reorganize the quantum group basics, via a series of books. Probably the best to stay updated on this is to check his website or his YouTube channel

# The saga ends …

I have now finished my class on random matrices. The last lecture motivated the notion of (asymptotic) freeness from the point of view of looking on independent GUE random matrices. So you might think that there should now be continuations on free probability and alike coming soon. But actually this part of the story was already written and recorded and if you don’t want to spoil the tension you should watch the series not in its historical but in its logical order:

1. Random Matrices (videos, homepage of class)
2. Free Probability Theory (videos, homepage of class)
3. Non-commutative Distributions (and Operator-Valued free Probability Theory) (videos, homepage of class)

More information, in particular the underlying script (sometimes in a handwritten version, sometimes in a more polished texed version), can be found on the corresponding home page of the lecture series.

May freeness be with you …

# Welcome to the Non-Commutative World!

About two weeks ago I posted with Tobias Mai on the archive the preprint “A Note on the Free and Cyclic Differential Calculus”. Here is what we say in the abstract:

In 2000, Voiculescu proved an algebraic characterization of cyclic gradients of noncommutative polynomials. We extend this remarkable result in two different directions: first, we obtain an analogous characterization of free gradients; second, we lift both of these results to Voiculescu’s fundamental framework of multivariable generalized difference quotient rings. For that purpose, we develop the concept of divergence operators, for both free and cyclic gradients, and study the associated (weak) grading and cyclic symmetrization operators, respectively. One the one hand, this puts a new complexion on the initial polynomial case, and on the other hand, it provides a uniform framework within which also other examples – such as a discrete version of the Ito stochastic integral – can be treated.

At the moment I am not in the mood to say more specifically about this preprint (maybe Tobias or I will do so later), but I want to take the opportunity — in particular as the first anniversary of this blog is also coming closer — to put this in a bigger context and mumble a bit about the bigger picture and our dreams … so actually about what this blog should be all about.

Free probability theory has come a long way. Whereas born in the subject of operator algebras, the realization that is also has to say quite a bit about random matrices paved the way to its use in many (and, in particular, also applied) subjects. Hence there are now also papers in statistics, like this one, or in deep learning, like this one or this one, which use tools from free probability for their problems. The last words on how far the use of free probability goes in those subjects are surely not yet spoken but I am looking forward to see more on this.

This is of course all great and nice for our subject, but on the other hand there is also a bigger picture in the background, where I would hope for some more fundamental uses of free probability.

This goes roughly like this. There is the classical world, where we are dealing with numbers and functions and everything commutes; then there is our non-commutative world, where we are dealing with operators and limits of random matrices and where on the basic level nothing commutes. That’s where quite a bit of maximal non-commutative mathematics has been (and is still being) developed from various points of views:

• free probability deals with a non-commutative notion of independence for non-commuting random variables;
• there is a version of a non-commutative differential calculus which allows to talk about derivatives in non-commutative variables; my paper with Tobias mentioned above is in this context and tries to formalize and put all this a bit further;
• free analysis (or free/non-commutative function theory) aims at a non-commutative version of classical complex analysis, i.e., a theory of analytic functions in non-commuting variables;
• free quantum groups provide the right kind of symmetries for such non-commuting variables.

The nice point is that all those subjects have their own source of motivation but it turns out that there are often relations between them which are non-commutative analogues of classical results.

So, again this is all great and nice, BUT apart from the commutative and our maximal non-commutative world there is actually the, maybe most important, quantum world. This is of course also non-commutative, but only up to some point. There operators don’t commute in general, but commutativity is replaced by some other relations, like the canonical commutation relations, and there are actually still a lot of operators which commute (for example, measurements which are at space-like positions are usually modeled by commuting operators). Because of this commutativity, basic concepts of free probability do not have a direct application there.

Here is a bit more concretely what I mean with that. In free probability we have free analogues of such basic concepts as entropy or Fisher information. There are a lot of nice statements and uses of those concepts and via random matrices they can also be seen as arising as a kind of large N limit of the corresponding classical concepts. However, in the classical world those concepts have usually also a kind of operational meaning by being the answer to fundamental questions. For example, the classical Shannon entropy is the answer to the question how much information one can transmit over classical channels. Now there are quantum channels and one can ask how much information one can transmit over them; again there are answers in terms of an entropy, but this is unfortunately not free entropy, but von Neumann entropy, a more commutative non-commutative cousin of classical entropy. There are just too many tensor products showing up in the quantum world which prevent a direct use of basic free probability concepts. But still, I am dreaming of finding some day operational meanings of free entropy and similar quantities.

Anyhow, I hope to continue to explain in this blog more of the concrete results and problems which we have in free probability and related subjects; but I just wanted to point out that there are also some bigger dreams in the background.

# Update on videos of lectures and talks

## Videos of talks at Fields Institute

The videos of my three talks in the distinguished lecture series at the Fields Institute are now online and can be found here.

In particular, the first talk, entitled Dan-Virgil Voiculescu: visionary operator algebraist and creator of free probability theory, was a talk for a public audience and gives not only a bit of information on Dan Voiculescu, but also a very high level idea of what free probability is all about. And it also has some movie references …

## Videos of lecture series “Non-commutative distributions”

The course on “Non-commutative distributions” has now finished; all 20 lectures are online and can be found on our video platform. My hand-written notes for the classes can be found here.

# Focus Program on Applications of Noncommutative Functions; Featuring a Celebration Banquet for Dan Voiculescu’s 70th Birthday

Today started the Focus Program on Applications of Noncommutative Functions at the Fields Institute in Toronto. There will be two workshops: this week on the “Developments and Technical Aspects of Free Noncommutative Functions” and next week on “Applications to Random Matrices and Free Probability of Free Noncommutative Functions”. Both workshops look interesting to me; unfortunately I will miss most of the first one as I will fly only on Wednesday to Toronto.

I will give a series of three talks on the relation between free probability and random matrices. The first talk will be quite general and is also intended for a public academic audience. Its main purpose is to celebrate the 70th birthday of Dan Voiculescu by giving an idea of Dan’s achievements and of free probability theory. The talk and the banquet will be on the very day of Dan’s birthday.

# Class on “Non-Commutative Distributions”: Hand-Written Notes are Up

I have now put up scans of my hand-written notes for the class, see here, and will update those irregularly.

The class is still running well and more or less according to plan. After generalities on non-commutative distributions, non-commutative (fully matricial) functions, and operator-valued Cauchy transforms we are now bringing some structure into our non-commutative distributions, by looking on operator-valued freeness. I plan to cover the basic part of the theory of operator-valued freeness, in particular, operator-valued additive convolution, both from a combinatorial and an analytic point of view. However, much of this is parallel to the scalar-valued theory from last term, so I will be quite brief on details (in particular, proofs) at many places – one should look back to and compare with the relevant parts from last term; in particular, Sections 2, 3, 4, 5 of the corresponding class notes.

# “Non-Commutative Distributions”, Summer Term 2019

My class on “Non-commutative distributions” started today. The first lecture is already online, see our video platform. Actually, we have a new video system, so the sound should now be better than last term. I am not sure, though, whether this also applies to the frames.

The class will run during our summer term, which will end mid July. Since I will travel quite a bit during term, there will be some cancellations and reschedulings of lectures; nevertheless, I still hope that we will have in the end again something like 25 lectures.

The general topic of the class is progress which was made in the last couple of years on non-commutative distributions, and which relies on advances in

• the operator-valued version of free probability theory (in particular, for its analytic description)
• free analysis or free non-commutative function theory
• relating analytic questions about operators in von Neumann algebras with the theory (of Cohn et al.) of non-commutative linear algebra or the free skew field (aka as non-commutative rational functions)
• using the linearization trick to relate non-linear scalar problems with linear operator-valued problems

All of the above will be explained in the lectures. So don’t worry if you have no idea what all this actually means.

Much of this progress was actually achieved in recent years in the context of my ERC-Advanced Grant on “Non-Commutative Distributions”. As this grant has finished now, the class can also be seen as kind of final report for this.

I will assume some familiarity with basic functional analysis and complex analysis. It is surely also helpful to know at least a bit about free probability theory, but this can also be acquired by watching along the way a few of the videos from last term or reading relevant parts of the corresponding class notes.

# Update on “Free Probability” Class

The lectures on our free probability class are over now. All 26 lectures are uploaded on the video platform, and the scans of my handwritten course notes can be found here.

I plan to continue next term with a class on “Non-Commutative Distributions”; this will in particular cover the operator-valued version of free probability and its use for dealing with polynomials in free variables, as well as addressing regularity properties of the distribution of such polynomials. There will be more cool stuff, but I still have to think about details. Our summer term starts in April, then I will be back with more information.

The plan is to continue with the recording of the lectures. If you have any suggestions on how to improve on this, please let me know.

# News on Lectures on “Free Probability Theory”

The lectures on free probability are back!

I have now started to put scans of my handwritten lecture notes online; they correspond more or less to what I write on the blackboard. As we are now, in the context of random matrix calculations, having a lot of indices hanging around it might in some cases be easier to decipher those from my notes than from the blackboard. Maybe sometimes in the future I will even tex them, but don’t count on this … In any case, most of the material I am presenting in this class is either from my book with Andu or from my book with Jamie, so that there exist already nicely written notes on this.

There is of course much more to say about random matrices. One issue is that in this class I cover only convergence and asymptotic freeness results in the averaged sense. Of course, almost sure versions of those results usually also exist. For more on those and other aspects of random matrices I refer to the random matrix literature (part of which you can find on the homepage of my class “Random Matrices” from last term). There exists also a nice tex-ed version of the lectures notes from my class on random matrices.